Fundamental Unimathematics (Mega-Overmathematics) as a System of Revolutions in Fundamental Mathematics:

Uniarithmetics, Quantialgebra, and Quantianalysis:

Uninumbers, Quantielements, Quantisets, and Uniquantities

with Quantioperations and Quantirelations

Mathematical Monograph

by

© Ph. D. & Dr. Sc. Lev Gelimson

Academic Institute for Creating Fundamental Sciences (Munich, Germany)

The "Collegium" All World Academy of Sciences Publishers

Munich (Germany)

12th Edition (2012)

11th Edition (2010)

10th Edition (2004)

9th Edition (2003)

8th Edition (2002)

7th Edition (2001)

6th Edition (2000)

5th Edition (1997)

4th Edition (1995)

3nd Edition (1994)

2nd Edition (1993)

1st Edition (1992)

Dedicated to the memory of Bernard Bolzano and Georg Cantor

Abstract

As shown, the real numbers evaluate even not every bounded quantity; the sets, fuzzy sets, multisets, and set operations express and form not all collections; the cardinalities, measures, and probabilities are not sufficiently sensitive to infinite sets and even to intersecting finite sets. Operations are typically considered for natural numbers or countable sets of operands only and cannot model any mixed magnitude. Exponentiation is noncommutative and well-defined for nonnegative bases only. Division by zero is considered when unnecessary, ever brings insolvable problems, and is never efficiently utilized.

In created unimathematics (mega-overmathematics) including uniarithmetics, quantialgebra, and quantianalysis both of the finite and of the infinite with quantioperations and quantirelations, the hyper-Archimedean structure-preserving extension of the real numbers by including some infinite cardinal numbers and possibly the reciprocal to zero as nonnumber inverse overinfinity gives the uninumbers. They evaluate and are interpreted by quantisets algebraically quantioperable with any quantity of each element and with universal, perfectly sensitive, and even uncountably algebraically additive uniquantities.

2010 Mathematics Subject Classification: primary 00A05; secondary 03E10, 03E17, 03E72, 26E30.

Keywords: fundamental mathematics, megascience, revolution, megamathematics, unimathematics, mega-overmathematics, unimathematical test fundamental metasciences system, knowledge, philosophy, strategy, tactic, analysis, synthesis, object, operation, relation, criterion, conclusion, evaluation, measurement, estimation, expression, modeling, processing, symmetry, invariance, bound, level, worst case, defect, mistake, error, reserve, reliability, risk, supplement, improvement, modernization, variation, modification, correction, transformation, generalization, replacement.

Mathematics is usually divided into pure, applied, and computational mathematics. Pure mathematics can be further divided into fundamental and advanced mathematics.

Classical mathematics, its concepts, approaches, methods, and theories are based on inflexible axiomatization, intentional search for artificial contradictions, and even their purposeful creation to desist from further research. These and other fundamental defects do not allow us to acceptably and adequately consider, formulate, and solve many classes of typical urgent problems in science, engineering, and life. Mathematicians select either set theory or mereology as if these were incompatible. The real numbers cannot fill the number line because of gaps between them and hence evaluate even not every bounded quantity. The sets, fuzzy sets, multisets, and set operations express and form not all collections. The cardinalities and measures are not sufficiently sensitive to infinite sets and even to intersecting finite sets due to absorption. No conservation law holds beyond the finite. Infinity seems to be a heap of very different infinities the cardinality only can very roughly discriminate and no tool can exactly measure. Known hypernumber systems, starting with nonstandard analysis, demonstrate the possibility of their construction and use to more intuitively prove well-known theorems but cannot namely quantitatively solve many classes of typical urgent problems. Operations are typically considered for natural numbers or countable sets of operands only and cannot model any mixed magnitude. Exponentiation is well-defined for nonnegative bases only. Exponentiation and further hyperoperations are noncommutative. Division by zero is considered when unnecessary, ever brings insolvable problems, and is never efficiently utilized. The probabilities not always existing cannot discriminate impossible and other zero-measure events differently possible. The absolute error is noninvariant and alone insufficient for quality estimation. The relative error applies to the simplest formal equalities of two numbers only and even then is ambiguous and can be infinite. Mathematical statistics and the least square method irreplaceable in overdetermined problems typical for data processing are based on the noninvariant absolute error and on the second degree analytically simplest but usually very insufficient. This method is unreliable and not invariant by equivalent transformations of a problem, makes no sense by noncoinciding physical dimensions (units) in a problem to be solved, and can give predictably inacceptable and even completely paradoxical outputs without any estimation and improvement. Artificial randomization brings unnecessary complications. One-source iteration with a rigid algorithm requires an explicit expression of the next approximation via the previous approximations with transformation contractivity and often leads to analytic difficulties, slow convergence, and even noncomputability. Real number computer modeling brings errors via built-in standard function rounding and finite signed computer infinities and zeroes, which usually excludes calculation exactness, limits research range and deepness, and can prevent executing calculation for which even the slightest inconsistencies are inadmissible, e.g. in accounting. The finite element method gives visually impressive "black box" results not verifiable and often unacceptable and inadequate.

Every new alternative mathematics can be considered as an external revolution in mathematics which becomes megamathematics. In any new alternative mathematics itself, creating its own cardinally new very fundamentals replacing the very fundamentals of classical mathematics can be considered as an internal revolution in alternative mathematics also if classical mathematics itself remains unchanged.

Mega-overmathematics (by the internal entity), or unimathematics (by the external phenomenon), created and developed has the character of a superstructure (with useful creative succession, or inheritance) over conventional mathematics as a basis without refusing any achievement of ordinary mathematics. Moreover, unimathematics even calls for usefully applying ordinary mathematics if possible, permissible, acceptable, and adequate.

In these names, the prefix "mega" means infinitely many distinct overmathematics with including different infinities and overinfinities into the real numbers.

The prefix "uni" is here associated both with the union, or the general system, of these infinitely many distinct overmathematics and with the universality of these union and system.

The prefix "over" here means:

1) the superstructural character of mega-overmathematics, or unimathematics, with respect to conventional mathematics;

2) the addional nature of new possibilities offered by mega-overmathematics besides the usual opportunities of ordinary mathematics;

3) overpossibilities as the qualitatively new features of mega-overmathematics in setting, considering, and solving whole classes of typical urgent problems so that these overpossibilities often have a much higher order of magnitude compared with the possibilities of conventional mathematics. For example, one of such overpossibilities is oversensitivity as perfect unlimited sensitivity with exactly satisfying universal conservation laws and with complete exclusion of any absorption so that infinitely or overinfinitely great magnitudes are exactly separated from one another even by infinitesimal or overinfinitesimal differences.

Unimathematics can be called not only universal and unified but also general, natural, physical, intuitive, nonrigorous, free, flexible, perfectly sensitive, practical, useful, exclusively constructive, creative, inventive, etc.

Mega-overmathematics is a system of infinitely many diverse overmathematics which differ by possible hyper-Archimedean structure-preserving extensions of the real numbers via including both specific subsets of some infinite cardinal numbers as canonic positive infinities and signed zeroes reciprocals as canonic overinfinities, which gives the uninumbers. They provide adequately and efficiently considering, setting, and namely quantitatively solving many typical urgent problems. In created uniarithmetics, quantialgebra, and quantianalysis of the finite, the infinite, and the overinfinite with quantioperations and quantirelations, the uninumbers evaluate, precisely measure, and are interpreted by quantisets algebraically quantioperable with any quantity of each element and with universal, perfectly sensitive, and even uncountably algebraically additive uniquantities so that universal conservation laws hold. Quantification builds quantielements, integer and fractional quantisets, mereologic quantiaggregates (quanticontents), and quantisystems with unifying mereology and set theory. Negativity conserving multiplication, base sign conserving exponentiation, exponentiation hyperefficiency, composite (combined) commutative exponentiation and hyperoperations, root-logarithmic overfunctions, self-root-logarithmic overfunctions, the voiding (emptifying) neutral element (operand), and operations with noninteger and uncountable quantities of operands are also introduced. Division by zero is regarded when necessary and useful only and is efficiently utilized to create overinfinities. Unielements, unisets, mereologic uniaggregates (unicontents), unisystems, unipositional unisets, unimappings, unisuccessions, unisuccessible unisets, uniorders, uniorderable unisets, unistructures, unicorrespondences, and unirelation unisystems are also introduced. The same holds for unitimes, potential uniinfinities, general uniinfinities, subcritical, critical, and supercritical unistates and uniprocesses, as well as quasicritical unirelations. Unidestructurizators, unidiscriminators, unicontrollers, unimeaners, unimean unisystems, unibounders, unibound unisystems, unitruncators, unilevelers, unilevel unisystems, unilimiters, uniseries uniestimators, unimeasurers, unimeasure unisystems, uniintegrators, uniintegral unisystems, uniprobabilers, uniprobability unisystems, and unicentral uniestimators efficiently provide unimeasuring and uniestimating. The universalizing separate similar (proportional) limiting reduction of objects, systems, and their models to their own similar (proportional) limits as units provides the commensurability and comparability of disproportionate and, therefore, not directly commensurable and comparable objects, systems, and their models. The unierror irreproachably corrects and generalizes the relative error. The unireserve, unireliability, and unirisk based on the unierror additionally estimate and discriminate exact objects, models, and solutions by the confidence in their exactness with avoiding unnecessary randomization. All these uniestimators for the first time evaluate and precisely measure both the possible inconsistency of a uniproblem (as a unisystem which includes unknown unisubsystems) and its pseudosolutions including quasisolutions, supersolutions, and antisolutions. Multiple-sources iterativity and especially intelligent iterativity (coherent, or sequential, approximativity) are much more efficient than common single-source iterativity. Intelligent iterability universalization leads to collective coherent reflectivity, definability, modelablity, expressibility, evaluability, determinability, estimability, approximability, comparability, solvability, and decisionability. This holds, in particular, in truly multidimensional and multicriterial systems of the expert definition, modeling, expression, evaluation, determination, estimation, approximation, and comparison of the qualities of objects, systems, and models which are disproportionate and hence incommensurable and not directly comparable, as well as in truly multidimensional and multicriterial decision-making systems. Sufficiently increasing the exponent in power mean theories and methods can bring adequate results. This holds for linear and nonlinear unibisector theories and methods with distance or unierror minimization, unireserve maximization, as well as for distance, unierror, and unireserve equalization, respectively. Unimathematical data coordinate and/or unibisector unipartitioning, unigrouping, unibounding, unileveling, scatter and trend unimeasurement and uniestimation very efficiently provide adequate data processing with efficiently utilizing outliers and even recovering true measurement information using incomplete changed data. Universal (in particular, infinite, overinfinite, infinitesimal, and overinfinitesimal) continualization provides perfect computer modeling of any uninumbers. Perfectioning built-in standard functions brings always feasible and proper computing. Universal transformation and solving algorithms ensure avoiding computer zeroes and infinities with computer intelligence and universal cryptography systems hierarchies. It becomes possible to adequately consider, model, express, measure, evaluate, estimate, overcome, and even efficiently utilize many complications such as contradictions, infringements, damages, hindrances, obstacles, restrictions, mistakes, distortions, errors, information incompleteness, variability, etc. Unimathematics (mega-overmathematics) also includes knowledge universal test and development fundamental metasciences.

Unimathematics as a megasystem of revolutions in mathematics is divided into fundamental, advanced, applied, and computational unimathematics as systems of revolutions in fundamental, advanced, applied, and computational mathematics.

Foreword. Megamathematics as Revolutions in Mathematics

There are separate scientific achievements of mankind but many of them often bring rather unsolvable problems than really improving human life quality. One of the reasons of such a situation is that the available level of classical science is clearly insufficient to adequately solve and even consider many urgent human problems. To provide creating and developing applicable and, moreover, adequate methods, theories, and sciences, we need their testing (Lev Tsvik [1975, 1978, 1995, 2001, 2002], Alexey Borisenko [2002]) via universal if possible, at least applicable and, moreover, adequate test metamethods, metatheories, and metasciences whose general level has to be high enough. Mathematics as a universal quantitative scientific language naturally has to play here a key role.

[Wikipedia Mathematics] notes that mathematics can be subdivided into arithmetic, algebra, geometry, and analysis building pure mathematics and studying quantity, structure, space, and change, respectively. Among additional subdivisions are logic, set theory (foundations), empirical mathematics of the various sciences (applied mathematics including computational mathematics), and the rigorous study of uncertainty.

It seems to be logical to fuzzily divide mathematics into fundamental mathematics, advanced mathematics, applied mathematics, and computational mathematics (with dividing pure mathematics into fundamental mathematics and advanced mathematics). But mathematics remains unified. Mark Burgin [2004] proposed named sets as unified foundations for mathematics.

There were a number of scientific revolutions (qualitatively radical constructive jump-like changes) in mathematics (Nicolas Bourbaki [1949], Arnold Kaufmann, Maurice Denis-Papin, Robert Faure [1964], [Encyclopaedia of Mathematics 1988]) such as:

discovering the existence of irrational numbers via proving the irrationality of 2^1/2 or, equivalently, that the diagonal of a square is incommensurable with its side by Pythagoreans (possibly by Hippasus in the 5th century BC);

creating axiomatic geometry and number theory by Euclid [1482] in c. 300 BC;

introducing negative numbers in the "Nine Chapters on the Mathematical Art" in China (100 BC - 50 BC) and using them by Fibonacci in the 13th century as debts and losses whereas many European mathematicians ignored them as meaningless even in the 17th-18th centuries;

using square roots of negative numbers by Heron in the 1st century AD, Niccolo Fontana Tartaglia and Gerolamo Cardano in the 16th century, naming imaginary numbers by René Descartes in the 17th century, introducing complex analysis by Abraham de Moivre and Leonhard Euler and geometrically interpreting complex numbers by Caspar Wessel in the 18th century, their general acceptance due to Carl Friedrich Gauss, Augustin Louis Cauchy, and Niels Henrik Abel, as well as introducing quaternions by Sir William Rowan Hamilton and octonions by John Thomas Graves in the 18th century;

creating analysis (calculus) by Gottfried Wilhelm Leibniz [1684] and Isaac Newton [1687];

creating non-Euclidean hyperbolic geometry by Nikolai Lobachevsky [1829] and non-Euclidean elliptic geometry in 1854 by Bernhard Riemann [1990] who constructed an infinite family of non-Euclidean geometries via Riemannian metrics on the unit ball in the Euclidean space;

creating the foundations of real analysis by Augustin Louis Cauchy [1882], Karl Theodor Wilhelm Weierstrass [1894], and Richard Dedekind [1930];

creating set theory by Georg Cantor [1932];

research axiomatization by David Hilbert [1899, 1932].

Mathematics has very many achievements and is both a universal scientific language and a basis for future research.

Rigorously axiomatized standard (classical) mathematics created by many famous mathematicians has already successfully solved very many scientific, engineering, educational, and life problems. Its future development will provide its very important role.

But there are very many typical urgent problems of our complicated world and time for which standard (classical) mathematics cannot propose its available adequate methods.

It occurs that in a whole series of key directions, by the level of thinking, modern classical mathematics corresponds to physics from the antique times to the 19th century, which also considered its atoms indivisible. In the 20th and 21st centuries, physics slowly deepens their final division into the component parts. It requires such research monsters as the Large Hadron Collider. And physics is indeed quite foremost natural science…

Practice let a number of scientists recognize and explicitly express such understanding.

Ruggero Maria Santilli [2008] wrote: "...there cannot be really new physical theories without really new mathematics, and there cannot be really new mathematics without new numbers".

Ivan Gandzha and Jerdsey Kadeisvily [2011] noted: "Santilli has repeatedly stated that: The origin of protracted controversies or unsolved problems in physics, chemistry, biology, and other sciences, is generally due to the use of mathematics basically insufficient for the quantitative treatment of the problem at hand, with consequential need to develop new appropriate mathematics".

Jakub Czajko [2004a] wrote: "Physics needs new mathematical foundations. Some problems of physics could be traced to hidden, unresolved issues in pure mathematics (PM), some of which are almost as old as the, allegedly impossible and therefore prohibited, division by zero... Mathematics may need an upgrade after a discovery is made in physics, for some old ideas may be irrelevant to new aspects of the physical reality. We need a synthetic mathematics (SM) to complement the classical analytic methods in mathematical research. Induction alone is insufficient for the syntheses needed to comprehend physics. Yet some new aspects of the physical reality could be deduced from experiments backed by the SM. Its inductive abstraction and strict rules of inference for deduction made mathematics the most exact of all exact sciences. Yet the PM emphasized its apriorical character and its ‘‘statutory’’ independence of any experimental evidence almost to the point of self-destruction. For by allowing the use of postulative method to define its primitive notions and fundamental objects, its splendid exactness is practically defeated... Many results of recent experiments and observations remain unexplained, because the PM still operates within the perimeter outlined by some ancient and medieval paradigms. Therefore we must upgrade abstract mathematics after significant breakthroughs in physics. It is not enough to justify former physical achievements by showing that mathematics complies with them. We should create quite new mathematics that goes far beyond and above of what past physics may have suggested. It is imperative thus to keep mathematics in sync with developments of new ideas in physics."

Classical mathematics [Encyclopaedia of Mathematics 1988] with hardened systems of axioms, intentional search for contradictions and even their purposeful creation cannot (and does not want to) regard very many problems in science, engineering, and life. This generally holds when solving valuation, estimation, discrimination, control, and optimization problems, as well as in particular by measuring very inhomogeneous objects and rapidly changeable processes. Lev Gelimson [1995a-g, 2001a, 2001b, 2001h, 2003f, 2004a, 2009a, 2009b] discovered that many classical fundamental mathematical theories, methods, and concepts [Encyclopaedia of Mathematics 1988] are insufficient for adequately solving and even considering many typical urgent problems.

It is clear that further scientific revolutions in mathematics must follow. There can be different ways for them, e.g.:

internal revolutions in standard (classical) mathematics itself;

scientific revolutions external with respect to standard (classical) mathematics itself but internal with respect to whole mathematics via creating a number of alternative mathematics so that whole mathematics becomes megamathematics.

Already either accepting or rejecting the axiom of choice bisects standard (classical) mathematics into two partially different alternative mathematics even if they have very much in common.

Along with classical (standard) mathematics, a number of other (alternative, nonstandard, nonclassical) mathematics can be possible and useful. Among them are, e.g.:

megamathematics based on alternative set theories by Petr Vopěnka [1979], Karel Hrbacek [2009b], etc.;

megamathematics based on mereology by Edmund Husserl [1901], Stanisław Leśniewski [1916], etc.;

non-Archimedean megamathematics by Abraham Robinson [1966], John Horton Conway [1976], Mark Burgin [2002, 2012], etc.;

megamathematics based on multisets by Richard Dedekind [1930], fuzzy sets by Lotfi Zadeh [1965] and Dieter Klaua [1965, 1966a, 1966b, 1967], rough sets by Zdzisław Pawlak [1982], fuzzy multisets by Ronald R. Yager [1986], etc.;

megamathematics by Ruggero Maria Santilli [1985a, 1985b, 1993a, 1993b, 1999]: "We therefore outline three sequential generalized mathematics introduced by the author under the name of iso-, geno- and hyper-mathematics which are based on generalized, Hermitean, non-Hermitean and multi-valued units, respectively";

mega-overmathematics by Lev Gelimson [1995a-g, 2001a, 2001b, 2001h, 2003f, 2004a, 2009a, 2009b, 2011c] first named "basic new mathematics" [1995a, 1995b] and then [ [2003f, 2004a] "elastic mathematics" are based on their uninumbers, quantielements, quantisets, and uniquantities with quantioperations and quantirelations. They provide universally and adequately modeling, expressing, measuring, evaluating, and estimating general objects. This all creates the basis for science unimathematical test fundamental metasciences systems by Lev Gelimson [2011b] which are universal.

The present monograph is dedicated to mega-overmathematics as revolutions in fundamental mathematics dealing with entirely abstract fundamental concepts [Encyclopaedia of Mathematics 1988] such as numbers, sets, their cardinalities and measures. These revolutions mean (in their general sense) creating fundamental mega-overmathematics alternative to classical (standard) fundamental mathematics and (in their special sense) creating own unified fundamentals for mega-overmathematics but DO NOT mean any change in classical mathematics itself.

Uniarithmetics, quantialgebra, and quantianalysis revolutionarily replace the very fundamentals of classical mathematics in mega-overmathematics.

Unfortunately, there is no possibility to refer to all the thousands of monographs and articles used by the author. He presents his sincere apologies and is much obliged to all those scientists, as well as to all the readers for their attention and desire to understand and use the following concepts that can help them solve their urgent problems and possibly give them the joy of touching some new knowledge.

The purpose of this work is a very "naive" joint presentations of some interconnected new concepts with many examples of their applications useful for the reader. No existence, consistancy, and uniqueness question is considered; sets, semisets, and classes are not distinguished; each complex uninumber is regarded as a separated number system. The following constructions are based on the real numbers, sets, and infinite cardinal numbers.

0. Introduction. Megamathematics and Mega-Overmathematics

0.1. Fundamental Science Unimathematical Test Fundamental Metasciences System

The fundamental science unimathematical test fundamental metasciences system by Lev Gelimson [2011b] in his mega-overmathematics is one of his science unimathematical test fundamental metasciences systems and can efficiently, universally, adequately, strategically, unimathematically test any fundamental science. This system includes:

fundamental metascience of fundamental science test philosophy, strategy, and tactic including fundamental science test philosophy metatheory, fundamental science test strategy metatheory, and fundamental science test tactic metatheory;

fundamental metascience of fundamental science consideration including science fundamentals determination metatheory, fundamental science approaches determination metatheory, fundamental science methods determination metatheory, and fundamental science conclusions determination metatheory;

fundamental metascience of fundamental science analysis including fundamental subscience analysis metatheory, science fundamentals analysis metatheory, fundamental science approaches analysis metatheory, fundamental science methods analysis metatheory, and fundamental science conclusions analysis metatheory;

fundamental metascience of fundamental science synthesis including science fundamentals synthesis metatheory, fundamental science approaches synthesis metatheory, fundamental science methods synthesis metatheory, and fundamental science conclusions synthesis metatheory;

fundamental metascience of fundamental science objects, operations, relations, and criteria including fundamental science object metatheory, fundamental science operation metatheory, fundamental science relation metatheory, and fundamental science criterion metatheory;

fundamental metascience of fundamental science evaluation, measurement, and estimation including fundamental science evaluation metatheory, fundamental science measurement metatheory, and fundamental science estimation metatheory;

fundamental metascience of fundamental science expression, modeling, and processing including fundamental science expression metatheory, fundamental science modeling metatheory, and fundamental science processing metatheory;

fundamental metascience of fundamental science symmetry and invariance including fundamental science symmetry metatheory and fundamental science invariance metatheory;

fundamental metascience of fundamental science bounds and levels including fundamental science bound metatheory and fundamental science level metatheory;

fundamental metascience of fundamental science directed test systems including fundamental science test direction metatheory and fundamental science test step metatheory;

fundamental metascience of fundamental science tolerably simplest limiting, critical, and worst cases analysis and synthesis including fundamental science tolerably simplest limiting cases analysis and synthesis metatheories, fundamental science tolerably simplest critical cases analysis and synthesis metatheories, fundamental science tolerably simplest worst cases analysis and synthesis metatheories, and fundamental science tolerably simplest limiting, critical, and worst cases counterexamples building metatheories;

fundamental metascience of fundamental science defects, mistakes, errors, reserves, reliability, and risk including fundamental science defect metatheory, fundamental science mistake metatheory, fundamental science error metatheory, fundamental science reserve metatheory, fundamental science reliability metatheory, and fundamental science risk metatheory;

fundamental metascience of fundamental science test result evaluation, measurement, estimation, and conclusion including fundamental science test result evaluation metatheory, fundamental science test result measurement metatheory, fundamental science test result estimation metatheory, and fundamental science test result conclusion metatheory;

fundamental metascience of fundamental science supplement, improvement, modernization, variation, modification, correction, transformation, generalization, and replacement including fundamental science supplement metatheory, fundamental science improvement metatheory, fundamental science modernization metatheory, fundamental science variation metatheory, fundamental science modification metatheory, fundamental science correction metatheory, fundamental science transformation metatheory, fundamental science generalization metatheory, and fundamental science replacement metatheory.

The fundamental science unimathematical test fundamental metasciences system in mega-overmathematics is universal and very efficient.

0.2. Fundamental Defects of Fundamental Mathematics

In particular, apply the fundamental science unimathematical test fundamental metasciences system by Lev Gelimson [2011b] in his mega-overmathematics to classical fundamental mathematics (Nicolas Bourbaki [1949], Arnold Kaufmann, Maurice Denis-Papin, Robert Faure [1964], [Encyclopaedia of Mathematics 1988]).

Even the very fundamentals of classical fundamental mathematics have evident cardinal defects of principle.

0.2.1. Real Numbers, Infinity, and Infinitesimals

Classical mathematics (Nicolas Bourbaki [1949], Arnold Kaufmann, Maurice Denis-Papin, Robert Faure [1964], [Encyclopaedia of Mathematics 1988]) believes there are no gaps between the real numbers whose set is denoted by R . But the probabilities of many typical reasonable events do not exist, for example the same probability

p_n = p

of the choice of a certain natural number n (e.g. 7) in the set

N = {1, 2, ...}.

This might be proved as follows. If that probability were positive, then the infinite sum of those probabilities for all natural numbers would be +∞. If that probability were zero, then the countable sum of those probabilities for all natural numbers would vanish because each partial sum would be zero and the limit of the sequence of these sums would vanish. But this infinite countable sum is the probability of the reliable event, that some natural number is chosen, and must be exactly 1.

Further the probabilities of many typical possible events vanish (e.g. that of the choice of a certain point on a segment of a straight line or curve), as if those were impossible events.

In ancient times, it was counted as follows: 1, 2, many; the concept "many" played the role of a lot.

It is known [Wikipedia Infinity] that already the Hellenistic Greeks used mathematical and philosophical infinity, e.g. Zeno in his paradoxes. Aristotle distinguished the potential infinity (implicitly used by Euclid and Archimedes) from the actual infinity (implicitly used by Archimedes). Early Indian mathematicians distinguished lowest, intermediate, and highest enumerable; nearly innumerable, truly innumerable, and innumerably innumerable; nearly infinite, truly infinite, and infinitely infinite numbers.

John Wallis [1655] introduced the infinity symbol, ∞ , with using infinitesimal 1/∞ by determining areas and distinguishing ∞ and ∞/2.

Gottfried Wilhelm Leibniz [1684, 1686] and Isaac Newton [1687] intensively used infinitesimals by creating infinitesimal (both differential and integral) calculus. Gottfried Wilhelm Leibniz [1714, 1716, 1846] also developed the ancient concept of both elementary and composite monads possibly infinitely and self-similarly divisible with the properties of their infinitesimal parts. This is also very important for modern mathematics [Encyclopaedia of Mathematics 1988] and physics including quantum physics [Encyclopaedia of Physics 1973], chaos theory (Ilya Prigogine [1993, 1997]), and fractal theory (Benoît Mandelbrot [1975, 1977, 1982], Alexey Stakhov [2009], and Sergey Abachiev [2012]), as well as Cantor sets [Encyclopaedia of Mathematics 1988] and Cantorian spacetime theory (Mohammed El Naschie [2009] and Jakub Czajko [2004a]).

Bernard le Bovier de Fontenelle [1727] freely and much operated with +∞ and -∞ for estimating only (because of absorption like ∞ ± a = ∞ and -∞ ± a = -∞ for any real number a , as well as ∞^a ± ∞^b = ∞^a for any rational a > b) and even considered ∞^∞ and ∞^1/∞ .

Jakub Czajko [2004b] also used ∞ for estimating other than the Cantor cardinality for which the Cantor generalized continuum hypothesis could be considered.

Georg Cantor [1932] introduced in 1874-1884 cardinal numbers to roughly discriminate very different infinities only. For example, the cardinal numbers of the segment [0, 1] and of the complete three-dimensional space are both equal to the same continuum cardinality ∁ . The cause is that, in contrast to the real numbers, each infinite cardinal number absorbs all the less and even equal cardinal numbers.

[Wikipedia Infinitesimal] notes that the Greek mathematician Archimedes (c.287 BC - c.212 BC) logically rigorously defined both infinities and infinitesimals via Archimedean property, as well as Archimedean number systems without any infinite or infinitesimal members [Archimedes 1912]. A non-Archimedean system cannot be complete because the reals are the unique complete ordered field up to isomorphism. There are three levels at which a non-Archimedean number system could have first-order properties compatible with those of the reals:

the usual axiomatic level;

the level of all the first-order properties involving the basic ordered-field relations + , * , and ≤ ;

the level of all the first-order properties.

Joseph-Louis Lagrange [1811] justified using infinitesimals: "When we have grasped the spirit of the infinitesimal method and have verified the exactness of its results either by the geometrical method of prime and ultimate ratios, or by the analytical method of derived functions, we may employ infinitely small quantities as a sure and valuable means of shortening and simplifying our proofs."

Augustin Louis Cauchy [1882] defined in 1821 an infinitesimal as a sequence tending to zero and represented in 1827 a unit impulse as an infinitely tall and narrow Dirac-type delta function δ_α via an infinitesimal α .

Guiseppe Veronese [1891] and David Hilbert [1899] proposed non-Archimedean geometric systems of the infinite and infinitesimal.

Tullio Levi-Civita [1892, 1898] considered his non-Archimedean ordered field containing real-number, infinite, and infinitesimal quantities as series of integer-exponent powers of a basic positive infinitesimal ε as an unspecified symbol.

Hans Hahn [1907] discovered non-Archimedean completeness due to replacing the reals via his field.

Kurt Gödel [1929, 1930] proved the countable compactness theorem. Anatoly Maltsev [1936] proved the uncountable compactness theorem. It proves the existence of infinitesimals via possibility to formalize them.

Albert Thoralf Skolem [1970] developed in 1934 the first non-standard models of arithmetic.

Edwin Hewitt [1948] provided the ultrapower construction of the hyperreals.

Jerzy Łoś [1955] proved his theorem that any first-order formula is true in an ultraproduct if and only if it is true in "most" factors. This first-order logic allows logical quantifications over elements only but not over sets.

Abraham Robinson [1966] developed in 1961 his nonstandard analysis. The hyperreals build an infinitesimal-enriched continuum. The transfer principle provides Leibniz's law of continuity. The standard part function implements Fermat's adequality. Keith Duncan Stroyan [1972, 1997] invented infinitesimal microscopes and infinite telescopes. Howard Jerome Keisler [1976a, 1976b], David Tall [1980b], and Lorenzo Magnani [2001] improved them.

John Horton Conway [1976] created his systems (with a tree structure) of subsequently producing both the surreal (also exotic) numbers building an ordered field and the more general games without these properties, see also Philip Ehrlich [1994, 2006, 2012].

Edward Nelson [1977] extended the real numbers axioms (or language) to make infinitesimals in the real numbers themselves. Karel Hrbacek [2006, 2009a, 2009b] inproved this approach stratifying the real numbers in infinitely many levels.

David Tall [1979] expanded the real numbers to the superreal numbers as the field of Laurent series in an infinitesimal ε with a finite number of negative-power terms. David Tall [1980a] interpreted quite different continual sets estimations via measuring (with estimating a point via a positive infinitesimal) and cardinal approaches.

H. Garth Dales and William Hugh Woodin [1996] gave a more abstract super-real field as another extension of the real numbers.

John Lane Bell [2009] introduced smooth infinitesimal analysis based on the concept of the nilpotent nonzero infinitesimal whose square is zero. Then the set of infinitesimals is

I = {x | x² = 0}.

Jonathan W. Hoyle [2007] showed that the hyperreal continuum is Dedekind incomplete and cannot be completed but its internal subsets with an upper bound have a least upper bound due to the transfer principle. Therefore, the set I of all the infinitesimals, as well as N and R (each of them has an upper bound but no least upper bound), must be external.

Ruggero Maria Santilli [1985a, 1985b, 1993a, 1993b, 1999] created his own isomathematics, genomathematics, and hypermathematics based on his isonumbers, genonumbers, and hypernumbers for treating matter, as well as on their anti-Hermitian versions, namely isodual isonumbers, isodual genonumbers, and isodual hypernumbers for treating antimatter, respectively.

Mark Burgin [2002, 2012] added to the real and complex numbers infinitely big and oscillating numbers as external objects without changing the inner structure of the real and complex numbers spaces via injecting into them infinitely small numbers and other nonstandard entities as nonstandard analysis does. His real hypernumbers are sets of equivalent sequences of real numbers like real numbers are sets of equivalent fundamental sequences of rational numbers. In the universe of his hypernumbers, all sequences and series of real and complex numbers, as well as definite integrals of continuous functions, have values. They are either ordinary (real and complex) numbers for convergent sequences, series, and integrals, or (infinite and oscillating) hypernumbers for divergent sequences, series, and integrals. This simplifies control and management of divergent processes.

Therefore, infinitely small infinitesimal numbers have been already introduced and justified. But their known systems do not allow to conveniently express each required amount, e.g. that probability of the choice of a certain natural number. Belonging and existence answers ("This is an infinitesimal number"; "There is such an infinitesimal number") bring no satisfaction because all the necessary operations should be also applicable to the appropriate real number generalization.

If there were no equality such as

i² = -1,

then situation with the imaginary numbers would be similar. It would be impossible to both determine specific solutions to many very important equations and operate on such numbers. Instead of this, one could say: "That is an imaginary number", "There is such an imaginary number".

Firther in real analysis, both +∞ and -∞ represent a lot of many also very different infinities hardly distinguishable.

In complex analysis, the only infinity ∞ without any direction (in particular, sign) is used to provide

1/∞ = 0

and the one-point compactification of the complex plane [Encyclopaedia of Mathematics 1988].

Notata bene.

1. If x is a real variable, then

lim_x→+∞x = +∞

where function x is considered real. But if function x is considered complex, then

lim_x→+∞x = ∞ .

2. If x is a real variable, then

lim_x→+∞(x + xi) = lim_x→+∞(x - xi) = lim_x→+∞(- x + xi) = lim_x→+∞(- x - xi)

= lim_x→+∞x = lim_x→+∞(- x) = lim_x→+∞xi = lim_x→+∞(- xi) = ∞

where functions x and (- x) are considered complex.

Classical mathematics [Encyclopaedia of Mathematics 1988] cannot precisely measure different infinities at all. For example, any classical mathematical measure of each segment or interval on a straight line or a curve is independent of whether or not that includes its endpoints.

Bernard Bolzano [1851] stated his dissatisfaction with such circumstances and tried to do something in the particular case of a natural-number length.

Conclusions

1. The real numbers R evaluate no unbounded quantity and, because of gaps, not all bounded quantities. The same probability p_n = p of the random sampling of a certain n ∈ N = {0, 1, 2, ...} does not exist in R , since ∑_n∈N p_n is either 0 for p = 0 or +∞ for p > 0.

2. Using infinite and infinitesimal quantities has been already justified.

3. There is no clear way to explicitly express many typical specific infinite and infinitesimal quantities in urgent problems via well-known mathematics.

4. It is urgent to exactly express (in some suitable extension of the real numbers) all the infinitesimal, finite, infinite, and combined pure (dimensionless) amounts, e.g. such a p for any countable or uncountable set, as well as distributions and distribution functions on any sets of infinite measures, and conveniently operate on them.

5. Lev Gelimson first explicitly directly expressed infinitesimal, finite, infinite, and combined pure (dimensionless) amounts namely required (desired) for setting and solving many typical urgent problems via introducing canonic infinitesimals (namely signed zeroes) [1994c] and canonic infinities [1994c, 1995a, 1995c, 1997a, 1997b] exactly expressing the also uncountably algebraically additive quantities of the elements of some suitable canonic sets. From the beginning of 2001, the further editions [Gelimson 2001a, 2001b, 2001c, 2001d, 2001e, 2001h] of these and many other scientific publications are available on the Internet. Giovanni Giuseppe Nicosia placed in his Doctor Thesis [2001] in February, 2001, a reference to [Gelimson 2001d], as well as in [Nicosia 2005]. Scientist Vuara published [Gelimson 2001h] along with the references of some academicians and professors to Lev Gelimson's monographs, articles, and other scientific publications in [Wikibooks Hyperanalysis-MeasurementTheory]. [Hypernumber Blogspot] noted: "Other kinds of hypernumber are defined differently by Mark Burgin, Rugerro Maria Santilli and" the author. Armahedi Mahzar [Hypernumbers Group] wrote: "Other kinds of hypernumbers are the internal extensions of real numbers created by making the axis of real number more dense. Examples of such internal hypernumbers are the hyperreal numbers of Robinson, surreal numbers of Conway, hypernumbers of Mark Burgin and" the author. "This group will explore such existing kinds of hypernumbers and beyond."

0.2.2. Operations

In classical mathematics [Encyclopaedia of Mathematics 1988], in each concrete (mixed) physical magnitude, e.g. 5 liter fuel, the operation unifying "5 L" and "fuel" is not obvious.

It is impossible to consider either "fuel multiplied by 5 liter" or, all the more, "5 liter multiplied by fuel". These both pure theoretical possibilities are not reasonable at all. And classical mathematics [Encyclopaedia of Mathematics 1988] cannot propose nothing else.

Hence for any concrete (mixed) physical magnitude (quantity with a measurement unit), there is no suitable mathematical model and no known suitable operation.

Nota bene: Multiplication is the evident operation between the number "5" and the measurement unit "L".

Further even the pure number operations in classical mathematics [Encyclopaedia of Mathematics 1988] are considered to be at most countable, which makes the range of mathematical models very narrow.

John Wallis [1656] extended power exponents from positive integers to rational numbers.

But such a finite pure number operation as raising a negative number to a power is well-defined for even positive integer exponents only. See counterexamples

(-1)³ = -1 ≠ 1 = [(-1)⁶]^1/2 = (-1)^6/2 ,

(-1)^1/3 = -1 ≠ 1 = [(-1)²]^1/6 = (-1)^2/6 .

In classical mathematics [Encyclopaedia of Mathematics 1988], division by zero is undefined and hence avoided in real-number arithmetic, algebra, and analysis.

[Wikipedia Division_by_zero] notes that by formal operations in formal calculation using rules of arithmetic without consideration of whether the result of the calculation is well-defined, it is sometimes useful to think of a/0, where a ≠ 0, as being ∞ . This infinity can be either positive, negative, or unsigned, depending on context. The real projective line, the Riemann sphere, the extended non-negative real number line, and complex analysis bring here nothing new. In the known hyperreal and surreal numbers, division by zero is still impossible unlike division by nonzero infinitesimals. In computer arithmetic, the IEEE floating-point standard specifies that every floating point arithmetic operation, including division by zero, has a well-defined result. To provide this, the standard supports signed zeroes (positive zero +0 and negative zero -0), signed infinities, and NaN (not a number):

dividing a by +0 is positive infinity when a is positive, negative infinity when a is negative, and NaN when a = ±0;

dividing a by -0 is negative infinity when a is positive, positive infinity when a is negative, and NaN when a = ±0.

For 1/0, most calculators return either an error or undefined state. Some of them give (1/0)² = ∞ .

Nota bene: Here a/0 = ∞ independently of a or, at least, of its modulus (absolute value) |a|, e.g. 1/0 = ∞ = 10¹⁰/0.

[Wikipedia Signed_zero] gives further rules of the IEEE floating-point standard:

-0/|x| = -0 (x ≠ 0),

(-0)(-0) = +0,

|x|(-0) = -0,

x + (-0) = x + (+0) = x ,

(-0) + (-0) = (-0) - (+0) = -0,

(+0) + (+0) = (+0) - (-0) = +0,

x - x = x + (-x) = +0

(for any finite x , -0 when rounding toward negative).

Because of negative zero (and only because of it), the statements

z = -(x - y)

and

z = (-x) - (-y),

for floating-point variables x , y , and z , cannot be optimized to

z = y - x .

Some other special rules:

(-0)^1/2 = -0,

(-0)/(-∞) = +0 (follows the sign rule for division),

|x|/(-0) = -∞ (for non-zero x, follows the sign rule for division),

(±0)(±∞) = NaN ,

(±0)/(±0) = NaN .

Nota bene: Conservation law does not hold here.

Jakub Czajko [2004a] proposed distinguishing a/0 and b/0 by a ≠ b but rejected 0 × 0 = 0 with replacing this precise equality via approximate equality 0 × 0 ≈ 0 because he considered

0 × 0 = 1/∞ × 1/∞ = 1/∞² ≈ 0.

Robert Goldblatt [1998] built the hyperreals *R including both infinitesimal and infinite elements via extending standard arithmetic operations to R^∞ , see also Jonathan W. Hoyle [2007]. For a = <a(0), a(1), ...> and b = <b(0), b(1), ...>, they defined, in particular,

a/b = <a(0)/b(0), a(1)/b(1), …>,

a^b = <a(0)^b(0), a(1)^b(1), …>

with undefined sequences indices for which these operations are not defined. Therefore, also by building the nonstandard universe, division by zero and raising a negative number to a power can lead to problems.

It is usual to consider in classical mathematics [Encyclopaedia of Mathematics 1988] that the empty sum equals 0 whereas the empty product equals 1. The both rules work in the best cases only because both adding a number a to the empty sum and multiplying the empty product with a number a give the correct result (value, output) a . Otherwise, the results are incorrect as a rule. It would be better to consider:

the empty sum equals 0 if and only if namely addition is the only further operation;

the empty product equals 1 if and only if namely multiplication is the only further operation.

But any dependence of the already performed operations output on any further operation proves that such a result makes no objective sense at all.

Moreover, both the empty sum and the empty product are particular cases of the "result" of performing no operations (hence on no operands, arguments, or inputs). This "result" may not depend on any particular operation and must be universal. But classical mathematics [Encyclopaedia of Mathematics 1988] cannot provide any universal value of this output. In particular, neither 0 nor 1 can provide such universality.

Conclusions

1. The system of operations in classical mathematics has gaps because it cannot mathematically model any concrete (mixed) physical magnitude (quantity with a measurement unit).

2. Even the pure number operations in classical mathematics are considered to be at most countable.

3. Even finite pure number operations in classical mathematics can have very narrow correct definition domains.

4. It is urgent to define raising a negative number to a power.

5. It is urgent to exactly express (in some suitable extension of the real numbers) division by zero.

6. Lev Gelimson [1994c, 1995a] first explicitly directly expressed division by zero to further extend all the infinitesimal, finite, infinite, and combined pure (dimensionless) amounts and to conveniently operate on them with holding conservation law and introduced the empting (voiding) operation transforming any object to the empty (void) object (element) # (or the empty set ∅ so that # ∈ ∅ and # = ∅). Using the empty (void) operand # (or ∅) excludes (drops) any operation on this operand so that this operand neutralizes any operation. Then the "result" of performing no operations (hence on no operands, arguments, or inputs) equals namely # (or ∅), which is universal. Further zero 0 may be considered to be nonnumber which does not belong to the natural numbers N , to the integer numbers Z , to the real numbers R , to the complex numbers C , etc.

0.2.3. Set Theory vs. Mereology

The set theory of Georg Cantor [1932] is the basis of contemporary classical mathematics [Encyclopaedia of Mathematics 1988]. Intuitive approach leads to paradoxes [Encyclopaedia of Mathematics 1988]. To avoid them, axiomatic set theory systems [Encyclopaedia of Mathematics 1988] are used.

The classical Cantor sets with either unit or zero quantities of their possible elements may contain any object as an element either once or not at all, and its further repetitions are ignored. 0 and 1 are the only values of the indicator functions in the classical Cantor sets. No other quantities of the existing elements are considered. For example, the following two sets are exactly equal to each other: the first set consists of a million of 1 Euro coins conditionally indistinguishable, and the second set consists of one 1 Euro coin. That is, a millionaire is equivalent to a poor person. Of course, in many cases, the Cantor set theory gives much healthier models than we often see. But frequently it does not work at all. And the fundamental conservation laws are broken. Hence this theory cannot model processes with holding these universal laws. No other quantities of the existing elements are considered also by Cantor set relations and operations. Those set operations with absorption are only restrictedly invertible. The simplest equations

X ∪ A = B

and

X ∩ A = B

in X are solvable only by

A ⊆ B

and

A ⊇ B ,

respectively (uniquely by

A = ∅

and

A = B = U

(a universal set), respectively). The equations

X ∪ A = B

and

X = B \ A

are equivalent by

A = ∅

only.

Richard Dedekind [1930] in 1888 considered sets with element repetitions. They are admissible in ordered sets such as tuples, sequences, vectors, permutations, arrangements, etc. (G. A. Korn and T. M. Korn [1968], Martin Aigner [1979], [Encyclopaedia of Mathematics 1988], as well as I. N. Bronstein and K. A. Semendjajew [1989] considering unordered combinations with repetitions as no sets but equivalence classes of equipartite arrangements with repetitions). Donald Ervin Knuth [1997] noted that Indian mathematician Bhascara Acharya (circa 1150) studied permutations of a multiset (named by Nicolaas Govert de Bruijn in the 1970s) with element repetitions. In multisets (Wayne D. Blizard [1991], Apostolos Syropoulos [Syropoulos 2001]), element repetitions are usually taken into account with any natural numbers as multiplicities.

Alexey Petrovsky [1992, 1994, 2003] developed an axiomatic approach to the metrization of the multiset space.

Daniel Loeb [1992] generalized multisets via hybrid sets with any integer multiplicities.

Lotfi Zadeh [1965] and Dieter Klaua [1965, 1966a, 1966b, 1967] introduced fuzzy sets, see also Didier Dubois and Henri Prade [1980]. In a fuzzy set, usually the membership function of each element may also lie strictly between these ultimate values 0 and 1 in the case of uncertainty only.

Ronald R. Yager [1986] introduced fuzzy multisets combining fuzzy sets and multisets via mapping each element to more than one fuzzy membership function over the unit interval, see also Sadaaki Miyamoto [2001, 2004].

Zdzisław Pawlak [1982] introduced rough sets.

Didier Dubois and Henri Prade [1990] unified rough fuzzy sets and fuzzy rough sets.

None of these well-known set generalizations can express and model many typical unordered objects collections (without structure), e.g. that of half an apple and a quarter pear. Even if 1/2 and 1/4 are pure numbers between 0 and 1 (the only values of the indicator functions in the classical Cantor sets) like membership functions in fuzzy sets, the fuzzy sets cannot model this unordered objects collection because it consists of exactly half an apple and a quarter pear and hence is nonfuzzy (crisp, precise, exact, definite, well-defined, strictly defined).

In many unordered collections (e.g. of bonds and coins), namely the quantity (multiplicity, number, amount) of each element is decisive and can be arbitrary, for example 3.5 kg, 1 loaf (bread), - 45 € (Euro) by buying and those with opposite signs by selling.

Bernard Bolzano [1851] expressed his dissatisfaction with the fact that classical mathematics is powerless to quantitatively reflect many finite and even infinite changes in the infinite sets. He attempted to make something in the very special case of rectangles with the integral sides only. But all these attempts were later declared to be misbelief with mistakes.

[Wikipedia Mereology] notes that mereology treats the wholes and their parts. The meronomic relation between entities is closer than both the element-set membership relation and the set inclusion relation. Plato and Aristotle used informal part-whole reasoning. Edmund Husserl [1901] investigated part-whole relation. Stanisław Leśniewski [1916] initiated and then created mereology but rejected (together with his successors) set theory. Nearly all philosophers and mathematicians avoid mereology.

Conclusions

1. The classical Cantor sets with either unit or zero quantities (0 and 1 are the only values of the indicator functions in the classical Cantor sets) of their possible elements may contain any object as an element either once or not at all with otherwise ignoring its true quantity.

2. The Cantor set relations and operations only restrictedly reversible and allowing absorption contradict universal conservation laws of nature because of ignoring element quantities and hinder constructing any universal degrees of quantity.

3. Both mereology and well-known set generalizations such as multisets, fuzzy sets, and rough sets, as well as ordered sets (tuples, sequences, vectors, permutations, arrangements, etc.) and unordered combinations with repetitions, cannot express and model many typical objects collections (without structure).

4. It is urgent to further generalize Cantor sets, their relations and operations, as well as to unify the approaches both of generalized set theory and mereology.

5. Lev Gelimson [1995a] first introduced (nonlogical) quantification and quantity determination operations, unified the membership, inclusion, and part-whole relations with providing also uncountable operations, and introduced general quantisystems. They generalize systems as mereology objects. The general quanticontents of general quantisystems further generalize quantisets in which the quantity of each element may be any general object, e.g. a general quantisystem. An element with its individual quantity builds a quantielement.

0.2.4. Cardinalities

Cardinal number theory of Georg Cantor [1932] introduced cardinal numbers to roughly discriminate very different infinities only as the first instrument of comparing and measuring distinct infinities. This tool remains the only in classical mathematics [Encyclopaedia of Mathematics 1988] and is useful. However, it is low-sensitive. When we either add to an infinite set still an equivalent set or divide an initial infinite set into two equivalent infinite sets and leave only one of them, their cardinalities remain the same and are equal to each other. Moreover, the unit interval [0, 1], the entire three-dimensional infinite space, and even any spaces of countable dimensionalities have the same continuum cardinality ∁ . The cause is that, in contrast to the real numbers, each infinite cardinal number absorbs all the less and even equal cardinal numbers. Infinity simply remains a heap of infinities completely distinct and extremely roughly (by cardinal numbers only and nothing more) divided into the classes. The two of them are especially important: the class of countable sets with their common cardinal number denote by aleph with index zero and the class of the sets of the power of continuum with their common cardinal number ∁ . The ancient numerical scale is recalled: one, two, many… Here the concept "many" unifies all the further natural numbers (positive integers) beginning with 3 without their separation.

Conclusions

1. The cardinality is sensitive to finite unions of disjoint finite sets only but not sufficiently sensitive to infinite sets and even to intersecting finite sets (because of absorption). It gives the same continuum cardinality ∁ for clearly very distinct point sets in a Cartesian coordinate system between two parallel lines or planes differently distant from one another.

2. It is urgent to provide exactly discriminating any noncoinciding possibly infinitely great or small objects or models by holding universal conservation laws.

3. Lev Gelimson [1994c, 1995a] first generalized exactly counting all the elements of any also infinite set via its uniquantity with universalization even for infinitesimal differences of element quantities of any infinite quantisets and with providing exactly (perfectly sensitively) discriminating any noncoinciding possibly infinitely great or small objects or models by holding the universal conservation law.

0.2.5. Measures

Measure theory in classical mathematics ([Bourbaki 1949], [Encyclopaedia of Mathematics 1988]) typically uses measures of Johann Radon and Henri Lebesgue along with integrals of Bernhard Riemann and Henri Lebesgue. Usual measures may take nonnegative values or +∞ . Signed measures including charges may also take negative values or -∞ .

Felix Hausdorff [1919] proposed his measure already before [1935]. It always exists along with his dimension which can be noninteger and is widely used in fractal theory (Benoît Mandelbrot [1975, 1977, 1982], Alexey Stakhov [2009], and Sergey Abachiev [2012]), as well as Cantor sets [Encyclopaedia of Mathematics 1988] and Cantorian spacetime theory (Mohammed El Naschie [2009] and Jakub Czajko [2004a]).

The counting measure slightly generalizes counting and is defined either as the number of elements in any finite Cantor set or +∞ for any infinite Cantor set, which is natural but brings nothing new.

Therefore, well known measures are sensitive only restrictedly, namely in the limits of the specific dimensionality, within them completely ignore even uncountable zero-measure changes, and give either 0 or +∞ for distinct point sets between two parallel lines or planes differently distant from one another.

Any measure of each segment or interval on a straight line or a curve is independent of whether or not that includes its endpoints.

In classical mathematics ([Bourbaki 1949], [Encyclopaedia of Mathematics 1988]), there are no sensitive common measures for any even bounded sets of mixed dimensions, i.e. sets simultaneously including parts of different dimensions such as separate points, intervals, as well as bounded parts of surfaces and spaces.

Bernard Bolzano [1851] stated his dissatisfaction with such circumstances and tried to do something in the particular case of a natural-number length.

Conclusions

1. Well-known measures are only finitely sensitive within a certain dimensionality, give either 0 or +∞ for distinct point sets between two parallel lines or planes differently distant from one another, and cannot discriminate the empty set ∅ and null sets, namely zero-measure sets.

2. There are no sensitive common measures for any even bounded sets of mixed dimensions.

3. It is urgent to exactly measure any possibly infinitely great or small objects or models by holding universal conservation laws.

4. Lev Gelimson [1994c, 1995a] first generalized exactly counting all the elements of any infinite set via its uniquantity with universalization even for infinitesimal differences of element quantities of any infinite quantisets. This provided exactly (perfectly sensitively) measuring any possibly infinitely great or small objects or models by holding universal conservation laws for setting and solving many typical urgent problems. He introduced canonic infinitesimals (namely signed zeroes) [1994d] and canonic infinities [1994c, 1995a, 1995c, 1997a, 1997b] exactly expressing the also uncountably algebraically additive quantities of the elements of some suitable canonic sets. From the beginning of 2001, the further editions [Gelimson 2001a, 2001b, 2001c, 2001d, 2001e, 2001h] of these and many other scientific publications are available on the Internet. Giovanni Giuseppe Nicosia placed in his Doctor Thesis [2001] in February, 2001, a reference to [Gelimson 2001d], as well as in [Nicosia 2005]. Scientist Vuara has published [Gelimson 2001h] along with the references of some academicians and professors to Lev Gelimson's monographs, articles, and other scientific publications in [Wikibooks Hyperanalysis-MeasurementTheory]. [Hypernumber Blogspot] noted: "Other kinds of hypernumber are defined differently by Mark Burgin, Rugerro Maria Santilli and" the author. Armahedi Mahzar [Hypernumbers Group] wrote: "Other kinds of hypernumbers are the internal extensions of real numbers created by making the axis of real number more dense. Examples of such internal hypernumbers are the hyperreal numbers of Robinson, surreal numbers of Conway, hypernumbers of Mark Burgin and" the author. "This group will explore such existing kinds of hypernumbers and beyond." [Gelimson 2003a, 2004a] further developed his uniquantities building an explicit universal exact counting measure perfectly sensitive even to infinitesimal differences of element quantities of any infinite quantisets.

Addition

Alexey Petrovsky [2003] investigated new types of the spaces of measurable sets and multisets, as well as the general properties of set and multiset measures.

Mark Burgin [2005] proposed his hypermeasures via adding infinitely big and oscillating numbers as external objects to the real and complex numbers without changing the inner structure of the real and complex numbers spaces via injecting into them infinitely small numbers and other nonstandard entities as nonstandard analysis does. His real hypernumbers are sets of equivalent sequences of real numbers like real numbers are sets of equivalent fundamental sequences of rational numbers. In the universe of his hypernumbers, all sequences and series of real and complex numbers, as well as definite integrals of continuous functions, have values. They are either ordinary (real and complex) numbers for convergent sequences, series, and integrals, or (infinite and oscillating) hypernumbers for divergent sequences, series, and integrals.

0.2.6. Probabilities

Any probability measure takes values from the closed unit interval [0, 1] only and is a particular case of a measure.

In classical mathematics [Encyclopaedia of Mathematics 1988], real numbers having gaps between them cannot express not only unlimited, but also many limited quantities (e.g. the probability of selecting one given number from all natural numbers).

Let us assume that there are 10 balls with ciphers (digits) 0, 1, ... , 9, respectively, in a bag. Precisely one of the balls is picked out blindly (randomly, without any extrasensory abilities). What is the probability that the picked ball has namely a given (predefined) cipher (digit), for example 7? The total number of all the possible outcomes is 10. By only one of them, the desired event occurs. That is why the desired probability by its classical definition [Encyclopaedia of Mathematics 1988] is 1/10.

Let us now consider a more complicated problem. Imagine that we select exactly one number from countably many nonnegative integers 0, 1, 2, ... , 10, ... , 100, ... , 1000, ... with equal probability of selecting any of them. What is this probability or, equally, the probability that we have selected namely a given (predefined) number, for example, 7?

Classical mathematics [Encyclopaedia of Mathematics 1988] declares that this probability does not exist at all because of the following "proof". Ad absurdum, suppose that this probability exists. Then it has to be either 0 or positive. If it were 0, then the total probability of selecting any of nonnegative integers whose total number is finite would also vanish. The same would hold for all the nonnegative integers, which is proved via the corresponding limiting process. But the total probability of selecting any of all the nonnegative integers has to be precisely 1 as the probability of a certain event. Indeed, exactly one of all the nonnegative integers is selected. If the desired probability were, on the contrary, any positive number, then divide 1 by this number and take any nonnegative integer which is greater than this quotient. The so-called axiom of Archimedes [Encyclopaedia of Mathematics 1988] provides that there exist infinitely many such nonnegative integers. For definity, take the least from these nonnegative integers. Then the total probability of selecting any of nonnegative integers from 0 to the taken nonnegative integer would be greater than 1, which is impossible for any probability at all. Moreover, in this case of any positive desired probability, the corresponding limiting process would even give plus infinity for the total probability of selecting any of all the nonnegative integers instead of 1 as the probability of this certain event. In this way, classical mathematics [Encyclopaedia of Mathematics 1988] leads to the conclusion that the desired probability does not exist at all.

Note that the Archimedes axiom [Encyclopaedia of Mathematics 1988] in mathematics is an invention, which is typical for entire mathematics itself, whereas the Archimedes law on a buoyant force [Encyclopaedia of Mathematics 1988] much better known is a law of nature fully objective and hence is a discovery, which is typical for natural sciences.

Classical mathematics [Encyclopaedia of Mathematics 1988] also declares without any explanations that if the probability of selecting anyone of the elements of any uncountable set, for example an interval, a straight line, a rectangle, a plane, or a space, is the same, then it vanishes, as if that would be an impossible event.

But these and many other typical events are fully reasonable and possible and hence must have certain positive probabilities. And if classical mathematics cannot indicate them, then its real number system is clearly insufficient and has gaps.

It is possible to assume that such probabilities are indeterminate infinitesimals.

Let us give the following analogy. It is necessary to solve a certain equation. How much benefit would we achieve due to the conclusion that the solutions to this equation are some undetermined imaginary numbers without their determination and clear indication? It is not difficult to guess the school grade for such an answer…

Conclusions

1. Any probability measure is a particular case of a measure.

2. The probabilities cannot discriminate impossible and some differently possible events.

3. The probabilities of reasonable possible events can be nonexising at all.

4. There is no known universal exact perfectly sensitive measure for any possibly infinitely great or small objects or models by holding universal conservation laws.

5. The real numbers having gaps between them cannot express not only unlimited, but also many limited quantities.

6. It is urgent to exactly measure any probability so that each fully reasonable and possible event has a certain namely positive probability.

7. Lev Gelimson [1994c, 1995a] first generalized exactly counting all the elements of any infinite set with universalization even for infinitesimal differences of element quantities of any infinite quantisets and with providing a certain namely positive probability of any fully reasonable and possible event.

0.2.7. Contradictory Objects, Systems, and Models

In classical mathematics [Encyclopaedia of Mathematics 1988], contradictory objects, systems, and models are declared nonexisting at all and are completely ignored along with contradictory problems even if they are urgent. It intentionally avoids, ignores, and cannot (and possibly hence does not want to) adequately consider, model, express, measure, evaluate, and estimate many complications. Among them are contradictions, infringements, damages, hindrances, obstacles, restrictions, mistakes, distortions, errors, information incompleteness, multivariant approach, etc.

Nota bene: All existing objects and systems in nature, society, and thinking have complications, e.g. contradictoriness, and hence exist without adequate models in classical mathematics [Encyclopaedia of Mathematics 1988].

Conclusions

1. Classical mathematics intentionally avoids, ignores, and cannot (and possibly hence does not want to) adequately consider, model, express, measure, evaluate, and estimate many complications such as contradictions, infringements, damages, hindrances, obstacles, restrictions, mistakes, distortions, errors, information incompleteness, multivariant approach, etc.

2. All existing objects and systems in nature, society, and thinking have complications, e.g. contradictoriness.

3. It is urgent to adequately model also contradictory objects and systems.

4. Lev Gelimson [1995a, 1995b] created his own mathematics named "basic new mathematics" allowing and efficiently using also contradictory objects, systems, and models existing at least in the symbolic sense as "black boxes". He also unified the membership, inclusion, and part-whole relations, which alone can often avoid contradictions and excludes many paradoxes. From the beginning of 2001, the further editions [Gelimson 2001a, 2001b, 2001c, 2001d, 2001h] of these and many other scientific publications are available on the Internet. Giovanni Giuseppe Nicosia placed in his Doctor Thesis [2001] in February, 2001, a reference to [Gelimson 2001d], as well as in [Nicosia 2005]. Scientist Vuara has published [Gelimson 2001h] along with the references of some academicians and professors to Lev Gelimson's monographs, articles, and other scientific publications in [Wikibooks Hyperanalysis-MeasurementTheory]. [Gelimson 2003a, 2003b, 2003d, 2003e, 2003f, 2004a] further developed his own mathematics then named "elastic mathematics".

Addition

Alexey Petrovsky [2001a, 2001b] constructed a general decision rule for contradictory expert classification of multiattribute objects and multiple criteria project selection based on contradictory sorting rules.

0.3. Classical Fundamental Mathematics Revolutions Necessity

Naturally, along with the above, there are very many further lacks and shortcomings of classical fundamental mathematics (Nicolas Bourbaki [1949], Arnold Kaufmann, Maurice Denis-Papin, Robert Faure [1964], [Encyclopaedia of Mathematics 1988]).

Therefore, the very fundamentals of classical fundamental mathematics have a lot of obviously deep and even cardinal defects of principle.

Consequently, to make classical fundamental mathematics adequate, its evolutionarily locally correcting, improving, and developing which can be useful are, unfortunately, fully insufficient. Classical fundamental mathematics needs revolutionarily replacing its inadequate very fundamentals via adequate very fundamentals.

Conclusions

1. The very fundamentals of classical fundamental mathematics have a lot of obviously deep and even cardinal defects of principle.

2. Mathematics is the only science whose very fundamentals remain almost unchanged for more than a century whereas there were many revolutions in all natural, technical, and humanitarian sciences natural, technical, and humanitarian sciences in this time.

3. It is urgent to create and develop alternative mathematics with revolutionarily replacing the inadequate very fundamentals of classical fundamental mathematics via adequate very fundamentals.

4. Lev Gelimson [1995a, 1995b] created his own mathematics named "basic new mathematics" with revolutionarily replacing the inadequate very fundamentals of classical fundamental mathematics via adequate very fundamentals. From the beginning of 2001, the further editions [Gelimson 2001a, 2001b, 2001c, 2001d, 2001h] of these and many other scientific publications are available on the Internet. Giovanni Giuseppe Nicosia placed in his Doctor Thesis [2001] in February, 2001, a reference to [Gelimson 2001d], as well as in [Nicosia 2005]. Scientist Vuara has published [Gelimson 2001h] along with the references of some academicians and professors to Lev Gelimson's monographs, articles, and other scientific publications in [Wikibooks Hyperanalysis-MeasurementTheory]. [Gelimson 2003a, 2003b, 2003d, 2003e, 2003f, 2004a] further developed his own mathematics then named "elastic mathematics".

0.4. Classical Fundamental Mathematics Revolutions Possibility

Natural, technical, and humanitarian sciences discover and model real, objective laws of nature, technology, thinking, and society. It is impossible and inadmissible to devise any imaginable dependencies instead of such real laws and to try to impose these dependencies to the real nature, technology, and society. Each of these sciences (with its object) is unique. It is difficult to imagine any alternative general physics another than classical general physics because the nature is the same and unique. However, also in general physics, there are different particular approaches, methods, theories, and especially hypotheses, for example, the corpuscular and wave theories of light.

On the contrary, mathematics is an expedient fabrication, absolute invention, and a result of fantasy sufficiently free. Where can be found, e.g., number 2 or a rectangle themselves (NOT their images completely material, for example, drawn on a chalk board) in the nature? Symbol 2 is relative at all. It is inherent in the numeration system with the base not less than 3 in the Arab (Indian) numeration. In the binary system, this is symbol 10. In the Roman numeration, this is symbol II. I.e., symbol 2 is conditional. In general, it is not possible to noticeably depict the contour of a rectangle. Indeed, a one-dimensional line possesses only a length and has zero width. I.e., mathematics completely consists of inventions. But it is not completely freely devised. It is the universal language of sciences which simulates real objects and their relations. Any real two-element sets such as 2 apples, 2 lions, etc. (which can be placed in the one-to-one correspondence), have their real quantity in common. And number 2 has been devised to express this real quantity which is objective, i.e. independent of our consciousness. And a rectangle is an ideal model for the description of, e.g., the faces of bricks. Carl Friedrich Gauss [1863] said and wrote: "Mathematics is the queen of sciences, and number theory is the queen of mathematics".

The sole limitations of fantasy by creating mathematics are its convenience and suitibility for sufficiently adequate mathematical simulation of real (natural, technical, and social) objects and systems, as well as for solving other problems important for these objects and systems.

In principle, each mathematical science allows entire full-valuable alternatives. A classical example could be of Euclidean, Lobachevskian, and Riemannian geometries. Different whole and complete mathematics (plural) as distinct sciences are also well-possible, well-considerable, and well-creatable. In particular, this holds for overmathematics advanced, created, and developed by Lev Gelimson [1995a-2012]. Consequently, questions of the type "What right has the author to advance, create, and develop his overmathematics?" are fully absurd. The sole limitation should be furthermore the expedience: "Why is overmathematics necessary and useful under conditions that classical mathematics is already available, well-known, and well-developed for millenia?"

Conclusions

1. There is a fundamental difference of mathematics as a pure invention of imaginable models from natural, technical, and humanitarian sciences discovering and modeling true laws of nature, engineering, thinking, and society.

2. Pluralism in mathematics already takes place and is always admissible.

3. It is possible and admissible to create and develop alternative mathematics.

4. Lev Gelimson [1995a, 1995b] created his own mathematics named "basic new mathematics" with revolutionarily replacing the inadequate very fundamentals of classical fundamental mathematics via adequate very fundamentals. From the beginning of 2001, the further editions [Gelimson 2001a, 2001b, 2001c, 2001d, 2001h] of these and many other scientific publications are available on the Internet. Giovanni Giuseppe Nicosia placed in his Doctor Thesis [2001] in February, 2001, a reference to [Gelimson 2001d], as well as in [Nicosia 2005]. Scientist Vuara has published [Gelimson 2001h] along with the references of some academicians and professors to Lev Gelimson's monographs, articles, and other scientific publications in [Wikibooks Hyperanalysis-MeasurementTheory]. [Gelimson 2003a, 2003b, 2003d, 2003e, 2003f, 2004a] further developed his own mathematics then named "elastic mathematics".

0.5. Classical Fundamental Mathematics Revolutions Usefulness

First of all, it is necessary to show essential differences of revolutionary fundamental mathematics from classical mathematics [1]. Moreover, these differences have to take place namely in the fundamentals of revolutionary fundamental mathematics versus classical fundamental mathematics. The reason is that it is NOT a matter of any particular theory. Revolutionary fundamental mathematics has to be entire science based on its own principles and to include a whole synergistic system of many concepts, theories, and methods.

Further, revolutionary fundamental mathematics has to give many new possibilities (as compared to those in classical fundamental mathematics) to consider and solve new classes of problems (including the simulation of objects) very important for real life and science but NOT considerable and NOT solvable by classical mathematics. Or, at least, revolutionary mathematics has to do this essentially better than classical mathematics does.

Conclusions

1. Revolutionary fundamental mathematics should both have a fundamental difference from classicalal fundamental mathematics and has to give many new possibilities (as compared to those in classicala fundamental mathematics) to consider and solve new classes of problems (including the simulation of objects) very important for real life and science but NOT considerable and NOT solvable by classical mathematics. Or, at least, revolutionary fundamental mathematics has to do this essentially better than classical fundamental mathematics does.

2. Lev Gelimson [1995a, 1995b] created his own mathematics named "basic new mathematics" with revolutionarily replacing the inadequate very fundamentals of classical fundamental mathematics via adequate very fundamentals. From the beginning of 2001, the further editions [Gelimson 2001a, 2001b, 2001c, 2001d, 2001h] of these and many other scientific publications are available on the Internet. Giovanni Giuseppe Nicosia placed in his Doctor Thesis [2001] in February, 2001, a reference to [Gelimson 2001d], as well as in [Nicosia 2005]. Scientist Vuara has published [Gelimson 2001h] along with the references of some academicians and professors to Lev Gelimson's monographs, articles, and other scientific publications in [Wikibooks Hyperanalysis-MeasurementTheory]. [Gelimson 2003a, 2003b, 2003d, 2003e, 2003f, 2004a] further developed his own mathematics then named "elastic mathematics".

0.6. Fundamental Mega-Overmathematics Revolutions Directions

0.6.1. Exclusively Constructive Creative Philosophy Principles

0.6.1.1. Uniphilosophy (Exclusively Constructive Creative Philosophy) Principles as a System of Revolutions in Philosophy

Fundamental principles of uniphilosophy (exclusively constructive creative philosophy) build a fundamental system of revolutions in philosophy, in particular, the following subsystems.

0.6.1.1.1. Fundamental Principles of Uniphilosophy as a Fundamental Subsystem of Revolutions in Philosophy

The fundamental subsystem of revolutions in philosophy includes the following fundamental principles of exclusively constructive creative philosophy:

1. Exceptional natural constructivism (with the complete absence of artificial destructiveness).

2. Free efficient creativity (exclusively practically purposeful, verified, and efficient unlimitedly free creativity, intuition, and phantasy flight).

3. Scientific optimism and duty (each urgent problem can and must be solved adequately and efficiently enough).

4. Complication utilization (creating, considering, and efficiently utilizing only necessary and useful also contradictory objects and models, as well as difficulties, problems, and other complications).

5. Symbolic feasibility (at least symbolic existence of all the necessary and useful even contradictory objects and models).

0.6.1.1.2. Advanced Principles of Uniphilosophy as an Advanced Subsystem of Revolutions in Philosophy

The advanced subsystem of revolutions in philosophy includes the following advanced principles of exclusively constructive creative philosophy.

1. Exclusively efficient intuitive evidence and provability (reasonable fuzziness, intuitive ideas without axiomatic rigor if necessary and useful).

2. Unrestrictedly flexible constructivism (if necessary even creating new knowledge (concepts, approaches, methods, theories, doctrines, and even sciences) to adequately set, consider, and solve urgent problems).

3. Tolerable simplicity (choosing the best in the not evidently unacceptable simplest).

4. Perfect sensitivity, or conservation laws universality (no uncompensated change in a general object conserves its universal measures).

5. Exact discrimination of noncoinciding objects and models (possibly infinitely or overinfinitely large with infinitesimal or overinfinitesimal distinctions and differences).

6. Separate similar (proportional) limiting universalizability (the reduction of objects, systems, and their models to their own similar (proportional) limits as units).

7. Collective coherent reflectivity, definability, modelablity, expressibility, evaluability, determinability, estimability, approximability, comparability, solvability, and decisionability (in particular, in truly multidimensional and multicriterial systems of the expert definition, modeling, expression, evaluation, determination, estimation, approximation, and comparison of objects, systems, and models qualities which are disproportionate and hence incommensurable and not directly comparable, as well as in truly multidimensional and multicriterial decision-making systems).

0.6.1.2. Some Other Principles of Uniphilosophy

Among other principles of uniphilosophy are the following:

1. Truth priority (primacy of practically verified purely scientific truths and criteria prior to commonly accepted dogmas, views, agreements, and authority, with all due respect to them).

2. Peaceful pluralism (with peaceful development of scientific and life diversity).

3. Efficient creative inheritance (efficiently using, analyzing, estimating, and developing already available knowledge and information).

4. Efficient constructive freedom (unrestrictedly free exclusively constructive and useful self-determination and activity, in particular, in knowledge and information research, creation, and development).

5. Fundamentality priority (primacy of conceptual and methodological fundamentals).

6. Knowledge efficiency (only useful quality (acceptability, adequacy, depth, accuracy, etc.) and amount (volume, completeness, etc.) of knowledge, information, data, as well as creation, analysis, synthesis, verification, testing, structuring, systematization, hierarchization, generalization, universalization, modeling, measurement, evaluation, estimation, utilization, improvement, and development of objects, models, knowledge, information, and data along with intelligent management and self-management of activity).

7. Mutual definability and generalizability (relating successive generalization of concepts in definitions with optional linear sequence in knowledge construction).

8. Efficient unificability of opposites only conditionally distinguished (such as real/potential, real/ideal, specific/abstract, exact/inexact, definitively/possibly, pure/applied, theory/experiment/practice, nature/life/science, for example, the generally inaccurate includes the accurate as the limiting particular case with the zero error).

9. Partial laws sufficiency (if there are no known more general laws).

10. Focus on discoveries and inventions (dualistic unity and harmony of academic quality and originality, discovering phenomena of essence, inventive climbing, helpful knowledge bridges, creative multilingualism, scientific art, anti-envy, learnability, teachability, and terminology development).

0.6.2. Principles of Unimathematics as a System of Revolutions in the Principles of Mathematics

The principles of exclusively constructive creative unimathematics (mega-overmathematics) constitute a system of scientific revolutions in the principles of mathematics including the following subsystems.

0.6.2.1. Fundamental Principles of Unimathematics as a Fundamental Subsystem of Revolutions in the Principles of Mathematics

The fundamental subsystem of revolutions in the principles of mathematics includes the following principles of unimathematics:

1. Typical urgent problems priority and exclusiveness (adequately setting and solving and efficiently using urgent problems only with completely avoiding unnecessary considerations is the only criterion of the necessity and usefulness of creating and developing new knowledge including concepts, approaches, methods, theories, doctrines, and sciences).

2. Intuitive conceptual and methodological fundamentality priority (creating and efficiently using unified knowledge foundation due to fundamental general systems including objects, models, and intuitive fuzzy principles, concepts, and methodology).

3. Collective coherent reflectivity, definability, modelablity, expressibility, evaluability, determinability, estimability, approximability, comparability, solvability, and decisionability (in particular, in constructing nonlinear conceptual systems of knowledge and in truly multidimensional and multicriterial systems of the expert definition, modeling, expression, evaluation, determination, estimation, approximation, and comparison of objects, systems, and models qualities which are disproportionate and hence incommensurable and not directly comparable, as well as in truly multidimensional and multicriterial decision-making systems).

4. Reasonable fuzziness with useful rigor only (exclusively practically useful axiomatization, deductivity, and rigorously proving, as well as intuitive ideas without axiomatic strictness if necessary and useful).

5. Unrestrictedly flexible constructivism (even creating new sciences to adequately set, consider, and solve typical urgent problems).

0.6.2.2. Noncontradictoriness Principles of Unimathematics as a Noncontradictoriness Subsystem of Revolutions in the Principles of Mathematics

The noncontradictoriness subsystem of revolutions in principles of mathematics includes the following principles of unimathematics:

1. The unificability of membership, inclusion, and part-whole relations.

2. Necessary and useful creativity exclusiveness (efficiently and intelligently creating and considering exclusively necessary and useful objects and models with completely ignoring any artificial contradictions typical in classical mathematics).

3. The efficient utilizability of contradictoriness and other complications (creating, considering, and efficiently utilizing exclusively necessary and useful contradictory objects and models, as well as difficulties, problems, and other complications).

4. Symbolic feasibility (at least symbolic existence of all the necessary and useful even contradictory objects and models).

5. Decision-making delayability (if necessary and useful, e.g. by estimating existence and sense with a possible further revaluation in the course of review).

0.6.2.3. Universalizability Principles of Unimathematics as a Universalization Subsystem of Revolutions in the Principles of Mathematics

The universalizability subsystem of revolutions in principles of mathematics includes the following principles of unimathematics:

1. Infinite cardinals canonizability (infinite cardinal numbers as canonical positive infinities namely real but not potential).

2. Zeroes reciprocals overinfinities canonizability (signed zeroes reciprocals as canonical overinfinities namely real but not potential).

3. Hyper-Archimedean axiomability (naturally generalizing the Archimedes axiom to the infinite and the overinfinite).

4. Exactness of the infinite and the overinfinite (perfectly sensitive, invariant, and universal infinite and overinfinite, infinitesimal and overinfinitesimal generalization of the numbers by the uninumbers with exact measurement generalizing counting, unlimited (possibly even noninteger and uncountable) manipulation and operability, as well as exact discrimination in the infinite and the overinfinite even by infinitesimal and overinfinitesimal distinctions and differences).

5. General (nonlogical) quantificability (assignment, definition, determination, and measurement of the individual quantity of a element becoming a quantielement and of the individual quantities of elements in a set which becomes a quantiset).

6. Separate similar (proportional) limiting universalizability (the reduction of objects, systems, and their models to their own similar (proportional) limits as units, in particular, of magnitudes to the moduli of their own unidirectional limits with the same signs).

7. Perfect manipulability (perfectly sensitive, invariant, and universal useful modeling, expression, evaluation, counting measurement, estimation, and essential generalization of urgent objects, relations, structures, systems, and their contents extending sets and quantisets).

8. Conservation laws universalizability (in the overinfinitesimal, the infinitesimal, the finite, the infinite, and the overinfinite).

0.6.2.4. Efficiency Principles of Unimathematics as an Efficiency Subsystem of Revolutions in the Principles of Mathematics

The efficiency subsystem of revolutions in principles of mathematics includes the following principles of unimathematics:

1. Uniproblem unisolvability (existence and expressibility of the best quasisolution, solution, and supersolution among possibly inexact meaningful pseudosolutions to any urgent uniproblem with setting as a unisystem with unknown unisubsystems).

2. Tolerable simplicity (selecting the best in the class of not evidently unacceptable simplest meaningful pseudosolutions).

3. Efficient knowledge (efficient quality (acceptability, adequacy, profundity, exactness, structurality, systematization, inheritance, universality, invariance, strength, stability, reliability, flexibility, etc.) and quantity (volume, completeness, etc.) of objects, models, knowledge, information, data, and their perfectly sensitive creation, analysis, synthesis, verification, testing, structuring, systematization, hierarchization, generalization, universalization, modeling, evaluation, measurement, estimation, utilization, improvement, development, and reasonable control).

4. Free intuitive intelligent iterativity (coherent, or sequential, approximativity) (possibly with many sources and directions, unrestrictedly flexible universal algorithms with avoiding computer zeroes and infinities and independent of analytic solvability with providing mapping contractivity).

5. Collective coherent reflectivity, definability, modelablity, expressibility, evaluability, determinability, estimability, approximability, comparability, solvability, and decisionability (in particular, in truly multidimensional and multicriterial systems of the expert definition, modeling, expression, evaluation, determination, estimation, approximation, and comparison of objects, systems, and models qualities which are disproportionate and hence incommensurable and not directly comparable, as well as in truly multidimensional and multicriterial decision-making systems).

6. General noncriticality (subcritical, critical, and supercritical states, processes, and phenomena in a general structured system which are defined and determined by generally noncritical relationships).

7. General nonlimitability (underlimiting, limiting, and overlimiting states, processes, and phenomena in a general structured system which are defined and determined by generally nonlimiting relationships).

Each new alternative mathematics can be regarded as an external (with respect to this alternative mathematics) revolution in mathematics (as a whole) becoming megamathematics. In alternative mathematics itself, creating its own very fundamentals completely new in comparison with the bases of ordinary mathematics can be regarded as an internal (in relation to this alternative mathematics) revolution in mathematics (as a whole). Even in such a case, ordinary mathematics itself can continue its evolution independently of these both and any other revolutions.

Mega-overmathematics (unimathematics) as a whole is a system which consists of infinitely many separate overmathematics that differ from one another by the sets of canonical infinities (signed infinite cardinal numbers) and canonical overinfinities (signed zeroes reciprocals) included into the real numbers, as well as by the nature of such inclusion, of the use of these infinities and overinfinities (in particular, by choosing canonical sets whose uniquantities embody these canonical infinities, and of the features and of the use of operations on infinities and overinfinities).

Along with such natural division of mega-overmathematics (unimathematics) as a whole by its common origin, nature, and essence into infinitely many separate overmathematics, mega-overmathematics (unimathematics) as a whole can be also conditionally divided into its following separate parts accordingly to their characters and roles:

1) fundamental unimathematics;

2) advanced unimathematics;

3) applied unimathematics;

4) computational unimathematics.

The principles of fundamental unimathematics build the universalizability system of revolutions in the principles of mathematics and include:

1. Infinite cardinals canonizability (infinite cardinal numbers as canonical positive infinities).

2. Zeroes reciprocals overinfinities canonizability (signed zeroes reciprocals as canonical overinfinities).

3. Hyper-Archimedean axiomability (a natural generalization of the Archimedes axiom to the infinite and the overinfinite).

4. Exactness of the infinite and the overinfinite (perfectly sensitive, invariant, and universal infinite and overinfinite, infinitesimal and overinfinitesimal generalization of the numbers by the uninumbers with exact measurement generalizing counting, unlimited (possibly even noninteger and uncountable) manipulation and operability, as well as exact discrimination in the infinite and the overinfinite even by infinitesimal and overinfinitesimal distinctions and differences).

5. General (nonlogical) quantificability (assignment, definition, determination, and unimeasurement of the individual quantities of separate elements becoming quantielements and of the elements of a set which becomes a quantiset).

6. Separate similar (proportional) limiting universalizability (the reduction of objects, systems, and their models to their own similar (proportional) limits as units, in particular, of magnitudes to the moduli of their own unidirectional limits with the same signs).

7. Perfect manipulability (perfectly sensitive, invariant, and universal useful modeling, expression, evaluation, counting measurement, estimation, and essential generalization of urgent objects, relations, structures, systems, and their contents extending sets and quantisets).

8. Conservation laws universalizability (in the overinfinitesimal, the infinitesimal, the finite, the infinite, and the overinfinite).

Fundamental unimathematics includes uniarithmetics, quantialgebra (quantitative algebra) and quantianalysis (quantitative analysis) as the unimathematics basis, namely:

1) fundamental science on uninumbers truly universal in the overinfinite, infinite, finite, infinitesimal, and overinfinitesimal, as well as in any sums combined of such terms with conservation laws universality. For the first time, the uninumbers perfectly subtly model the entire universes of the infinite and the first invented and discovered overinfinite, discover their mysteries, exactly express and discriminate infinities or overinfinities even with any infinitesimal or overinfinitesimal differences and distinctions, in particular, provide namely a positive probability of any possible event (with interpreting probability distributions in the non-Euclidean Lobachevskian hyperbolic geometry);

2) fundamental sciences on

quantielements, or elements with their quantities,

quantisets (quantitative sets operable like numbers) whose elements quantities can be arbitrary objects (this is a profound generalization of Cantor's set theory as the foundation of modern standard mathematics) including overinfinities, infinities, infinitesimals, and overinfinitesimals without absorption, conservation laws universally holding (Bolzano's dream previously impossible),

quantioperations,

quantirelations,

quantiaggregates,

quantistructures,

quantisystems,

quantistates,

quantiprocesses,

quantilaws,

all of them being based the on such operations of general (nonlogical) quantification as quantity definition, determination, and assignment;

3) fundamental sciences on

unioperations further generalizing quantioperations and applicable both to noninteger numbers of operands and even to uncountable sets of operands,

uniquantities as truly universal, invariant, and perfectly sensitive measures with conservation laws universality (while Cantor's cardinal numbers are finitely sensitive to the finite and extremely insensitive to the infinite and the overinfinite, and any known measure is finitely sensitive within a particular dimension only, so that there is no reasonable, applicable, suitable, and adequate measure of any set of mixed dimensions; in addition, there is absorption, so that conservation laws are violated).

0.6.3. Mega-Overmathematics Quantisystem, Its Quantisubsystems, Quanticomponents, and Quanticontent (Quanti-Aggregate)

General objects and systems (wholes) including their subsystems (parts) and components (elements) unify and generalize the approaches both of:

1) mereology which treats the whole-part meronomic relation and reasoning by Plato, Aristotle, Edmund Husserl [1901] and Stanisław Leśniewski [1916];

2) set theory by Georg Cantor [1932] both the element-set membership relation and the set inclusion relation and has been generalized by Lev Gelimson [1995a] who first introduced general (nonlogical) quantification and quantity determination operations, unified the membership, inclusion, and part-whole relations with providing also uncountable operations, and introduced general quantisystems in which an element with its individual quantity builds a quantielement.

Quantisystems and their quantisubsystems generalize systems (wholes) and subsystems (parts) as mereology objects.

The general quanticontents (quanti-aggregates) of general quantisystems further generalize quantisets in which the quantity of each element may be any general object, e.g. a general quantisystem. An element with its individual quantity builds a quantielement.

Quanticomponents further generalize both quantielements and parts.

General quantisystems can any general objects, models, considerations, approaches, principles, algorithms (see e.g. Lev Tsvik [1975]), methods, theories, sciences, their application, development, correction, generalization, education, etc. (see e.g. Lev Tsvik [1975, 1978, 1995, 2001, 2002]) and have any abstraction level.

Examples:

the universe, the sun, etc.;

a deformable solid, e.g. the earth, a building, etc.;

computer and Internet hardware and/or software;

number coding, cryptography, etc.;

the finite element system for a deformable solid;

the finite element method application;

the finite element method;

individual and collective psychologies, as well as psychological science;

language, literature, art, music, sport, etc.;

a person, communication, society, science, industry, finance, politics, ideology, methodology, etc.

Lev Gelimson [1995a] introduced systematically placing the individual quantity q of an element a as its left subscript in a quantielement _qa . In particular, this individual quantity q can also be a quantielement _rb . Note that power exponents are commonly placed as right superscripts and can be multiple. Multilevel placing multiple subscripts and superscripts brings many typesetting difficulties and misunderstanding, especially by text transformation via software including browsers.

Mega-overmathematics by Lev Gelimson [1987-2012] proposed suitable notation providing single-level placing multiple subscripts and superscripts with naturally using backslash \ and double slash // (not to mistake slash / for division sign) and using parentheses to show the correct sequence of operations.

Examples:

_r\ba is a quantielement whose element is a and whose individual quantity is a quantielement _rb ;

_r\b\ac is a quantielement whose element is c and whose individual quantity is a quantielement _r\ba ;

raising a base b to a power with a multiple power exponent c^d gives b^c\d ;

raising a base a to a power with a multiple power exponent b^c\d gives a^b\c\d ;

a variable a with a multiple index m_n as the right superscript is a_m//n ;

a variable a with a multiple index m_n//p as the right subscript is a_m//n//p ;

a variable a with a multiple index m_n as the left subscript is a_m//n ;

a variable a with a multiple index m_n//p as the left superscript is ^m//n//pa ;

raising a base b to a power with an indexed power exponent m_n gives b^m//n ;

raising a base a to a power with a multiply indexed power exponent m_n//p gives a^m//n//p ;

raising a base b to a power with a multiply indexed power exponent m_n//p and taking the obtained power with an individual quantity which is a quantielement _r\ba gives _r\b\a(b^m//n//p).

0.6.4. Mega-Overmathematics Operations

0.6.4.1. Sign-Conserving Multiplication

In classical mathematics (Nicolas Bourbaki [1949], Arnold Kaufmann, Maurice Denis-Papin, Robert Faure [1964], G. A. Korn, T. M. Korn [1968], I. N. Bronstein, K. A. Semendjajew [1989], [Encyclopaedia of Mathematics 1988]), usual multiplication is well-known. If all the factors to be multiplied are positive, then their product is positive, too, which is natural. If some factors to be multiplied are positive whereas the remaining factors to be multiplied are negative, then their product is positive when the number of negative factors is even whereas their product is negative when the number of negative factors is odd. This seems to be rather artificial than natural. Its origin (source) is that in the (real or complex) numbers, multiplication of real numbers distributes over addition in the well-known concepts of a ring and a field which both are commutative.

Nota bene: Both in mathematical logic and in set theory, the both distributive laws (of multiplication over addition and of addition over multiplication) hold. In mathematical logic, namely in Boolean algebra, logical disjunction ∨ plays the role of addition whereas logical conjunction ∧ plays the role of multiplication, and for any sentences (propositions that may be true or false) A , B , and C , we have both

(A ∨ B) ∧ C = (A ∧ C) ∨ (B ∧ C)

and

(A ∧ B) ∨ C = (A ∨ C) ∧ (B ∨ C).

In set theory, namely in set algebra, unification ∪ plays the role of addition whereas intersection ∩ plays the role of multiplication, for any sets A , B , and C , we have both

(A ∪ B) ∩ C = (A ∩ C) ∪ (B ∩ C)

and

(A ∩ B) ∪ C = (A ∪ C) ∩ (B ∪ C).

In real or complex algebra, for any numbers A , B , and C , multiplication distributes over addition

(A + B)C = AC + BC

whereas addition does not distribute over multiplication because generally (as a rule)

AB + C ≠ (A + C)(B + C).

Therefore, already in the well-known concepts of a ring and a field which both are commutative, one of the both distributive laws (of multiplication over addition and of addition over multiplication) does not hold. Hence it is possible and at least equally natural to additionally consider completely nondistributive rings and fields along with well-known rings and fields which are partially nondistributive.

Nota bene: Common multiplication naturally leads to noncommutative common power and exponential functions well-defined by negative bases if and only if exponents are integer whereas sign-conserving multiplication naturally leads to commutative sign-conserving power and exponential functions well-defined by any real-number bases and exponents.

The fundamental ideas of sign-conserving multiplication of real numbers are as follows:

the modulus (absolute value) of the sign-conserving product (as a result of sign-conserving multiplication) of real numbers equals the modulus (absolute value) of the usual product of these numbers;

the value of the sign function of the sign-conserving product of real numbers vanishes if and only if the value of the sign function of the usual product of these numbers vanishes, i.e. if and only if at least one of these numbers vanishes;

the value of the sign function of the sign-conserving product of real numbers equals 1 if and only if all these numbers are positive;

the value of the sign function of the sign-conserving product of real numbers equals -1 if and only if at least one of these numbers is negative and none of these numbers vanishes.

To denote sign-conserving multiplication, simply use the parenthesis " either instead of a multiplication sign if it is implicit (i.e. omitted) or to the left of a multiplication sign (e.g. × , • , Π , etc.) if it is explicitly used.

Analytically, for any index set J , all indices j ∈ J , and any indexed real numbers building a set {a_j | j ∈ J}, their sign-conserving product

"Π_j∈J a_j = min(sign a_j | j ∈ J) |Π_j∈J a_j|

so that

|"Π_j∈J a_j| = |Π_j∈J a_j|

and

sign(Π_j∈J a_j|) = sign|Π_j∈J a_j| min(sign a_j | j ∈ J) = min(sign|a_j| | j ∈ J) min(sign a_j | j ∈ J).

Example: For any real numbers a , b , and c ,

a"b"c = a "× b "× c = a "• b "• c

= min(sign a , sign b , sign c) |abc|

so that

|a"b"c| = |abc|

and

sign(a"b"c) = sign|abc| min(sign a , sign b , sign c)

= min(sign|a|, sign|b|, sign|c|) min(sign a , sign b , sign c).

Nota bene: Introducing an additional factor, e.g.

sign|abc| = min(sign|a|, sign|b|, sign|c|),

is here necessary to provide

sign(a"b"c) = 0

if at least one of these numbers a , b , and c vanishes whereas then

a"b"c = 0

due to vanishing the factor |abc| which is absent in sign(a"b"c).

0.6.4.2. Negative Base Power Theory

If a < 0 and we want to consider the real numbers R only, then we may consider by m ∈ Z , n ∈ N

a^{(2m + 1)/(2n)} := a^{2(2m + 1)/(4n)}

giving real sense to a^b by any irrational b , too. Using modulus (absolute value) |a| gives the same results but is much less natural because

a ≠ |a|,

(2m + 1)/(2n) = 2(2m + 1)/(4n).

Mega-overmathematics by Lev Gelimson [1987-2011c] naturally introduces many further (also uncountable) quantioperations and quantirelations. Among them is sign-conserving power function

a"^b = |a|^b sign a

defined by any real numbers a ≠ 0 and b , as well as by a = 0 and any b > 0. Then we have, e.g.,

a"² = a² sign a ,

(-1)"³ = -1 = [(-1)"⁶]"^1/2 = (-1)"^6/2 ,

(-1)"^1/3 = -1 = [(-1)"²]"^1/6 = (-1)"^2/6 .

Nota bene: Fundamental, advanced, applied, and/or computational mathematical considerations also belong to fundamental, advanced, applied, and/or computational overmathematics, respectively, if and only if such considerations directly and explicitly use namely overmathematical very fundamentals revolutionarily replacing the inadequate very fundamentals of classical mathematics.

Examples:

1. Sign-conserving raising a specific nonnegative number to a real power belongs to mathematics but not to overmathematics because simply raising this specific nonnegative number to this real power in classical mathematics brings the same result. Hence replacing simply raising a specific nonnegative number to a real power with sign-conserving raising a specific nonnegative number to a real power brings nothing new. Therefore, raising a specific nonnegative number to a real power does not require overmathematical sign-conserving raising to a real power.

2. Sign-conserving raising a specific negative number to a real power belongs both to mathematics and to overmathematics because simply raising any negative number to any real power in classical mathematics is always ill-defined [Encyclopaedia of Mathematics 1988] because there are infinitely many irrational power exponents arbitrarily near to the given real power exponent so that for them, such a power is indefinite at all. All the more, for any even integer power exponent considered isolated, such a power is well-defined but brings the opposite result:

a^b = |a|^b ,

a"^b = |a|^b sign a = -|a|^b (a < 0).

For any odd integer power exponent, such a power (also considered isolated) is always ill-defined:

a^2z+1 = -|a|^2z+1 ≠ |a|^2z+1 = a^2(2z+1)/2 = [a^2(2z+1)]^1/2 (a < 0; z =0, ±1, ±2, ...).

Hence replacing simply raising any negative number to any real power with sign-conserving raising this negative number to this real power always brings a new result. Therefore, raising any negative number to any real power does require namely overmathematical sign-conserving raising to a real power.

3. Sign-conserving real power function as whole belongs both to mathematics and to overmathematics because simply raising any negative number to any real power in classical mathematics is always ill-defined [Encyclopaedia of Mathematics 1988] because there are infinitely many real power exponents arbitrarily near to the given real power exponent so that for them, such a power is indefinite at all. All the more, for any even integer power exponent, such a power is well-defined but brings the opposite result. For any odd integer power exponent, such a power is ill-defined. Hence replacing simply raising any negative number to any real power with sign-conserving raising this negative number to this real power always brings a new result. Therefore, raising any negative number to any real power does require namely overmathematical sign-conserving raising to a real power.

Notata bene:

1. For any polarly represented complex (also imaginary) power base

a = re^iφ

where r is a nonnegative number (modulus, or polar radius), unique polar argument φ belongs to half-opened segment [0, 2π[ (0 included but 2π excluded), and

i² = -1,

naturally generalize the sign function with direction function

dir a = e^iφ = cos φ + i sin φ

and the above sign-conserving real power function with direction-conserving complex-base real-exponent power function

a"^b = |a|^b dir a = r^b dir a

with a complex power base a and a real power exponent b .

2. For any polarly represented complex (also imaginary) power base

a = re^iφ

where r is a nonnegative number (modulus, or polar radius), unique polar argument φ belongs to half-opened segment [0, 2π[ (0 included but 2π excluded), and

i² = -1,

as well as for any complex (also imaginary) power exponent

b = c + di

where c and d are real numbers,

further naturally generalize the above direction-conserving complex-base real-exponent power function

a"^b = |a|^b dir a = r^b dir a

with direction-adding complex power function

a"^b = a"^c+di = |a|^c+di dir a = r^c+di e^iφ = r^cr^di e^iφ = r^ce^{id ln r} e^iφ = r^ce^{i(d ln r + φ)} .

Nota bene: Use " in a"^b if necessary only.

0.6.5. Mega-Overmathematics Zero Overinfinity Nature and Utilization

Mega-overmathematics by Lev Gelimson [1994c, 1995a] first explicitly directly expressed division by zero to further extend all the infinitesimal, finite, infinite, and combined pure (dimensionless) amounts and to conveniently operate on them with holding conservation law. Further zero 0 may be considered to be nonnumber which does not belong to the natural numbers N , to the integer numbers Z , to the real numbers R , to the complex numbers C , etc.

To avoid problems with the signs of the both signed zeros ±0, use their common modulus |±0| and introduce the positive canonic zero overinfinity

Φ = 1/|0| = 1/|±0|

and the negative canonic zero overinfinity

-Φ = -1/|0| = -1/|±0|.

Nota bene: In computer science, namely in the IEEE floating-point standard, positive zero +0 and negative zero -0 are used. But these designations are not suitable because parentheses are almost always needed. In mega-overmathematics by Lev Gelimson [1994c, 1995a], designations 0⁺ for the positive zero and 0^- for the negative zero are used. Now also introduce designations

Θ = +0 = 0⁺

for the positive zero and

-Θ = -0 = 0^-

for the negative zero.

Therefore,

Φ = 1/Θ ,

Θ = 1/Φ ,

ΘΦ = 1.

Nota bene: The positive canonic zero overinfinity Φ and its finite positive powers are greater than any cardinal number, nonstandard number, surreal number, hyperreal number, hyper-real number, etc. because their products with zero, the positive zero, and the negative zero vanish.

0.6.6. Mega-Overmathematics Uninumber Scale Finite Interpretation

Classical mathematics (Nicolas Bourbaki [1949], Arnold Kaufmann, Maurice Denis-Papin, Robert Faure [1964], G. A. Korn, T. M. Korn [1968], I. N. Bronstein, K. A. Semendjajew [1989], [Encyclopaedia of Mathematics 1988]) uses logarithmic and other nonhomogeneous scales. In nonstandard analysis by Abraham Robinson [1966], Keith Duncan Stroyan [1972, 1997] invented infinitesimal microscopes and infinite telescopes, Howard Jerome Keisler [1976a, 1976b], David Tall [1980b], and Lorenzo Magnani [2001] improved them. None of the known nonhomogeneous scales, infinitesimal microscopes, and infinite telescopes can provide a suitable finite interpretation of the hypernumbers including both infinities and infinitesimals. In particular, this holds for the uninumbers in mega-overmathematics.

0.6.6.1. Finite Scale Interpretation of the Uninumbers Including Zero, Quasizeroes, Infinitesimals, Nonzero Real Numbers, Infinities, and Overinfinities

There are many reasonable algorithms providing a suitable finite interpretation of the hypernumbers including zero, quasizeroes, infinitesimals, nonzero real numbers, infinities, and overinfinities, in particular, of the uninumber scale in mega-overmathematics, e.g. the following algorithm (Fig. 1) using interval (-5, 5) on the homogeneous s-axis:

Fig. 1. Finite Scale Interpretation of the Uninumbers Including Zero, Quasizeroes, Infinitesimals, Nonzero Real Numbers, Infinities, and Overinfinities

1) use interval (-5, -4) for negative zero overinfinities, place the canonic negative zero overinfinity x = -Φ at the interval midpoint s = -4.5 dividing this interval into two intervals (-5, -4.5) and (-4.5, -4) of equal lengths, place negative zero overinfinity x = -Φ² at the midpoint s = -4.75 of interval (-5, -4.5) and negative zero overinfinity x = -Φ^1/2 at the midpoint s = -4.25 of interval (-4.5, -4), etc., using logarithmic approach (a geometric progression of negative zero overinfinities x corresponds to an arithmetic progression in homogeneous coordinate values s);

2) use interval (-4, -3] for the opposites (additive inverses) -ω₀ , -ω₁ , -ω₂ , ... of infinite cardinal numbers (or their minimal ordinal numbers), place x = -ω₀ at s = -3, x = -ω₁ at the interval midpoint s = -3.5 dividing this interval into two intervals (-4, -3.5) and (-3.5, -3) of equal lengths, x = -ω₂ at the midpoint s = -3.75 of interval (-4, -3.5), etc., the left half-interval being divided into two intervals of equal lengths and the opposite (additive inverse) of the next infinite cardinal number (or its minimal ordinal number) being placed at the midpoint of the last divided interval;

3) use interval (-3, -2) for negative real numbers, place x = -1 at the interval midpoint s = -2.5 dividing this interval into two intervals (-3, -2.5) and (-2.5, -2) of equal lengths, place x = -2 at the midpoint s = -2.75 of interval (-3, -2.5) and x = -1/2 at the midpoint s = -2.25 of interval (-2.5, -2), etc., using logarithmic approach (a geometric progression of negative real numbers x corresponds to an arithmetic progression in homogeneous coordinate values s);

4) use interval [-2, -1) for the reciprocals (multiplicative inverses) -θ₀ = -1/ω₀ , -θ₁ = -1/ω₁ , -θ₂ = -1/ω₂ , ... of the opposites (additive inverses) -ω₀ , -ω₁ , -ω₂ , ... of infinite cardinal numbers (or their minimal ordinal numbers), place x = -θ₀ at s = -2, x = -θ₁ at the interval midpoint s = -1.5 dividing this interval into two intervals (-2, -1.5) and (-1.5, -1) of equal lengths, x = -θ₂ at the midpoint s = -1.25 of interval (-1.5, -1), etc., the right half-interval being divided into two intervals of equal lengths and the reciprocal (multiplicative inverse) of the opposite (additive inverse) of the next infinite cardinal number (or its minimal ordinal number) being placed at the midpoint of the last divided interval;

5) use interval (-1, 0) for negative quasizeroes, place the canonic negative quasizero x = -Θ at the interval midpoint s = -0.5 dividing this interval into two intervals (-1, -0.5) and (-0.5, 0) of equal lengths, place negative quasizero x = -Θ² at the midpoint s = -0.25 of interval (-0.5, 0) and negative quasizero x = -Θ^1/2 at the midpoint s = -0.75 of interval (-1, -0.5), etc., using logarithmic approach (a geometric progression of negative quasizeroes x corresponds to an arithmetic progression in homogeneous coordinate values s);

6) place x = 0 at s = 0;

7) use interval (0, 1) for positive quasizeroes, place the canonic positive quasizero x = Θ at the interval midpoint s = 0.5 dividing this interval into two intervals (0, 0.5) and (0.5, 1) of equal lengths, place positive quasizero x = Θ² at the midpoint s = 0.25 of interval (0, 0.5) and positive quasizero x = Θ^1/2 at the midpoint s = 0.75 of interval (0.5, 1), etc., using logarithmic approach (a geometric progression of positive quasizeroes x corresponds to an arithmetic progression in homogeneous coordinate values s);

8) use interval [1, 2) for the reciprocals (multiplicative inverses) θ₀ = 1/ω₀ , θ₁ = 1/ω₁ , θ₂ = 1/ω₂ , ... of infinite cardinal numbers (or their minimal ordinal numbers) ω₀ , ω₁ , ω₂ , ... , place x = θ₀ at s = 2, x = θ₁ at the interval midpoint s = 1.5 dividing this interval into two intervals (1, 1.5) and (1.5, 2) of equal lengths, x = θ₂ at the midpoint s = 1.25 of interval (1, 1.5), etc., the left half-interval being divided into two intervals of equal lengths and the reciprocal (multiplicative inverse) of the next infinite cardinal number (or its minimal ordinal number) being placed at the midpoint of the last divided interval;

9) use interval (2, 3) for positive real numbers, place x = 1 at the interval midpoint s = 2.5 dividing this interval into two intervals (2, 2.5) and (2.5, 3) of equal lengths, place x = 2 at the midpoint s = 2.75 of interval (2.5, 3) and x = 1/2 at the midpoint s = 2.25 of interval (2, 2.5), etc., using logarithmic approach (a geometric progression of positive real numbers x corresponds to an arithmetic progression in homogeneous coordinate values s);

10) use interval [3, 4) for the infinite cardinal numbers (or their minimal ordinal numbers) ω₀ , ω₁ , ω₂ , ... , place x = ω₀ at s = 3, x = ω₁ at the interval midpoint s = 3.5 dividing this interval into two intervals [3, 3.5) and (3.5, 4) of equal lengths, x = ω₂ at the midpoint s = 3.75 of interval (3.5, 4), etc., the right half-interval being divided into two intervals of equal lengths and the next infinite cardinal number (or its minimal ordinal number) being placed at the midpoint of the last divided interval;

11) use interval (4, 5) for positive zero overinfinities, place the canonic positive zero overinfinity x = Φ at the interval midpoint s = 4.5 dividing this interval into two intervals (4, 4.5) and (4.5, 5) of equal lengths, place positive zero overinfinity x = Φ² at the midpoint s = 4.75 of interval (4.5, 5) and positive zero overinfinity x = Φ^1/2 at the midpoint s = 4.25 of interval (4, 4.5), etc., using logarithmic approach (a geometric progression of positive zero overinfinities x corresponds to an arithmetic progression in homogeneous coordinate values s).

0.6.6.2. Finite Scale Interpretation of the Uninumbers Including Zero, Infinitesimals, Nonzero Real Numbers, and Infinities

There are many reasonable algorithms providing a suitable finite interpretation of the hypernumbers including zero, infinitesimals, nonzero real numbers, and infinities, in particular, of the uninumber scale in mega-overmathematics, e.g. the following algorithm (Fig. 2) using interval (-3, 3) on the homogeneous s-axis:

Fig. 2. Finite Scale Interpretation of the Uninumbers Including Zero, Infinitesimals, Nonzero Real Numbers, and Infinities

1) use interval (-3, -2] for the opposites (additive inverses) -ω₀ , -ω₁ , -ω₂ , ... of infinite cardinal numbers (or their minimal ordinal numbers), place x = -ω₀ at s = -2, x = -ω₁ at the interval midpoint s = -2.5 dividing this interval into two intervals (-3, -2.5) and (-2.5, -2) of equal lengths, x = -ω₂ at the midpoint s = -2.75 of interval (-3, -2.5), etc., the left half-interval being divided into two intervals of equal lengths and the opposite (additive inverse) of the next infinite cardinal number (or its minimal ordinal number) being placed at the midpoint of the last divided interval;

2) use interval (-2, -1) for negative real numbers, place x = -1 at the interval midpoint s = -1.5 dividing this interval into two intervals (-2, -1.5) and (-1.5, -1) of equal lengths, place x = -2 at the midpoint s = -1.75 of interval (-2, -1.5) and x = -1/2 at the midpoint s = -1.25 of interval (-1.5, -1), etc., using logarithmic approach (a geometric progression of negative real numbers x corresponds to an arithmetic progression in homogeneous coordinate values s);

3) use interval [-1, 0) for the reciprocals (multiplicative inverses) -θ₀ = -1/ω₀ , -θ₁ = -1/ω₁ , -θ₂ = -1/ω₂ , ... of the opposites (additive inverses) -ω₀ , -ω₁ , -ω₂ , ... of infinite cardinal numbers (or their minimal ordinal numbers), place x = -θ₀ at s = -1, x = -θ₁ at the interval midpoint s = -0.5 dividing this interval into two intervals (-1, -0.5) and (-0.5, 0) of equal lengths, x = -θ₂ at the midpoint s = -0.25 of interval (-0.5, 0), etc., the right half-interval being divided into two intervals of equal lengths and the reciprocal (multiplicative inverse) of the opposite (additive inverse) of the next infinite cardinal number (or its minimal ordinal number) being placed at the midpoint of the last divided interval;

4) place x = 0 at s = 0;

5) use interval [0, 1) for the reciprocals (multiplicative inverses) θ₀ = 1/ω₀ , θ₁ = 1/ω₁ , θ₂ = 1/ω₂ , ... of infinite cardinal numbers (or their minimal ordinal numbers) ω₀ , ω₁ , ω₂ , ... , place x = θ₀ at s = 1, x = θ₁ at the interval midpoint s = 0.5 dividing this interval into two intervals (0, 0.5) and (0.5, 1) of equal lengths, x = θ₂ at the midpoint s = 0.25 of interval (0, 0.5), etc., the left half-interval being divided into two intervals of equal lengths and the reciprocal (multiplicative inverse) of the next infinite cardinal number (or its minimal ordinal number) being placed at the midpoint of the last divided interval;

6) use interval (1, 2) for positive real numbers, place x = 1 at the interval midpoint s = 1.5 dividing this interval into two intervals (1, 1.5) and (1.5, 2) of equal lengths, place x = 2 at the midpoint s = 1.75 of interval (1.5, 2) and x = 1/2 at the midpoint s = 1.25 of interval (1, 1.5), etc., using logarithmic approach (a geometric progression of positive real numbers x corresponds to an arithmetic progression in homogeneous coordinate values s);

7) use interval [2, 3) for the infinite cardinal numbers (or their minimal ordinal numbers) ω₀ , ω₁ , ω₂ , ... , place x = ω₀ at s = 2, x = ω₁ at the interval midpoint s = 2.5 dividing this interval into two intervals [2, 2.5) and (2.5, 3) of equal lengths, x = ω₂ at the midpoint s = 2.75 of interval (2.5, 3), etc., the right half-interval being divided into two intervals of equal lengths and the next infinite cardinal number (or its minimal ordinal number) being placed at the midpoint of the last divided interval.

0.6.6.3. Finite Scale Interpretation of Zero and Nonzero Real Numbers

There are many reasonable algorithms providing a suitable finite interpretation of zero and nonzero real numbers, e.g. the following algorithm (Fig. 3) using interval (-1, 1) on the homogeneous s-axis:

Fig. 3. Finite Scale Interpretation of Zero and Nonzero Real Numbers

1) use interval (-1, 0) for negative real numbers, place x = -1 at the interval midpoint s = -0.5 dividing this interval into two intervals (-1, -0.5) and (-0.5, 0) of equal lengths, place x = -2 at the midpoint s = -0.75 of interval (-1, -0.5) and x = -1/2 at the midpoint s = -0.25 of interval (-0.5, 0), etc., using logarithmic approach (a geometric progression of negative real numbers x corresponds to an arithmetic progression in homogeneous coordinate values s);

2) place x = 0 at s = 0;

3) use interval (0, 1) for positive real numbers, place x = 1 at the interval midpoint s = 0.5 dividing this interval into two intervals (0, 0.5) and (0.5, 1) of equal lengths, place x = 2 at the midpoint s = 0.75 of interval (0.5, 1) and x = 1/2 at the midpoint s = 0.25 of interval (0, 0.5), etc., using logarithmic approach (a geometric progression of positive real numbers x corresponds to an arithmetic progression in homogeneous coordinate values s).

0.6.7. Mega-Overmathematics Canonic Sets

In mega-overmathematics, it is possible to anyways select distinct canonic sets whose quantities equal canonic infinite cardinal numbers (or their minimal ordinal numbers) ω₀ , ω₁ , ω₂ , ... . But it is reasonable that such a choice provides a suitable evaluation of infinities necessary and useful for solving a specific given problem.

For solving typical urgent problems, the first two canonic infinite cardinal numbers (or their minimal ordinal numbers) ω₀ and ω₁ . It is suitable to simply denote them with

ω = ω₀ ,

Ω = ω₁ .

To provide their set implementation so that the quantities of some chosen canonic sets equal canonic infinite cardinal numbers (or their minimal ordinal numbers) ω₀ and ω₁ , it is reasonable to select the natural numbers

N = {1, 2 , ... }

and unit interval

(0, 1],

respectively, so that

Q(N) = ω ,

Q(0, 1] = Ω .

0.6.8. Mega-Overmathematics Spectrum (Palette)

For solving a given urgent problem, it is possible to efficiently use many essentially distinct overmathematics specified by variants of considering zero as a source of overinfinities and introducing canonic infinite cardinal numbers (or their minimal ordinal numbers).

Hence there are many essentially distinct overmathematics, e.g.:

1) finite overmathematics which avoids using quasizeroes, infinitesimals, infinities, and overinfinities;

2) ω-overmathematics which uses the first canonic infinite cardinal number (or its minimal ordinal number) ω = ω₀ , avoids using any quasizeroes and overinfinities, other infinities and the corresponding infinitesimals, and is suitable for solving typical urgent problems with countable sets;

3) Ω-overmathematics which uses the second canonic infinite cardinal number (or its minimal ordinal number) Ω = ω₁ , avoids using any quasizeroes and overinfinities, other infinities and the corresponding infinitesimals, and is suitable for solving typical urgent problems with bounded continual sets;

4) Φ-overmathematics which uses quasizeroes and overinfinities, avoids using infinitesimals and infinities, and is suitable for solving typical urgent problems with division by zero;

5) ω-Ω-overmathematics which uses the first two canonic infinite cardinal numbers (or their minimal ordinal numbers) ω = ω₁ and Ω = ω₁ , avoids using any quasizeroes and overinfinities, other infinities and the corresponding infinitesimals, and is suitable for solving typical urgent problems with unbounded continual sets;

6) ω-Φ-overmathematics which uses quasizeroes and overinfinities, the first canonic infinite cardinal number (or its minimal ordinal number) ω = ω₀ , avoids using any other infinities and the corresponding infinitesimals, and is suitable for solving typical urgent problems with countable sets and division by zero;

7) Ω-Φ-overmathematics which uses quasizeroes and overinfinities, the second canonic infinite cardinal number (or its minimal ordinal number) Ω = ω₁ , avoids using any other infinities and the corresponding infinitesimals, and is suitable for solving typical urgent problems with bounded continual sets and division by zero;

8) ω-Ω-Φ-overmathematics which uses quasizeroes and overinfinities, the first two canonic infinite cardinal numbers (or their minimal ordinal numbers) ω = ω₁ and Ω = ω₁ , avoids using any other infinities and the corresponding infinitesimals, and is suitable for solving typical urgent problems with unbounded continual sets and division by zero.

By the principle of efficient tolerable simplicity, if a given urgent problem cannot be adequately and efficiently setted, considered and solved using classical mathematics, then select in mega-overmathematics the possibly simplest sufficient overmathematics.

Part 1. Uniarithmetics, Quantialgebra,

and Quantianalysis of Uninumbers

Chapter 1.1. Uninumbers and Uninumber Systems

1.1.1. Uninumbers

To extend and refine the scale of the numbers, we shall supplement them with the infinite cardinal numbers by preserving the properties of the usual operations and relations.

Notation 1.1.1.1. For any number set S , an S-number is any element of S, an S-operation is an operation in S under which S is closed, an S-relation is a relation in S .

Examples 1.1.1.2. -3 is a Z-number [Z the integers], -2/11 a Q-number [Q the rational numbers];

+ , × , and -_≥ [subtracting a relatively not greater number] are N-operations;

= , > , and < [as well as their "unions" ≥ and ≤] are N- , Z- , Q- , and R-relations.

Notation 1.1.1.3. An S-number system is a number set S with all S-operations and S-relations.

Example 1.1.1.4. The rational-number algebraic system [linearly ordered field] is the set Q with the usual operations + , - , × , and / (or : ) [division], as well as with the usual relations = , > , and < (along with the "combined" relations ≥ and ≤).

Definition 1.1.1.5. An S-uninumber algebraic system is the smallest structure-preserving extension of an S-number system with including all the infinite cardinal numbers. They are regarded as some symbols [letters] with certain number values for which the usual strict inequalities

0 < 1 < 2 < ... < ω < ω₁ < ω₂ < ...

between the finite and infinite ordinal numbers identified with the corresponding cardinal numbers for which

0 < 1 < 2 < ... < ℵ₀ < ℵ₁ < ℵ₂ < ...

are valid. All the properties of the S-operations and of the S-relations, except the Archimedean property of multiplication by [finite] S-numbers, [formally] hold for any terms possibly including infinite cardinal numbers.

Remark 1.1.1.6. For S = Q , the field and order properties hold. Relations valid in set theory but formally contradicting the properties of an S-number system do not hold in an S-uninumber system. For example, relations

ℵ₀ = ℵ₀ + 1 = 2ℵ₀ = nℵ₀ = (ℵ₀)² = (ℵ₀)ⁿ (n ∈ N),

∁² ≤ ∁ ,

or, by another notation regarding the finite and infinite ordinal numbers namely as the corresponding cardinal numbers,

ω = ω + 1 = 2ω = nω = ω² = ωⁿ (n ∈ N),

Ω² ≤ Ω

(where

Ω = ω₁ = ℵ₁ = ∁),

respectively, hold in classical mathematics but are false in the natural-uninumber system [monoid] N , the uniinteger system [integral domain] Z , the rational-uninumber system Q , and the real-uninumber system R . In our constructing a new sensitive degree of quantity by using the known objects [the numbers and infinite cardinal numbers] only, we have to change some familiar relations between them. Every measure also gives other results than the cardinality does. Each other relation between terms possibly including infinite cardinal numbers [which satisfies Definition 1.1.1.5] holds in an S-uninumber system, e.g.

(1 + ω)² = ω² + 2ω + 1

or the nonstrict inequality

Ω + 1 ≥ Ω

whose strict "subinequality"

Ω + 1 > Ω

satisfies the algebraic rules for the usual numbers and hence Definition 1.1.1.5 [in set theory, on the contrary,

Ω + 1 = Ω].

Each S-uninumber system is clearly non-Archimedean but hyper-Archimedean with respect to the four arithmetic operations by

Definitions 1.1.1.7.

1.1.1.7.1. A partially ordered set M with a partial binary operation

f : M₂ → M ,

where

M₂ ⊆ M × M ,

is called hyper-Archimedean with respect to f if for any a ∈ M such that there are nonidentical elements of M in the Cantorian set union

{f(m , a) | m∈M} ∪ {f(a , m) | m ∈ M}

and for any b ∈ M , there exists a c ∈ M for which

f(c , a) > b

or

f(a , c) > b .

1.1.1.7.2. A partially ordered set M is called hyper-Archimedean over a Cantorian set L with respect to a partial binary operation

g : L₁ × M₁ ∪ M₂ × L₂ → M ,

where

M₁ , M₂ ⊆ M ,

L₁ , L₂ ⊆ L ,

if for any a ∈ M with nonidentical elements of M in the Cantorian set union

{g(l , a) | l ∈ L₁} ∪ {g(a , l) | l ∈ L₂}

and for any b ∈ M , there exists an l ∈ L with

g(l , a) > b

or

g(a , l) > b .

1.1.1.7.3. A variable, mapping, or correspondence with range M is called hyper-Archimedean over a Cantorian set L with respect to g if for M the same holds.

Remark 1.1.1.8. Such a hyper-Archimedean extension of S to S depends on a choice of the S-operations [forming an S-uninumber set S] as well as of the S-relations and can be various.

Examples 1.1.1.9. The addition, subtraction, and multiplication of any numbers, division by finite numbers (or also including the limits of all the Cauchy sequences [1]) lead to the ordered commutative ring [no field] Q (or R, respectively). Such a uninumber is a sum of a uniinteger and a rational or real number, respectively, giving a [translational] number scale extension without condensation. The addition, subtraction, multiplication, and division of any numbers [without restriction] make the inclusion of these limits unnecessary and give the same ordered commutative field of all rational or, equivalently, real uninumbers, whose carrier is the set

Q° = R°

with condensation on any interval in comparison with Q and R.

Definition 1.1.1.10. An S-uninumber is an element from the set S being the carrier [support] of an S-uninumber system and the extension of S [see Definition 1.1.1.5].

Definition 1.1.1.11. An S-quantioperation is the extension of an S-operation from S to S in an S-uninumber system.

Definition 1.1.1.12. An S-quantirelation is the extension of an S-relation from S to S in an S-uninumber system.

Definition 1.1.1.13. An S-ultranumber is an S-uninumber that is no S-number. The set of all S-ultranumbers

S = S \ S .

Definition 1.1.1.14. An S-overnumber is an S-uninumber [namely an S-ultranumber] with modulus greater than that of any S-number. The set of all S-overnumbers is denoted by S .

Definition 1.1.1.15. An S-undernumber is an S-uninumber [0 or an S-ultranumber] with modulus less than that of any nonzero S-number. The set of all S-undernumbers is denoted by S.

Definition 1.1.1.16. An S-internumber is an S-ultranumber that is no S-overnumber and no S-undernumber. The set of all S-internumbers is denoted by S .

Corollary 1.1.1.17.

S = S \ S \ S

but

S ≠ S ∪ S ∪ S

[since 0 ∈ S and 0 ∉ S ].

Examples 1.1.1.18. Ω and other infinite cardinal numbers, their sums and products are N-, Z-, Q-, and R-overnumbers;

their reciprocals such as Ω^-1 [belonging to

Q° = R°]

are undernumbers;

2/Ω - 1, other sums of a nonzero S-undernumber and of a nonzero S-number are S-internumbers.

Examples 1.1.1.19. For

S ∈ {N , Z , Q , R} :

S = S = S \ S ,

S = {0} ,

S = ∅ .

1.1.2. Separated Uninumber Systems

Such considering complex [uni]numbers, [uni]number matrices, etc. is convenient.

Notation 1.1.2.1. A system is an object consisting of its elements between which (and possibly elements of other objects) there can be certain internal (external or mixed, respectively) relations and correspondences.

Notation 1.1.2.2. A separated system is a system for which, as a whole, equality relation is componentwise and whose elements are strictly separated and vary independently of each other.

Notation 1.1.2.3. A separated S-number (S-uninumber) system is a separated system of S-numbers (S-uninumbers, respectively).

Examples 1.1.2.4. A complex S-number (S-uninumber), an S-quaternion (S-uniquaternion), etc., and an S-number (S-uninumber) vector, sequence, and matrix are separated S-number (S-uninumber, respectively) systems;

a rational number and the decimal expansion of a real number are no separated number systems.

Definition 1.1.2.5. A separated S-uninumber system operation (relation) is a componentwise extension of a separated S-number system operation (relation, respectively) from the S-numbers to the S-uninumbers.

Examples 1.1.2.6. By representing a complex number

a + bi ∈ C

(the set of the complex numbers) in the form (a , b) ,

C = R × R ,

C = R × R ,

C = R × R ∪ R × R ,

C = R × R ∪ R × R ,

C = R × R ,

C = R × (R \ R) ∪ (R \ R) × R .

1.1.3. Equivalence Quantirelations of Uninumbers

Along with quantiequality quantirelation, we can introduce four additional equivalence quantirelations.

Definition 1.1.3.1. S-uninumbers a and b are called:

(0) equal (a = b) if and only if a - b = 0,

(1) close (a ≅ b) if and only if a - b is an S-undernumber,

(2) near (a ⇔ b) if and only if a = b = 0 or (a - b is no S-overnumber and (a - b)/(a + b) is an S-undernumber),

(3) similar (a ≈ b) if and only if a = b = 0 or (a - b)/(a + b) is an S-undernumber,

(4) equiorderal (a ∼ b) if and only if a = b = 0 or both a/b and b/a are no S-undernumbers.

Theorem 1.1.3.2. These five quantirelations are equivalence relations.

Proof. The reflexivity and symmetry of each of them are obvious, as well as the transitivity of (0), (1), and (4) always and of (2) and (3) for

a = b = 0.

It is clear that if a - b and b - c are no S-overnumbers, then a - c is no S-overnumber, since

a - c = (a - b) + (b - c) .

It remains to prove that if

(a - b)/(a + b)

and

(b - c)/(b + c)

are S-undernumbers, then

(a - c)/(a + c)

is also an S-undernumber. For any real number ε > 0, let us take the real number

δ = ε/(2 + 3ε) ∈ ]0, 1/3[.

We have

|a - b|/|a + b| < δ

and

|b - c|/|b + c| < δ

where b ≠ 0 (otherwise

a = b = c = 0

and the transitivity is obvious). We shall obtain

|a - c|/|a + c| < ε ,

since by putting

α = a - b ,

γ = c - b

we have

|a - c| = |α - γ| ≤ |α| + |γ| <

δ|2b + α| + δ|2b + γ| ≤

δ|2b| + δ|α| + δ|2b| + δ|γ| =

4δ|b| + δ|α| + δ|γ| <

ε(|2b| - |α| - |γ|) ≤

ε|2b + α + γ|.

The penultimate inequality can be proved in the following way:

δ = ε/(2 + 3ε),

2δ + 3εδ = ε,

2εδ + 2δ² = ε - 2δ - εδ + 2δ²,

4δ(ε + δ)/(1 - δ) = 2ε - 4δ,

(ε + δ)(|α| + |γ|) <

(ε + δ)(2|b| δ/(1 - δ) + 2|b|δ/(1 - δ)) = (2ε - 4δ)|b|

because

|α| < δ|2b + α| ≤ 2|b|δ + δ|α|

implies

|α| < 2|b|δ/(1 - δ)

and similarly

|γ| < 2|b|δ/(1 - δ),

4δ|b| + δ|α| + δ|γ| < ε (2|b| - |α| - |γ|).

Examples 1.1.3.3.

(ω - Ω)² = (Ω - ω)²,

Ω + 2/ω ≅ - Ω^-1/2 + Ω ,

ω + π ⇔ - (e - Ω^-1 - ω)

where e is the Euler number [1],

Ω + ω/2 ≈ Ω - 2ω + 1,

Ω/10 + ω² ∼ - πΩ + 2Ω^-3.

Corollary 1.1.3.4. Equal S-uninumbers are close, near, similar, and equiorderal. Near S-uninumbers are similar and equiorderal. The remaining pairs of these relations are independent of each other [e.g. Ω^-1 and Ω^-2 are close only].

Definition 1.1.3.5. The S-monad a_S of an S-uninumber a is the set of all the S-uninumbers b close to a. All such b with b - a > 0 form the positive submonad a⁺, with b - a < 0 the negative submonad a^- . A unireal monad is

a = a^- ∪ {a} ∪ a⁺.

Corollary 1.1.3.6.

a ∈ a_S ,

a ∈ a ,

a ∉ a^- ,

a ∉ a⁺ .

Chapter 1.2. Uninumber Bases and Representations

1.2.1. Uninumber Bases

A uninumber can be interpreted positionally and geometrically like a complex number.

Definition 1.2.1.1. An S-uninumber basis is a set of certain S-uninumbers including a nonzero S-number or an S-internumber and such that each ratio of its [distinct] elements is either an S-overnumber or an [of course, nonzero] S-undernumber.

Example 1.2.1.2.

{ω^αΩ^β | α , β ∈ R}

is a real-uninumber basis.

Definition 1.2.1.3. S-uninumber bases are called equal, close, near, similar, or equiorderal if there is their one-to-one mapping in which any preimage and its image are such.

Definition 1.2.1.4. The restriction of a set of certain S-uninumbers including a nonzero S-number or an S-internumber to an S-uninumber basis is leaving anyone [exactly one] element in every class of equiorderal S-uninumbers that are partitioned by this equivalence relation.

Definition 1.2.1.5. An S-uninumber of order 0 is an S-number.

Definition 1.2.1.6. An S-uninumber of order n + 1 (n ∈ N) over an S-uninumber basis B is a homogeneous linear quanticombination of B

a = ∑_j∈J a_j b_j

(b_j ∈ B ; J is any set)

whose coefficients a_j (j ∈ J) are S-uninumbers of orders not greater than n over B.

Corollary 1.2.1.7. For any n ∈ N , an S-uninumber of order n is that of order n + 1.

Definition 1.2.1.8. A common S-uninumber basis to certain S-uninumbers is an S-uninumber basis such that each of them is a homogeneous linear quanticombination of this basis with coefficients that are S-uninumbers of any orders.

Corollary 1.2.1.9. A restricted union of the individual bases to certain S-uninumbers is a common S-uninumber basis to them.

1.2.2. Uninumber Representations

There are different number systems (Solomon Feferman [1964]) including a number system with an irrational base, namely the golden ratio

a = (5^1/2 + 1)/2,

by George Bergman [1957] with generalizations by Alexey Stakhov [2009] and Sergey Abachiev [2012], as well as binomial enumeration theory by Alexey Borisenko [1991, 2004a, 2004b, 2009].

Known number representations in classical mathematics (Nicolas Bourbaki [1949], Arnold Kaufmann, Maurice Denis-Papin, Robert Faure [1964], G. A. Korn, T. M. Korn [1968], I. N. Bronstein, K. A. Semendjajew [1989], [Encyclopaedia of Mathematics 1988]) ignore infinities and infinitesimals, can be non-unique (e.g.

1 = 0.99...),

and satisfy no conservation law.

Indeterminate nonstandard numbers by Abraham Robinson [1966] have no explicit representation, even if nonstandard analysis can give new proofs.

Definition 1.2.2.1. The separation of a uninumber a is the quantioperation whose result (of the same name) is

a = }a{ + )a( = [a] + {a} + )a( = [a].{a}.)a(

where

}a{ is the noninfinitesimal (maybe infinite) part of a ,

[a] is the quantiintegral part of a , or [}a{] (using the entier function [1]),

{a} is the fractional part of a ,

{a} ∈ [0, 1[ ,

)a( is the infinitesimal part of a .

Examples 1.2.2.2. Using canonical infinities ω , Ω and the general neutralizator ⅁ which equals the empty element # :

a = ω - Ω² - Ω^-1 :

}a{ = [a] = ω - Ω²,

{a} = 0 ,

)a( = Ω^-1;

a = ω/b (b ∈ R):

}a{ = [a] = ω/b ,

{a} = 0 ,

)a( = ⅁ .

Definition 1.2.2.3. A nonfuzzy [m-ary (m ∈ N \ {1})] continued quantifraction is the quantilimit of the corresponding sequence if there is some explicit or latent law of possibly correcting the last digit/denominator taken into account in truncating.

Definition 1.2.2.4. A fuzzy [m-ary (m ∈ N \ {1})] continued quantifraction is the quantilimit of the corresponding sequence if there is no explicit or latent law of possibly correcting the last digit/denominator taken into account in truncating.

Notation 1.2.2.5.

ω.1_-1₊... = (ω.1, ω.12, …);

-Ω.1928... = (-Ω.2 , -Ω.19, -Ω.193, …).

Examples 1.2.2.6.

0.111..._- = (0.1, 0.11, 0.111, ...) =

(1/9-10^-n/9 | n ∈ N) = 1/9 - 10^-ω/9 < 1/9;

0.23999..._- = (0.2, 0.23, 0.239, …) = 0.24 - 10^-ω < 0.24 < 0.24₊ =

(0.3, 0.25, 0.241, …) = 0.24 + 10^-ω ;

0.11...

is a fuzzy number in

[1/9 - 10^-ω/9, 1/9 + 8/9 × 10^-ω]

(included into the quantimonade 1/9 of 1/9 and is asymmetric);

a_m- = sign a × [|a| × m^ω]/m^ω ;

a_m+ = sign a × [|a| × m^ω + 1]/m^ω ;

a_m is a fuzzy number in

_{sign a}[a_m- , a_m+] ⊆ a

(the unimonade of a);

by using the Fibonacci numbers

F₁ = 1, F₂ = 1, F_n+2 = F_n+1 + F_n (n ∈ N = {1, 2, …})

and the Binet formula

F_n = (αⁿ + (-1)ⁿ⁺¹/αⁿ)/5^1/2

already known by Abraham de Moivre [Encyclopaedia of Mathematics 1988]

where

α = (5^1/2 + 1)/2,

the continued fractions

Ω + 1/(1 + 1/(1 + …))_- =

(Ω + 1/(1 + 1), Ω + 1/(1 + 1/(1 + 1)), Ω + 1/(1 + 1/(1 + 1/(1 + 1))), …) =

(Ω + F₂/F₃ , Ω + F₃/F₄ , Ω + F₄/F₅ , … , Ω + F_n+1/F_n+2 , …) =

Ω + (α^ω+1 + (-1)^ω+2/α^ω+1) / (α^ω+2 + (-1)^ω+3/α^ω+2),

Ω + 1/(1 + 1/(1 + …))₊ =

(Ω + 1/1, Ω + 1/(1 + 1/1), Ω + 1/(1 + 1/(1 + 1/1)), …) =

(Ω + F₁/F₂ , Ω + F₂/F₃ , Ω + F₃/F₄ , … , Ω + F_n/F_n+1 , …) =

Ω + (α^ω + (-1)^ω+1/α^ω) / (α^ω+1 + (-1)^ω+2/α^ω+1).

Definition 1.2.2.7. The total harmonic expansion of a uninumber a is

a_ht = [a] + ∑_n∈N\{1} c_n/n

where

c_n = [n({a} - ∑_k=2^n-1 c_k/k)] ∈ {0, 1}

with using the entier function [Encyclopaedia of Mathematics 1988].

Definition 1.2.2.8. The selective harmonic expansion of a uninumber a is

a_hs = [a] + ∑_k∈N 1/n_k

where

n_k = ]1/{a} - ∑_l=1 ^k-1 1/n_l[

and

]x[ ::= -[-x]

with using the entier function [Encyclopaedia of Mathematics 1988].

Remark 1.2.2.9. In the general case, a_ht and a_hs are distinct as a nonordinary and an ordinary successive sets and do not coincide with a .

Examples 1.2.2.10.

(2/3)_ht = 1/2 + 0/3 + 0/4 + 0/5 + 1/6 =

2/3 = 1/2 + 1/6 = (2/3)_hs ;

e_hs = 2 + 1/2! + 1/3! + … ;

(√2)_hs = 1 + 1/3 + 1/13 + 1/253 + 1/218204 + … ;

((2ω + 3)/(5ω - 5))_hs = 1/3 + 1/5 + 1/(ω - 1) =

(2ω + 3)/(5ω - 5).

Definition 1.2.2.11. A common quantifraction is a uninumber expressible by a uniquotient of uniintegers.

Remark 1.2.2.12.

a_ht = a_hs = a

if and only if a is a common quantifraction.

Definition 1.2.2.13. The canonical expansion of a uninumber is the corresponding homogeneous linear quanticombination of an R-uninumber basis including unit and canonical infinities and infinitesimals.

Example 1.2.2.14.

(5ωΩ - Ω - 5ω³ + ω² + 2ω + 1)/(5ω - 1) =

Ω - ω² + 2/5 + 7/10 × ω^-1 + 7/50 × ω^-2 + 7/250 × ω^-3 + … .

Definition 1.2.2.15. The geometric representation of an R-uninumber

a = ∑_j∈J a_j b_j

as a possible element of a separated S-uninumber system over an R-uninumber basis

B = (b_j | j ∈ J)

is the unipoint

A = (a_j | j ∈ J)

in the frame of reference

B = (b_j | j ∈ J).

Example 1.2.2.16.

0.111…_-

is represented by the unipoint

(1/9, -1/9)

in the frame of reference

(1, 10^-ω).

Definition 1.2.2.17. The positional representation of an S-uninumber

a = ∑_j∈J a_j b_j

over a fixed positional S-uninumber basis

B = (b_j | j ∈ J)

is the positional set

A = (a_j | j ∈ J).

Example 1.2.2.18.

0.111…_- = 9^-1.-9^-1 || (1, 10^-ω).

Remark 1.2.2.19. The unireal a and uniimaginary b parts of a complex uninumber a + bi can be separately represented.

Part 2. Quantialgebra and Quantianalysis

of Quantielements and Quantisets

with Quantioperations and Quantirelations

Chapter 2.1. Quantioperations, Quantirelations, and Quantielements

2.1.1. Quantioperations and Quantirelations

Suppose that there are:

A) arbitrary (perhaps variable, inaccurate, uncertain, partial identical) objects

a , b , c , d , ... ,

q , r , s , t , ...

of any nature, which can be also regarded as indivisible elements (operands etc.),

B) the usual algebraic and set operations and relations with their designations and properties (Georg Cantor [1932], Nicolas Bourbaki [1949], Arnold Kaufmann, Maurice Denis-Papin, Robert Faure [1964], G. A. Korn, T. M. Korn [1968], I. N. Bronstein, K. A. Semendjajew [1989], [Encyclopaedia of Mathematics 1988]).

Definitions 2.1.1.1.

2.1.1.1.1. A quantioperation is an operation with precise consideration of the whole quantity of each element in the operands of this operation. A quantiresult is the result of a quantioperation with precise consideration of the whole quantity of each element in this result.

2.1.1.1.2. A quantirelation is a relation with precise consideration of the whole quantity of each element in the objects of this relation.

2.1.1.1.3. A quantioperation and its quantiresult can be designated via the signs of an operation similar to this quantioperation and of the result of this operation.

2.1.1.1.4. A quantirelation can be designated via the sign of a relation similar to this quantirelation.

2.1.1.1.5. Adding a small circle ° on the top right to the sign of a quantioperation, its quantiresult, or a quantirelation explicitly indicates precisely considering the whole quantity of each element in them. This circle is optional if the corresponding usual operation or relation gives the same result or holds, respectively.

Examples 2.1.1.2.

Quantiaddition and quantisum +° or ∑° ,

quantisummary { }° ,

quantisubtraction and quantidifference -° ,

quantimultiplication and quantiproduct ×° or Π° ,

quantidivision and quantiquotient /° ,

quantiunification and quantiunion ∪° ,

quantiset quantisubtraction and quantiset quantidifference \° ,

symmetric quantiset quantidifference Δ° ,

quantiintersecting and quantiintersection ∩° ,

quanticomplementing and quanticomplementation C° ;

quantiequality =° ,

proper (strict) ⊂°

and nonstrict ⊆° quantiinclusion.

Examples 2.1.1.3.

{1, 1}° ≠° {1}° = {1} ,

{2} +° {1, 2} =° {1, 2, 2}° ,

{1, 1, 2}° \° {1, 2} =° {1} ,

{1, 1, 2}° Δ° {1, 2, 2}° =° {1, 2} ,

{1, 1, 1}° ∩° {1, 1, 2}° =° {1, 1}° ,

{2, 2}° ⊄° {2, 3} .

2.1.2. Quantification, Quantity Determination, and Quantielements

Quantielements are unifications of equal objects considered as elements.

Definition 2.1.2.1. Quantielement is an object which has the form

_qa

(read: quantity q of a , or a with quantity q , or subscript q before a , or index q before a , or deep q of a) for any two objects a (the basis of this quantielement, or the only element of or in this quantielement) and q (the total quantity of a in _qa).

Examples 2.1.2.2.

_{2.5 kg}meat,

_loafbread,

_{-100 € + 60 $}money,

₁a =° {a}° = {a} ,

_na =° {a, a, ... , a}°

(n times,

n ∈ N = Z⁺ = {1, 2, 3, ...}).

Definition 2.1.2.3. Quantifyng, or quantification, is a quantioperation _q with a parameter q

_q : a → _qa .

Definition 2.1.2.4. Determining quantity, or quantity determination, is a quantioperation

Q : _qa → q ,

Q(a ∈ _qa) = q

where

Q(b ∈ _qa) = 0

if

b ≠ a .

Definition 2.1.2.5. The empty quantielement is the empty set ∅ and is to be reduced to the final certain canonical form ₀# with

Q(a ∈ ₀#) = 0

and

Q(# ∈ ₀#) = 0

when it is represented in its intermediate uncertain and ambiguous forms

₀a =° _q#

where

0 is zero concerning an addition of quantities,

# is the empty element,

# ∈ ∅ ,

a ≠ # ,

q ≠ 0 .

Remark 2.1.2.6. Numbers and multiplicities as positive integers

n ∈ N

are special cases of quantity. Along with such generalization, it is a short way of writing results. This is also suitable for operations with repeating operands, for functions of repeating variables, and for ordered sets with repeating successive elements.

Examples 2.1.2.7.

+(_qa) = qa ,

×(_qa) = a^q ,

f(_qx) ,

(1, -1, -1, 1, 1, 1, -1, -1, -1, -1, ...) =

(1, ₂-1, ₃1, ₄-1, ...) = _n(-1)ⁿ⁺¹,

n ∈ N .

Definition 2.1.2.8. Nonempty quantielemens _qa and _rb are called similar

(_qa ≅° _rb)

by

a = b

and quantiequal

(_qa =° _rb)

by

a = b

and

q = r .

The empty quantielement is considered to be similar to each quantielement and quantiequal to itself only.

Theorem 2.1.2.9. If the both equality relations separately in the bases and in the quantities of certain quantielements are reflexive, symmetric, or transitive, or are equivalence relations, then the same is valid for the quantirelations of similarity (exceptionally in no chain of the forms

_qa ≅° ₀# ≅° _rb

where

a ≠ b

and

q ≠ 0 ≠ r)

and of quantiequality of the quantielements.

Proof obviously follows from 2.1.2.8. The conditions are necessary because equality relations of any objects can also be irreflexive (by variable objects), nonsymmetric (by latent relations of affiliation, inclusion, generalization, concrete definition, simplification, representation, designation, condition, evaluation, etc.), nontransitive (e.g. by limited tolerance) and therefore no equivalence relations.

Chapter 2.2. Quantialgebra and Quantianalysis of Quantielements

2.2.1. Algebraic Quantiaddition and Algebraic Quantimultiplication

Definition 2.2.1.1. An algebraic quantisum as the result of algebraic quantiadding (quantiaddition of) similar quantielements is called the quantielement with the same basis whose quantity is a suitable algebraic sum of all the quantities in the operands:

... +° _qa -° ... +° ₀# +° ... -° _ra +° ... =° _{... + q - ... - r + ...}a .

Theorem 2.2.1.2. If the quantities q in some quantielements _qa with the same basis a build a (possibly commutative) additive group with zero 0 and inverse -q to q , then:

the quantielements build a (commutative, respectively) additive group with zero ₀# and inverse _-qa ,

the two groups are isomorphic with natural isomorphism

q ↔ _qa .

Proof. This evident isomorphism leads to the associative and possibly commutative laws, zero, and the inverse (see 2.2.1.1).

Definition 2.2.1.3. An algebraic quantiproduct as the result of algebraic quantimultiplying (quantimultiplication of) quantielements is the quantielement whose basis and quantity are the suitable algebraic products of all the bases and quantities in the operands, respectively:

... ×° _qa /° ... /° _rb ×° ... =° _{... × q / ... / r × ...} (... × a / ... / b × ...).

Definition 2.2.1.4. A powerof a quantielement is a quantielement whose basis and quantity are the same powers of the basis and the quantity in the operand, respectively.

Remark 2.2.1.5. If the exponent is a natural number (positive integer), then 2.2.1.4 is a special case of 2.2.1.3.

Theorem 2.2.1.6. If both the bases and the quantities in some quantielements build two (possibly commutative) multiplicative groups with the units u and 1 and the inverses u/a and 1/q to every basis a and quantity q , respectively, then:

the quantielements build a (commutative, respectively) multiplicative group with the unit ₁u and the inverse _1/qu/a to every quantielement _qa , and

the correspondence

(a , q) ↔ _qa

between these three groups is bijective und homomorphic.

Proof. The correspondence ist bijective due to 2.1.2.3 and 2.1.2.8, naturally conserves multiplicativity due to 2.2.1.3, and leads to the associative and possibly commutative laws, zero, and the inverse (see 2.2.1.3).

Definition 2.2.1.7. Generalized isomorphism is a bijection of the algebraic structures of two systems with naturally conserving their algebraic operations.

Theorem 2.2.1.8. The bijection 2.2.1.6 is a generalized isomorphism.

Proof follows from 2.2.1.6 and 2.2.1.7.

Theorem 2.2.1.9. If multiplication of the quantities in some similar quantielements is distributive regarding to their addition, then quantimultiplication of the quantielements is distributive regarding to their quantiaddition.

Proof.

(_qa +° _ra) ×° _sb =°

_q+ra ×° _sb =°

_(q+r)s(ab) =°

_{qs + rs}(ab) =°

_qs(ab) +° _rs(ab) =°

_qa ×° _sb +° _ra ×° _sb.

Therefore,

(_qa +° _ra) ×° _sb =° _qa ×° _sb +° _ra ×° _sb

is valid.

The proof of

_sb ×° (_qa +° _ra) =° _sb ×° _qa +° _sb ×° _ra

is similar to the previous one:

_sb ×° (_qa +° _ra) =°

_sb ×° _q+ra =°

_s(q+r)(ba) =°

_{sq + sr}(ba) =°

_sq(ba) +° _sr(ba) =°

_sb ×° _qa +° _sb ×° _ra .

Theorem 2.2.1.10. If the quantities s in some quantielements _su with the unit basis u build a (possibly commutative) field S, then the quantielements _su build a (possibly commutative) field S .

Proof. The quantielements build a commutative additive group and – without the zero ₀u – a (commutative) multiplicative group (see 2.2.1.2) with the distributive laws (see 2.2.1.9).

Theorem 2.2.1.11. If additionally to Theorem 2.2.1.10 the quantities in some quantielements _qa with the same basis a build a commutative ring containing the field of quantities s, then scalars _su and vectors _qa build a vector space.

Proof. The scalars form a field S because of 2.2.1.10, and the vectors build a commutative additive group V because of 2.2.1.2. For any

_qa , _ra ∈ V ,

_su , _tu ∈ S ,

we have due to 2.2.1.1, 2.2.1.3 and 2.2.1.6

_su ×° _qa =° _sqa ∈ V ,

_su ×° (_qa +° _ra) =° _su ×° _qa +° _su ×° _ra ,

(_su +° _tu) ×° _qa =° _su ×° _qa +° _tu ×° _qa ,

(_su ×° _tu) ×° _qa =° _su ×° (_tu ×° _qa) ,

₁u ×° _qa =° _qa .

2.2.2. Algebraic and Quasialgebraic Quantiunification,

Quantiintersection, and Symmetric Quantisubtraction

Definition 2.2.2.1. A quantiunion and quantiintersection of similar quantielements with ordered quantities are the quantielements with the same basis whose quantities are the least upper bound and the greatest lower bound of the quantities in the quantielements-operands, respectively:

... ∪° _qa ∪° ... ∪° _ra ∪° ... =° _{sup{... , q, ... , r, ...}}a ,

... ∩° _qa ∩° ... ∩° _ra ∩° ... =° _{inf {... , q, ... , r, ...}}a .

Theorem 2.2.2.2. Ouantiunification and quantiintersection of similar quantielements with ordered quantities are mutually distributive quantioperations.

Proof. Because of their commutativity, it is enough to prove the following two identities:

A) (_qa ∪° _ra) ∩° _sa =° (_qa ∩° _sa) ∪° (_ra ∩° _sa).

Due to commutativity, we may suppose the inequality

q ≥ r .

By q ≥ r ≥ s , we have

_qa ∩° _sa =° _sa ∪° _sa

and

_sa =° _sa .

By q ≥ s ≥ r , we have

_qa ∩° _sa =° _sa ∪° _ra

and

_sa =° _sa .

By s ≥ q ≥ r , we have

_qa ∩° _sa =° _qa ∪° _ra

and

_qa =° _qa ;

B) _qa ∪° (_ra ∩° _sa) =° (_qa ∪° _ra) ∩° (_qa ∪° _sa).

Due to commutativity, we may suppose the inequality

r ≥ s .

By q ≥ r ≥ s , we have

_qa ∪° _sa =° _qa ∩° _qa

and

_qa =° _qa .

By r ≥ q ≥ s , we have

_qa ∪° _sa =° _ra ∩° _qa

and

_qa =° _qa .

By r ≥ s ≥ q , we have

_qa ∪° _sa =° _ra ∩° _sa

and

_sa =° _sa .

Theorem 2.2.2.3. Quantielements _qa with the same basis build a distributive algebra (not always complementary and Boolean). If the set of the quantities has its greatest lower inf{q} and/or least upper bound sup{q}, then this algebra has zero

_inf{q}a

and/or the unit

_sup{q}a .

If and only if the set has the both bounds, then this zero and this unit only are mutually complementary.

Proof. The quantioperations ∪°, ∩° are commutative and associative. 2.2.2.1 and 2.2.2.2 lead to the fusion laws

_qa ∪° (_qa ∩° _ra) =° _qa ,

_qa ∩° (_qa ∪° _ra) =° _qa .

Hence we have an algebra. It is distributive because of 2.2.2.2. The complementarity laws

_qa ∪° _Cqa =° _sup{q}a ,

_qa ∩° _Cqa =° _inf{q}a

(Cq is the complement to q) lead to the system of the equalities

max{q, Cq} =° sup{q},

min{q, Cq} =° inf{q}.

Therefore,

C(sup{q}) =° inf{q},

C(inf{q}) =° sup{q}.

By

inf{q} < q < sup{q},

we should have

Cq =° inf{q},

Cq =° sup{q}.

This distributive algebra is not always complementary and Boolean.

Definition 2.2.2.4. An algebra is called extremely complementary if it contains zero, unit, and a complement to every element extreme by an order. If an extremely complementary algebra is distributive, then it is called extremely Boolean.

Definition 2.2.2.5. A quantielement is called extreme among some similar quantielements with ordered quantities if his quantity is extreme (either the least or the greatest) among their quantities by this order.

Theorem 2.2.2.6. The similar quantielements whose quantities form an ordered set with the least element and the greatest element build an extremely Boolean algebra.

Proof. See 2.2.2.3, 2.2.2.4 und 2.2.2.5.

Definitions 2.2.2.7.

2.2.2.7.1. An algebraic quantiunion of similar quantielements with ordered quantities

... ∪° _qa \° ... \° _ra ∪° ... ∪° _sa \°... \° _ta ∪° ...

is the quantielement

_{a∪ - a\}a

where

a∪ = sup{... , q, ... , s, ...},

a\ = sup{... , r, ... , t, ...}.

2.2.2.7.2. A quasialgebraic quantiunion of similar quantielements with ordered quantities

... ∪° _qa \ ... \ _ra ∪° ... ∪° _sa \ ... \ _ta ∪° ...

is the quantielement

_a∪\a ,

where

a∪\ = 3/4 (a∪ - a\) + 1/4 |a∪ - a\| -

1/4 (1 - sign(a∪ - a\))(|a∪| - |a\|).

Remark 2.2.2.8. The last artificial formula provides generalizing both set subtraction by Georg Cantor [1932] and quantiunification.

Definition 2.2.2.9. An operation or a quantioperation (e.g. algebraic addition, quantiaddition, unification, and quantiunification (also quasi-algebraic)) is called commutative and/or associative if it becomes these properties after converting negative signs of an operation or a quantioperation into negative signs of operands or quantioperands.

Theorem 2.2.2.10. Algebraic and quasialgebraic quantiunifications of similar quantielements with ordered quantities are commutative and associative quantioperations.

Proof is evident see because of 2.2.2.7 and 2.2.2.9.

Definition 2.2.2.11. Symmetric quantidifference

Δ°{... , _qa , ... , _ra , ...}°°

of similar quantielements with ordered quantities is considered as a nonreducible quantiset (see later, namely 2.3.1.1) and is defined as quantielement

_{sup{... , q , ... , r , ...} - inf{... , q , ... , r , ...}}a .

Theorem 2.2.2.12. Symmetric quantisubtraction of similar quantielements with ordered quantities is commutative and associative quantioperation whose results have nonnegative quantities.

Proof obviously follows from 2.2.2.11.

Theorem 2.2.2.13. Quantiintersecting is distributive quantioperation neither with regard to symmetric quantisubtraction, nor with regard to algebraic quantiunification, nor with regard to quasialgebraic quantiunification of similar quantielements with ordered quantities.

Proof. The following three counterexamples suffice:

(₃a Δ° ₂a) ∩° ₁a ≠° (₃a ∩° ₁a) Δ° (₂a ∩° ₁a),

(₃a \° ₂a) ∩° ₁a ≠° (₃a ∩° ₁a) \° (₂a ∩° ₁a),

(₃a \ ₂a) ∩° ₁a ≠° (₃a ∩° ₁a) \ (₂a ∩° ₁a)

because of

₁a =° ₁a ∩° ₁a ≠° ₁a Δ° ₁a =° ₁a \° ₁a =° ₁a \ ₁a =° ₀# .

Definition 2.2.2.14. A quantielement _qa is a subquantielement of a quantielement _rb

(_qa ⊆° _rb)

with

b = a

if

b = a = # ,

else if

q ≤ r ,

and is in particular a proper subquantielement of a quantielement rb

(_qa ⊂° _rb)

if additionally

_qa ≠° _rb .

Remark 3.35. However, symmetric quantidifferences, algebraic quantisums and (also quasialgebraic) quantiunions of nonsimilar quantielements are no quantielements because of more than one basis. Therefore, the concept of a quantielement needs a suitable generalization by the permanence principle [Encyclopaedia of Mathematics 1988] as an expression of the principle of tolerable simplicity in constructive philosophy and overmathematics.

Chapter 2.3. Quantialgebra and Quantianalysis of Quantisets

2.3.1. Integral Quantisets

These are probably the most general nonstructured summaries of divisible objects.

Definition 2.3.1.1. A quantiset is a quantielement or a nonpositional quantisummary and equally a nonpositional quantisum of quantielements of the form _qa , each of them consisting of its element [basis], say a , with its own, or individual, quantity, say q , inside in the quantiset, the elements and element quantities being any objects [possibly fuzzy (Lotfi Zadeh [1965], Dieter Klaua [1965, 1966a, 1966b, 1967]), etc.]:

A =° {... , _qa , ... , _rb , ... , _sc , ...}° =°

...+°_qa +°...+°_rb +°...+°_sc +°... .

Quantifying is a set quantioperation of the form

_q : a → _qa .

The empty quantielement of the forms

₀a =° _q#

(# the empty element, # ∈ ∅) is the empty set ∅ and has to be reduced to its canonical form ₀# .

Excepting nonreducible quantisets of the form (see their examples in 2.2.2.11)

A° =° {... , _qa , ... , _rb , ...}°°,

a quantiset is to be reduced [collected] via quantiadding all similar quantielements, their quantities being added:

... +°_qa +°... +° _ra +° ... +°_sa +°... =°

_{... + q + ... + r + ... + s + ...}a .

Quantisets are quantiequal if, after the reduction, they contain all quantielements in common.

The quantity of an element in a quantiset is the quantity of this element in the unique quantielement similar to this element in the quantiset already reduced:

Q(a ∈ A) = q.

If there is no quantielement similar to this element in the quantiset already reduced, then the quantity of such an element is zero:

Q(d ∈ A) = 0

(see 2.1.2.4, 2.1.2.5).

Unit quantity is optional:

₁a =° a =° {₁a}° =° {a}° = {a}.

The uniquantity of a quantiset is the sum of the quantities of all nonempty elements in this quantiset:

Q(A) = ... + q + ...+ r + ... .

The uniquantity is the result of the quantity definition and the quantity determination of a quantiset and is independent of reducing its nonempty elements.

Outsidely quantifying a quantiset means multiplying the inside quantities in the quantiset by the outside quantity:

_tA =° _t{... , _qa , ... , _rb , ... , _sc , ...}° =° {... , _tqa , ... , _trb , ... , _tsc , ...}°,

Q(_sA) = sQ(A).

The graph of a reduced quantiset A is the pair and, perhaps, the point quantiset

G(A) =° {... , (a, q), ... , (b, r), ...}°.

Examples 2.3.1.2.

{₀a, _q#}° =° ₀# =° ∅ ,

{_qa, _-ra, _s#}° =° _{q - r}a ,

{₂0 , _π# , _-10, _e1}° =° {₁0, _e1}° =° {0, _e1}°.

Definition 2.3.1.3. An ordinary set is a reduced quantiset with unit element quantities only.

Remark 2.3.1.4. Quantisets naturally generalize Cantorian sets whose special case arises by zero or unit quantities only.

Remark 2.3.1.5. The linear notation of the nonstructured quantisets is used as the most comfortable one only. The Euler-Wenn diagrams [Encyclopaedia of Mathematics 1988] are also possible.

Remark 2.3.1.6. Even by

... , q , ... , r , ... ∈ [0, 1],

quantisets generalize fuzzy sets (Lotfi Zadeh [1965], Dieter Klaua [1965, 1966a, 1966b, 1967]) because a quantiset can also be fully sure and determinate evn by such quantities so limited. The quantiset

{_1/2apple, _1/4pear}°

can also consist of exactly half an apple and a quarter pear.

Remark 2.3.1.7. Negative, fractional, irrational, and imaginary numbers, as well as numbers with measure units, naturally appear in quantielements and quantisets by subtraction (e.g., as loss, debt, delivery, sale, and expense), division (shares), root extraction, and measurement.

Examples 2.3.1.8.

_{10 €}money - _{15 €}money =° _{-5 €}money

(e.g., an account to be settled in the case of a zero personal account; therefore, the empty set ∅ is not the (nonexistent) "emptiest" quantiset but the neutral one only like zero that is the neutral number but not the (nonexistent) smallest one);

(₃4)^1/2 =° _√32;

(_-11)^1/2 =° _±i(±1)

where i is an imaginary unit;

a housewife purchase result might be

{_{2 loaves}bread, _{1.5 kg}meat, _{case + 2}water-melons,

_{0.1 liter}linseed-oil, _{2 packets}detergent, _{3 boxes}matches,

_{-58.74 € - $ 5}money, _{-2 h}time, _{-1.5 liter}petrol}°.

Remark 2.3.1.9. If necessary, different units in a quantity can be (exactly, approximately, conditionally, time-dependently, etc.) transformed into a common unit with corresponding factors (here, e.g., via

1 case ⇔ 8,

1 loaf ⇔ 0.75 kg.

Definition 2.3.1.10. Quantisets are quantiequal if they in the reduced forms contain the same quantielements.

Theorem 2.3.1.11. If both equality relations separately in the bases and the quantities of certain quantisets are reflexive, symmetric, or transitive, or are equivalence relations, then the same is valid for the quantirelation of quantiequality of the quantisets.

Proof obviously follows from 2.1.2.8, 2.1.2.9, and 2.3.1.10. The conditions are necessary because equality relations of any objects can also be irreflexive (by variable objects), nonsymmetric (by latent relations of affiliation, inclusion, generalization, concrete definition, simplification, representation, designation, condition, evaluation, etc.), nontransitive (e.g., by limited tolerance) and therefore no equivalence relations.

2.3.2. Quantialgebra of Integral Quantisets

Definition 2.3.2.1. An algebraic quantisummary (or quantisum) of quantisets is a quantiset as the quantiresult of quantiadding all quantielements of the quantisets-operands, each quantity in the subtrahends changing sign:

... +° {... , _qa , ...}° -° ... -° {... , _rb , ...}° +°

... +° {... , _sc , ...}° -° ... -° {... , _td , ...}° +°... =°

{... , _qa , ... , _-rb , ... , _sc , ... , _-td , ...}°.

Example 2.3.2.2.

{₃1, 0, _π2}° -° {_q3, _e1, _-i0}° +° {_ei, _-π2, _i1}° =°

{_{1+ i}0, _{3 - e + i}1, _-q3, _ei}°.

Definition 2.3.2.3. An algebraic addition or unification is called algebraically commutative and/or associative if it becomes commutative and/or associative provided that each negative operation sign is avoided by changing own signs of the corresponding operands, say:

- 3 - 5 + 2 - (-4) + (-1) = (-3) + (-5) + 2 + 4 + (-1),

_qa \° _rb ∪° _sc \° _td =° _qa ∪° _-rb ∪° _sc ∪° _-td .

Definition 2.3.2.4. A quantiproduct as the quantiresult of quantimultiplication of quantisets is the quantiset which is the corresponding quantisum of all the quantiproducts including precisely one quantielement from every quantiset-operand.

Example 2.3.2.5.

{₂1, _-12}° ×° {_-32, ₆4}° =°

{₂1 ×° _-32, _-12 ×° _-32, ₂1 ×° ₆4, _-12 ×° ₆4}° =°

{_-62, ₃4, ₁₂4, _-68}° =° {_-62, ₁₅4, _-68}°.

Definition 2.3.2.6. A quantiquotient as the quantiresult of quantidivision of a quantiset-dividend by a quantielement-divisor is the quantiset which is the corresponding quantisum of all the quantiquotients of every quantielement of the quantiset by the quantielement-divisor.

Example 2.3.2.7.

{_-4-2, ₆9}° /° _-8-6 =° {_1/2(1/3), _-3/4(-3/2)}°.

Theorem 2.3.2.8. By quantimultiplication and algebraic quantiaddition of quantisets, as well as by quantidivision of a quantiset by a quantielement, the relation between their uniquantities with using the corresponding operation is valid.

Proof follows see from 2.3.1.1, 2.3.2.1, 2.3.2.4, and 2.3.2.6.

Corollary 2.3.2.9. If the quantities of each basis in certain quantisets form a commutative additive group then the algebraic quantiunification of the quantisets is an algebraically commutative and associative set quantioperation, and the quantisets form a commutative additive group with zero ₀# .

Theorem 2.3.2.10. If the bases a and quantities q in some quantisets build a (commutative) multiplicative semigroup H and a commutative ring R , respectively, then the quantisets build a (commutative) ring R . If H has unit u and R has unit 1, then R is a ring with unit and can have zero divisors even if R is free of those.

Proof. Quantiaddition +° in R is (along with addition + in R) commutative and associative. R includes zero ₀# and additive inverse

-A =° {... , _-qa , ... , _-rb , ...}°

to

A =° {... , _qa , ... , _rb , ...}°.

Quantimultiplication ×° in R is (along with multiplications × in H und R) associative and eventually commutative. The distributivity laws in R are valid due to 2.2.1.9. Because of 2.3.2.4,

₁u

is unit. If, e.g., every product in H is an arbitrary constant, then in R

A ×° B =° ₀#

is always valid by

Q(A) × Q(B) = Q(A)Q(B) = 0,

even if

A ≠ ₀# ≠ B .

Theorem 2.3.2.11. If additionally to Theorem 2.3.2.10 the quantities in some quantielements (scalars) of the form

_su

build a (commutative) ring

S ⊆ R

and R is commutative, then these scalars and the quantisets-vectors build a (commutative) algebra.

Proof. R is a commutative additive group. Consider mapping

×° : S × R → R

with using 2.3.2.4. For any

_su , _tu ∈ S ,

A , B ∈ R ,

we have

_su ×° (A +° B) =° _su ×° A +° _su ×° B ,

(_su +° _tu) ×° A =° _su ×° A +° _tu ×° A ,

(_su ×° _tu) ×° A =° _su ×° (_tu ×° A),

₁u ×° A =° A

due to 2.2.1.2, 2.2.1.11, 2.3.2.1, and 2.3.2.4. Therefore,

(S , R)

is a commutative vector space. Quantimultiplication

×° : R × R → R

is due to 2.3.2.4 associative and with regard to quantiaddition distributive. Finally, associativity gives

_su ×° (A ×° B) =° (_su ×° A) ×° B =° A ×° (_su ×° B)

(because of the properties of unit and commutativity of quantimultiplication in R). Therefore,

(S , R)

is an algebra. If H and S are commutative, then this algebra is obviously commutative, too.

Definitions 2.3.2.12.

2.3.2.12.1. A quantiunion of some reduced quantisets with ordered quantities of each basis is the quantiset in which the quantity of every basis in the quantisets-operands is the least upper bound of the quantities of the same basis in the quantisets-operands:

... ∪° A ∪° ... ∪° B ∪° ... =°

{... , _{sup{... , Q(a ∈ A), ... , Q(a ∈ B), ...}}a , ...}°.

2.3.2.12.2. A quantiintersection of some reduced quantisets with ordered quantities of each basis is the quantiset in which the quantity of every basis in the quantisets-operands is the greatest lower bound of the quantities of the same basis in the quantisets-operands:

... ∩° A ∩° ... ∩° B ∩° ... =°

{... , _{inf{... , Q(a ∈ A), ... , Q(a ∈ B), …}}a , ...}°.

Remark 2.3.2.13. If basis a does not appear in quantiset B , then

Q(a ∈ B) = 0

because of

₀a =° ₀# .

Examples 2.3.2.14.

{_-10, ₂1}° ∪° {1, 2}° =° {₂1, 2}°,

{_-10, ₂1}° ∩° {1, 2}° =° {_-10, 1}°.

Remarks 2.3.2.15.

2.3.2.15.1. In contrast to 2.3.2.1, 2.3.2.12 is a direct generalization of Cantorian set unification and intersection for quantisets.

2.3.2.15.2. Ouantiunification and quantiintersection of reduced quantisets with basis-wise ordered quantities (in similar quantielements) are mutually distributive quantioperations (see Theorem 2.2.2.2).

Theorem 2.3.2.16. Reduced quantisets with basis-wise ordered quantities (in similar quantielements) build (together with quantiunification and quantiintersection) a distributive algebra (not always complementary and Boolean). If the set of the quantities of every basis a has its greatest lower

inf({q}, a)

and/or least upper bound

sup({q}, a),

then this algebra has zero

inf A =° {... , _{inf({q}, a)}a , ...}°

and/or unit

sup A =° {... , _{sup({q}, a)}a , ...}°.

If and only if the set the set of the quantities of every basis a has the both bounds, then this zero and this unit only are mutually complementary and there is a unique complement to every quantiset only in which every quantity is base-wise extreme. This distributive algebra is not always complementary and Boolean.

Proof. Quantiunification ∪° and quantiintersection ∩° are commutative and associative quantioperations because the quantity sets in 2.3.2.12 are nonordered. The fusion laws

A ∪° (A ∩° B) =° A,

A ∩° (A ∪° B) =° A

are valid for any quantisets A and B because of 2.3.2.12. Hence we have an algebra. The distributivity laws

(A ∪° B) ∩° C =° (A ∩° C) ∪° (B ∩° C),

A ∪° (B ∩° C) =° (A ∪° B) ∩° (A ∪° C)

hold for any quantisets A , B , and C because of 2.3.2.12 and 2.2.2.2.

For every quantiset of the form

A_extr =° {... , _{sup({q}, a)}a , ... , _{inf({q}, b)}b , ...}°

with exclusively base-wise extreme quantities only, there is the unique complement

C°(A_extr) =° {... , _{inf({q}, a)}a , ... , _{sup({q}, b)}b , ...}°

(see 2.2.2.3). Hence this distributive algebra is not always complementary and Boolean.

Definition 2.3.2.17. A quantiset is extreme among some reduced quantisets in which the quantities of every basis form an ordered set with the least element and the greatest element, if every quantity in this quantiset is either the least element or the greatest element among the quantities of the same basis in these quantisets.

Theorem 2.3.2.18. Some reduced quantisets, whose quantities of every basis form an ordered set with the least element and the greatest element, build an extremely Boolean algebra.

Proof. See 2.2.2.4, 2.3.2.16 and 2.3.2.17.

Definitions 2.3.2.19.

2.3.2.19.1. An algebraic quantiunion of quantisets with basis-wise ordered quantities

... ∪° A \° ... \° B ∪° ... ∪° C \° ... \° D ∪° ...

is the quantiset

{... , _{a∪ - a\}a , ...}°

where

a∪ = sup{... , Q(a ∈ A), ... , Q(a ∈ C), ...},

a\ = sup{... , Q(a ∈ B), ... , Q(a ∈ D), ...}.

2.3.2.19.2. A quasialgebraic quantiunion of quantisets with basis-wise ordered quantities

... ∪° A \ ... \ B ∪° ... ∪° C \ ... \ D ∪° ...

is the quantiset

{... , a∪\a , ...}°

where

a∪\ = 3/4 (a∪ - a\) + 1/4 |a∪ - a\| -

1/4 (1 - sign(a∪ - a\))(|a∪| - |a\|).

Remark 2.3.2.20. The last artificial formula provides generalizing both set subtraction by Georg Cantor [1932] and quantiunification.

Theorem 2.3.2.21. Algebraic and quasialgebraic quantiunifications of quantisets with basis-wise ordered quantities are commutative and associative quantioperations.

Proof is evident see because of 2.2.2.9, 2.2.2.0, and 2.3.2.19.

Definition 2.3.2.22. Symmetric quantidifference of quantisets with basis-wise ordered quantities is quantiset (see 2.3.2.12)

Δ°{... , A , ... , B , ...}°° =°

{... , _{sup{... , Q(a ∈ A), ... , Q(a ∈ B), ...} - inf{... , Q(a ∈ A), ... , Q(a ∈ B), ...}}a , ...}°.

Theorem 2.3.2.23. Symmetric quantisubtraction of quantisets with basis-wise ordered quantities is commutative and associative quantioperation whose results have nonnegative quantities.

Proof obviously follows from 2.3.2.22.

Theorem 2.3.2.24. Quantiintersecting is distributive quantioperation neither with regard to symmetric quantisubtraction, nor with regard to algebraic quantiunification, nor with regard to quasialgebraic quantiunification of quantisets with basis-wise ordered quantities.

Proof. The three counterexamples 2.2.2.13 suffice here, too:

(₃a Δ° ₂a) ∩° ₁a ≠° (₃a ∩° ₁a) Δ° (₂a ∩° ₁a),

(₃a \° ₂a) ∩° ₁a ≠° (₃a ∩° ₁a) \° (₂a ∩° ₁a),

(₃a \ ₂a) ∩° ₁a ≠° (₃a ∩° ₁a) \ (₂a ∩° ₁a)

because of

₁a =° ₁a ∩° ₁a ≠° ₁a Δ° ₁a =° ₁a \° ₁a =° ₁a \ ₁a =° ₀# .

Definition 2.3.2.25. A quantiset A is a subquantiset of a quantiset B

(A ⊆° B)

if all the quantities in the reduced quantidifference B \° A are nonnegative, and is a proper subquantiset of a quantiset B

(A ⊂° B)

if

A ≠° B .

Examples 2.3.2.26.

{_i1, ₂-1}° ⊂° {_e-1, _0.12, _1+i1}°,

_-11 ⊂° ∅ ⊆° ₀# ⊆° ∅ ⊂ ₂2.

Remark 2.3.2.27. Quantidivision of a quantiset-dividend by a quantiset-divisor which is no quantielement can lead to complications which can be avoided by means of the additional consideration of such quantiquotients by the permanence principle [Encyclopaedia of Mathematics 1988].

2.3.3. Fractional Quantisets and Their Quantialgebra

Fractional quantisets are quantiquotients of quantisets.

Definition 2.3.3.1. A fractional quantiset is quantiquotient

A /° B

of two reduced quantisets A and B with

Q(B) ≠ 0.

Definition 2.3.3.2. The uniquantity

Q(A /° B)

of a fractional quantiset

A /° B

with

Q(B) ≠ 0

is quantiquotient

Q(A /° B) = Q(A) / Q(B).

Any outside nonzero quantification of a fractional quantiset applies to its quantiset-dividend and quantiset-divisor separately:

_q(A /° B) =° (_qA /° _qB),

q ≠ 0.

Definition 2.3.3.3. Fractional quantisets A /° B and C /° D are quantiequal if

A ×° D =° B ×° C =° D ×° A =° C ×° B .

Theorem 2.3.3.4. If the uniquantities of quantisets A , B , C , and D are nonzero, then the quantiequalities

A /° B =° C /° D ,

C /° D =° A /° B ,

B /° A =° D /° C ,

D /° C =° B /° A .

are equivalent.

Proof follows from 2.3.3.3.

Theorem 2.3.3.5. If some quantisets build a commutative ring without divisors of zero, then in the quantiquotients of these quantisets, quantiequality is equivalence quantirelation.

Proof. The reflexivity and symmetry of such quantiequality obviously follow from 2.3.3.3. If

A /° B =° C /° D ,

C /° D =° E /° F ,

then we have

A ×° D ×° F =° B ×° C ×° F =° B ×° D ×° E ,

A ×° F =° B ×° E ,

A /° B =° E /° F

along with the transitivity. Therefore, such quantiequality is equivalence quantirelation.

Theorem 2.3.3.6. Quantiequal fractional quantisets have equal uniquantities.

Proof follows from 2.3.3.2, 2.3.3.3, and 2.3.2.8.

Remark 2.3.3.7. In principle, it is possible to represent fractional quantisets an integral (nonfractional) quantisets if necessary. Nevertheless, such representations often depend on the choice of a main nonzero quantielement in a quantiset-divisor, are complicated and not always suitable like the power series expansions of linear fractional functions.

Example 2.3.3.8.

{1} /° {₁1, _qa , _rb}° =° {1, _-qa, _-rb, _(-q)(-q)(aa),

_(-q)(-r)(ab), _(-r)(-q)(ba), _(-r)(-r)(bb), ...}

(₁1

is chosen as a main nonzero quantielement).

Theorem 2.3.3.9. Every quantiset whose bases have a unit u in common, is a fractional quantiset.

Proof. For any quantiset A, we can represent

A =° A /° {u}.

Definition 2.3.3.10. An algebraic quantisum and an algebraic quantiproduct as quantiresults of algebraic quantiaddition and algebraic quantimultiplication, respectively, of fractional quantisets are called fractional quantisets determined according to the basic rules of operations with common fractions in mathematics:

... +° ( A /° B ) -° ... -° (C /° D) +° ... +° (E /° F) +°... =°

(... +° ... ×° A ×°...×° D ×°...×° F ×°... -°

...×° B ×°...×° C ×°...×° F ×°... +°

...×° B ×°...×° D ×°...×° E ×°... +° ... ) /°

(...×° B ×°...×° D ×°...×° F ×°...),

... ×° (A /° B) /° ... /° (C /° D) ×° (E /° F) ×°... =°

(...×° A ×°...×° D ×°...×° E ×°...) /°

(... ×° B ×° ... ×° C ×° ... ×°F ×° ...).

Example 2.3.3.11.

({_-32, ₅-4}° /° {₂-1, _-32}°) -°

({₄-3, ₂6}° /° {_-1-4, _-310}°) =°

({_-32, ₅-4}° ×° {_-1-4, _-310}° -°

{₂-1, _-32}° ×° {₄-3, ₂6}°) /°

({₂-1, _-32}° ×° {_-1-4, _-310}°) =°

({_-32 ×° _-1-4, ₅-4 ×° _-1-4,

_-32 ×° _-310, ₅-4 ×° _-310}° -°

{₂-1 ×° ₄-3, _-32 ×° ₄-3,

₂-1 ×° ₂6, _-32 ×° ₂6 }°) /°

{₂-1 ×° _-1-4, _-32 ×° _-1-4,

₂-1 ×° _-310, _-32 ×° _-310}° =°

({₃-8, _-516, ₉20, _-15-40}° -°

{₈3, _-12-6, ₄-6, _-612}°) /°

{_-24, ₃-8, _-6-10, ₉20}° =°

{_-15-40, ₃-8, ₈-6, _-83, ₆12, _-516, ₉20}° /°

{_-6-10, ₃-8, _-24, ₉20}°.

Theorem 2.3.3.12. Quantirelation using algebraic quantiaddition and/or algebraic quantimultiplication of fractional quantisets implies corresponding quantirelation using algebraic addition and/or algebraic multiplication of the uniquanties of these fractional quantisets.

Proof follows from 2.3.3.2, 2.3.3.6, 2.3.3.10, and 2.3.2.8.

Example 2.3.3.13. If

(A /° B) /° (C /° D -° E /° F) =° G /° H ,

then

Q(A /° B) /° (Q(C /° D) -° Q(E /° F)) =° Q(G /° H).

Theorem 2.3.3.14. If some integral quantisets build a commutative ring, then their quantiquotients (fractional quantisets) build a commutative field.

Proof. Quantiaddition and quantimultiplication of fractional quantisets are commutative and associative quantioperations. Quantimultiplication of fractional quantisets is distributive with regard to quantiaddition of them. These both statements obviously follows from 2.3.3.3 and 2.3.3.10.

Zero is

₀# /° ₁1 =° ₀# .

The additive inverse to A /° B is

(-A) /° B =° - (A /° B) .

Unit is

₁1 /° ₁1 =° ₁1.

The multiplicative inverse to

A /° B

is

B /° A .

Theorem 2.3.3.15. If in Theorem 2.3.3.14 the quantities in some quantielements-scalars _su build a commutative field which is a subset of the set of the quantities in the quantisets, then the scalars and the fractional quantisets as vectors build a commutative algebra.

Proof is similar to that in 2.3.2.11.

Definition 2.3.3.16. Quantiunification, quantiintersecting, algebraic and quasialgebraic quantiunification, and symmetric quantisubtraction of fractional quantisets are the same quantioperations with the of these fractional quantisets in their forms with a common denominator (quantiset-divisor).

Example 2.3.3.17.

({_-32, ₅-4}° /° {₂-1, _-32}°) Δ°

({₄-3, ₂6}° /° {_-1-4, _-310}°) =°

({₃-8, _-516, ₉20, _-15-40}° Δ°

{₈3, _-12-6, ₄-6, _-612}°) /°

{_-24, ₃-8, _-6-10, ₉20}° =°

{₁₅-40, ₃-8, ₈-6, ₈3, ₆12, ₅16, ₉20}° /°

{_-6-10, ₃-8, _-24, ₉20}°

(see 2.3.3.11).

Theorem 2.3.3.18. Fractional quantisets with basis-wise ordered quantities build (with quantiunification and quantiintersecting) a distributive algebra.

Proof. Quantiunification and quantiintersecting are evidently commutative and associative quantioperations. The fusion laws and the distributive laws follow from 2.3.3.16 and 2.3.2.16.

Theorem 2.3.3.19. Algebraic and quasialgebraic quantiunification and symmetric quantisubtraction of fractional quantisets are commutative und associative quantioperations.

Proof follows from 2.3.3.16, 2.3.2.21, and 2.3.2.23.

Theorem 2.3.3.20. Quantiintersecting is distributive quantioperation neither with regard to symmetric quantisubtraction, nor with regard to algebraic quantiunification, nor with regard to quasialgebraic quantiunification of fractional quantisets.

Proof. The three counterexamples 2.2.2.13 are valid also here.

Remark 2.3.3.21. Sets with any quantities of their elements become objects of algebra, geometry, analysis, differential and integral calculus and equations, probability calculus and statistics, error calculation, number representation, mechanics, strength science, physics, etc. (Lev Gelimson [1987-2011c]).

Notation 2.3.3.22. For a one-variable function of a basis (a completely algebraically additive one-variable function of a quantity), quantifying the preimage means quantifying (multiplying) the image, say

f(_qx) = _qf(x)

(Q(_xa) = xQ(a),

respectively).

2.3.4. Quantiintervals and Semiquantiintervals

We shall consider such particular (special) quantisets as quantiintervals and directed quantiintervals of any (e.g. infinite and/or negative) length with any quantities of their bounds. To briefly write many results of the same type, we shall introduce the initial and final semiquantiintervals [combining them] and a general notation of their brackets.

Definition 2.3.4.1. An initial and a final semiquantiintervals are symbol combinations of the forms

)_qa T

and

T _rb( ,

respectively, where the bounds a and b and their quantities q and r belong to a uninumber set S , the brackets ) and ( are any [possibly the same] elements of the set

{ ] , | , [ },

and the extension T is an ordered subset of S .

Definition 2.3.4.2. A quantiinterval as the commutative quantiunion of an initial semiquantiinterval and a final one, which have a common extension, say

)_qa T _rb) =° )_qa T +° T _rb) =° T _rb) +° )_qa T ,

is a quantiset containing both the ordinary set

]a T b[ ⊆ T

of all intermediate uninumbers t ∈ T , for which either

a < t < b

or

a > t > b ,

and each bound with its own quantity multiplied by 0 [always for a , b ∉ T], 1/2, or 1 in accordance with the adjacent bracket [with it, a bound has to be preserved even if its quantity vanishes:

(₀1 T ≠° (₀2 T ≠° (₀# T ] :

]_qa T =° ]a T ,

|_qa T =° _q/2a +° ]a T if a ∈ T ,

[_qa T =° _qa +° ]a T if a ∈ T ,

T _rb[ =° T b[ ,

T _rb| =° T b[ +° _r/2b if b ∈ T ,

T _rb] =° T b[ +° _rb if b ∈ T .

Corollary 2.3.4.3. The rearrangement of the bounds with their quantities by preserving the relative orientation of each bracket with reference to the adjacent bound of a quantiinterval does not affect it:

)_qa T _rb) =° (_rb T _qa( ,

say

[_qa T _rb| =° |_rb T _qa] .

Definition 2.3.4.4. Open, some partially open [partially closed], in particular, half-open [half-closed], and closed quantiintervals of zero length are (for a ∈ T):

]a T a[ =° _-1a ,

|a T a[ =° ]a T a| =° _-1/2a ,

[a T a[ =° |a T a| =° ]a T a] =° ₀a =° ₀# =° ∅,

|a T a] =° [a T a| =° _1/2a ,

[a T a] =° ₁a =° {a}.

Definition 2.3.4.5. For any

a , b , q , r ∈ R°,

T ∈ {R, R°}:

(1) a real quantiinterval is

)_qa , _rb( =° )_qa R _rb( ,

say a symmetric half-open real quantiinterval

|a , b| =° _1/2a +° ]a, b[ +° _1/2b ;

(2) a quantireal quantiinterval is

)_qa ,° _rb( =° )_qa R° _rb( ,

say

)a ,° b( =° +°_{d∈ )a , b(}d ;

(3) the following quantireal semiquantiintervals with a ≤ b to be combined are

[a ,° =° {a} +° a⁺ +° ]a ,° ,

|a ,° =° |a ,° =° _1/2a +° a⁺ +° ]a ,° ,

]a ,° =° a⁺ +° ]a ,° ,

, °b] =° , °b[ +° b^- +° {b},

, ° b| =° , °b| =° , °b[ +° b^- +° _1/2b ,

, ° b[ =° , °b[ +° b^- .

Definition 2.3.4.6. A directed quantiinterval, say

)_qa T _rb)

[the italic brackets, a the origin, b the end, q and r the quantities of these bounds], is a quantiset that consists of the elements of its generating quantiinterval

)_qa T _rb)

[the relative orientation of each bracket with reference to the adjacent bound is preserved] with the same (opposite) quantities if a ≤ b (a > b, respectively):

)_qa T _rb) =° )_qa T _rb) if a ≤ b ,

)_qa T _rb) =° _-1)_qa T _rb) if a > b .

Corollary 2.3.4.7. The rearrangement of the distinct bounds with their own quantities by preserving the relative orientation of each bracket with reference to the adjacent bound of a directed quantiinterval implies its additive inversion:

)_qa T _rb) =° _-1(_rb T _qa( ,

say

_[qa T _rb/ =° _-1/_rb T _qa] .

Notation 2.3.4.8. The complete algebraic additivity of a quantiset correspondence, say f, formally applies to the semiquantiintervals, quantiintervals, and directed quantiintervals, say:

f (|a ,) = f(_1/2a) + f(]a ,) = _1/2f(a) + f(]a ,).

Part 3. Uniarithmetics, Quantialgebra,

and Quantianalysis of Uniquantities

Chapter 3.1. Uniquantity Axioms and Advantages

3.1.1. Uniquantity Axioms

To introduce a universal, completely sensitive, and even uncountably algebraically additive degree of quantity, we need some suitable concepts like sets, set operations, and set relations.

Definition 3.1.1.1. The uniquantity of a quantiset is the [non-positional and possibly uncountable] quantisum of the own [inside] quantities of all elements [bases] of the quantiset

Q(A) = Q{... , _qa, ... , _rb, ... , _sc, ...}° =

... + q + ... + r + ... + s + ...

extending a point [zero-dimensional] measure [number of elements] by satisfying the axioms:

(A1) a uniquantity is completely algebraically additive, commutative, and associative, say

Q(... +° A -° ...-° B +° ... +° C -° ...-° D +° ...) =

... + Q(A) - ... - Q(B) + ... + Q(C) - ... - Q(D) + ... ,

(A2) quantifying a quantiset implies multiplying its uniquantity:

Q(tA) = tQ(A) ,

(A3) the uniquantity of a Cartesian product of quantisets is the product of their uniquantities:

Q(... × A × ... × B × ...) = ... × Q(A) × ... × Q(B) × ... ,

(A4) norming any bases in a quantiset does not affect its uniquantity, say

Q{... , _qa , ... , _rb , ... , _sc , ...}° = Q{... , _q||a||, ... , _rb, ... , _s||c||, ...}° ,

(A5) if all mutual distances between the bases of a quantiset are bounded above in common and a mapping of the quantiset preserves both the distances [is an isometry] and the own quantities of the bases in the quantiset, the mapping preserves its uniquantity,

(A6) uniquantity determination [a set quantioperation] commutes with a passage to the limit,

(A7) the uniquantities of some chosen canonical sets coincide with their cardinalities, in particular:

Q{a} = 1

for any object a ;

Q(N) = Q(Z⁺) = Q{1, 2, 3, ...} = ω

(instead of ℵ₀ for simplicity of notation);

Q|0, 1| = Ω .

Remark 3.1.1.2. Definition 3.1.1.1 cannot be exhaustive (see even Axiom A7) similarly to that of a probability. Each specific definition of a uniquantity is to be in agreement with this.

3.1.2. Advantages of Uniquantity

in Comparison with Cardinality and Measure

Remark 3.1.2.1. For disjoint sets, the cardinality and each measure are only countably additive. In the general case, because of set absorption, they are not algebraically additive and even nonadditive. The uniquantities of the quantisets remove these restrictions.

Examples 3.1.2.2. Usually, equalities like

µ[a, b] = ∑_{d∈[a , b]} µ(d)

[µ a measure] are not considered.

2 = card ({1, 2}∪{1}) ≠ card {1, 2} + card {1} = 3,

Q({1, 2}+°{1}) = Q{₂1, 2} = 2 + 1= 3;

2 = card({1, 2} \ {3}) ≠ card{1, 2} - card{3} = 1,

Q({1, 2} -° {3}) = Q{1, 2, _-13} = 1 + 1 - 1 = 1;

1 = card({1, 2, 3, ...} \ {2, 3, ...}) ≠

card{1, 2, 3, ...} - card{2, 3, ...} = ℵ₀ - ℵ₀ = 0,

Q({1, 2, 3, ...} \ {2, 3, ...}) =

Q(N) - Q(N -° {1}) = ω - (ω - 1) = 1;

3 = µ([1, 3] ∪ [2, 4]) ≠ µ[1, 3] + µ[2, 4] = 4,

Q([1, 3] +° [2, 4]) =

Q([1, 2[ +° 2[2, 3] +° ]3, 4]) = Ω + 2(Ω + 1) + Ω =

Q[1, 3] + Q[2, 4] = (2Ω + 1) + (2Ω + 1);

1 = µ([1, 3] \ [2, 4]) ≠ µ[1, 3] - µ[2, 4] = 0,

Q([1, 3] -° [2, 4]) =

Q([1, 2[ -° ]3, 4]) = Ω - Ω = 0 =

Q[1, 3[ -° Q]2, 4] = 2Ω - 2Ω = 0 (see A5);

∁ = card[1, 2] = card[1, 2[ = card]1, 2] = card]1, 2[ = card[1, +∞[,

1 = µ[1, 2] = µ[1, 2[ = µ]1, 2] = µ]1, 2[,

Q[1, 2] = Ω + 1 ≠ Q[1, 2[ = Q]1, 2] = Ω ≠ Q]1, 2[ = Ω - 1.

Chapter 3.2. Uniquantity Definition and Determination

3.2.1. Uniquantities of Finite and Countable Sets

Theorem 3.2.1.1. The uniquantity of a finite ordinary set is the number of its elements.

Proof.

Q{a₁ , a₂ , … , a_n} = Q({a₁} + {a₂} + ... + {a_n}) =

Q{a₁} + Q{a₂} + ... + Q{a_n} = n

(n ∈ N ; see Axioms A1, A7).

Theorem 3.2.1.2. The uniquantity of the empty set is zero.

Proof.

Q(∅) = Q({a} \ {a}) = Q{a} - Q{a} = 0

(see Axioms A1, A7).

Theorem 3.2.1.3. For the sets of all natural numbers N , positive integers Z⁺ (Z⁺ = N), negative integers Z^- , all nonzero integers Z^± , and all integers Z ,

Q(N) = ω ,

Q(Z⁺) = ω ,

Q(Z^-) = ω .

Q(Z^±) = 2ω ,

Q(Z) = 2ω + 1.

Proof. By Axiom A7,

Q(N) = Q(Z⁺) = Q{1, 2, 3, ...} = ω .

Axiom A4 gives

Q(Z^-) = Q{-1, -2, -3, ...} =

Q{|-1|, |-2|, |-3|, ...} =

Q{1, 2, 3, ...} =

Q(Z⁺) = Q(N) = ω .

By Axiom A1,

Q(Z^±) = Q{..., -3, -2, -1, 1, 2, 3, ...} =

Q({..., -3, -2, -1} +° {1, 2, 3, ...}) =

Q{..., -3, -2, -1} + Q{1, 2, 3, ...} =

Q(Z^-) + Q(Z⁺) =

ω + ω = 2ω ,

Q(Z) = Q{..., -3, -2, -1, 0, 1, 2, 3, ...} =

Q({..., -3, -2, -1} +° {0} +° {1, 2, 3, ...}) =

Q{..., -3, -2, -1} + Q{0} + Q{1, 2, 3, ...} =

Q(Z^-) + Q{0} + Q(Z⁺) =

ω + 1 + ω = 2ω + 1.

Theorem 3.2.1.4. Adding any integer z ∈ Z to each number in N or Z⁺ implies subtracting this integer from the uniquantity with counting also zero 0 if it arises:

Q{z + 1, z + 2, ...} = Q(N) - z = Q(Z⁺) - z = ω - z .

Adding any integer z ∈ Z to each number in Z^- implies adding this integer to the uniquantity with counting also zero 0 if it arises:

Q{... , z - 3, z - 2, z - 1} = Q(Z^-) + z = ω + z .

Proof. For z = 0, see Theorem 3.2.1.3.

For z ∈ N = Z⁺ ,

Q{z + 1, z + 2, ...} = Q({1, 2, ...} \ {1, 2, ... , z}) = Q(N \ {1, 2, ... , z}) = ω - z ,

Q{... , z - 3, z - 2, z - 1} = Q({... , -3, -2, -1} +° {0, 1, 2, ... , z - 1}) = Q(Z^-) + z = ω + z .

For z ∈ Z^- ,

Q{z + 1, z + 2, ...} = Q({z + 1, ... , 0} +° N) = (- z) + ω = ω - z ,

Q{... , z - 3, z - 2, z - 1} = Q({... , -3, -2, -1} -° {- z , - z + 1, ... , -3, -2, -1}) = Q(Z^-) - (- z) = ω + z .

Remark 3.2.1.5. For noninteger z , Definition 3.1.1.1 is insufficient (see Remark 3.1.1.2).

Definition 3.2.1.6.

Q{a + bn | n ∈ N} = ω/|b| - a/b - 1/2 + 1/(2|b|)

where

a , b ∈ R° ,

b ≠ 0.

Remark 3.2.1.7. First we verify Axiom A1 for b ∈ N = Z⁺ with distributing the set

N = Z⁺ = {1, 2, ... , (b - 1), b , b + 1, ...}

of the natural numbers (positive integers) over b sets

{-(b - 1) + bn | n ∈ N} = {1, b + 1, 2b + 1, ...},

{-(b - 2) + bn | n ∈ N} = {2, b + 2, 2b + 2, ...},

.......................................................................................

{-1 + bn | n ∈ N} = {b - 1, 2b - 1, 3b - 1, ...},

{0 + bn | n ∈ N} = {b , 2b , 3b , ...}

for

a ∈ {-(b - 1), -(b - 2), ... , -1, 0},

respectively. We have

ω = Q{1, 2, ... , (b - 1), b , b + 1, ...} =

Q({1, b + 1, 2b + 1, ...} +° {2, b + 2, 2b + 2, ...} +° ... +° {b - 1, 2b - 1, 3b - 1, ...} +° {b , 2b , 3b , ...}) =

Q({-(b - 1) + bn | n ∈ N} +° {-(b - 2) + bn | n ∈ N} +° ... +° {-1 + bn | n ∈ N} +° {0 + bn | n ∈ N}) =

∑_a=-(b-1)⁰ Q{a + bn | n ∈ N} =

∑_a=-(b-1)⁰ [ω/|b| - a/b - 1/2 + 1/(2|b|)] =

bω/b + [(b - 1) + (b - 2) + ... + 1 + 0]/b - b/2 + b/(2b) =

ω + b(b - 1)/(2b) - b/2 + 1/2 = ω ,

quod erat demonstrandum.

Checking Axiom A4 for -(a + bn) gives:

Q{-(a + bn) | n ∈ N} = Q{(-a) + (-b)n) | n ∈ N} =

ω/|-b| - (-a)/(-b) - 1/2 + 1/(2|-b|) =

ω/|b| - a/b - 1/2 + 1/(2|b|) =

Q{a + bn | n ∈ N}.

Finally we verify Axiom A7:

Q(N) = Q(Z⁺) = Q{1, 2, 3, ...} =

Q{0 + 1 × n | n ∈ N} =

ω/|1| - 0/1 - 1/2 + 1/(2|1|) = ω .

Hence Definition 3.2.1.6 does not contradict Definition 3.1.1.1.

Corollary 3.2.1.8.

Q{a + bn | n ∈ Z^-} = Q{a + (-b)n | n ∈ Z⁺} = Q{a + (-b)n | n ∈ N} =

ω/|-b| - a/(-b) - 1/2 + 1/(2|-b|) =

ω/|b| + a/b - 1/2 + 1/(2|b|),

Q{a + bn | n ∈ Z^±} =

Q{a + bn | n ∈ Z^-} + Q{a + bn | n ∈ Z⁺} =

ω/|b| + a/b - 1/2 + 1/(2|b|) +

ω/|b| - a/b - 1/2 + 1/(2|b|) =

2ω/|b| - 1 + 1/(|b|),

Q{a + bn | n ∈ Z} =

Q{a + bn | n ∈ Z^-} + Q{a + bn | n = 0} + Q{a + bn | n ∈ Z⁺} =

ω/|b| + a/b - 1/2 + 1/(2|b|) +

1 +

ω/|b| - a/b - 1/2 + 1/(2|b|) =

2ω/|b| + 1/(|b|).

Theorem 3.2.1.4 is extended to all z ∈ R°:

Q{z + 1, z + 2, ...} = Q{z + 1 × n | n ∈ N} = ω/|1| - z/1 - 1/2 + 1/(2|1|) = ω - z ,

Q{... , z - 3, z - 2, z - 1} = Q{z + 1 × n | n ∈ Z^-} = ω/|1| + z/1 - 1/2 + 1/(2|1|) = ω + z

and to regarding any arithmetic progression with common difference b ∈ R°, b ≠ 0. Namely, adding any z ∈ R° to a (and hence to every term of this progression) implies (with counting also zero 0 if it arises):

subtracting z/|b| from the uniquantity for n ∈ Z⁺ = N ,

adding z/|b| from the uniquantity for n ∈ Z^- = N ,

conserving the uniquantity for n ∈ Z^± or n ∈ Z due to mutually compensating the influences of adding any z ∈ R° to a on Z⁺ and Z^- because

Z^± = Z^- +° Z⁺ ,

Z = Z^- +° {0} +° Z⁺ .

In particular:

for the sets of all odd natural numbers,

Q{1, 3, 5, ...} = Q{-1 + 2n | n ∈ N} = ω/|2| - (-1)/2 - 1/2 + 1/(2|2|) = ω/2 + 1/4;

for the sets of all even natural numbers,

Q{2, 4, 6, ...} = Q{0 + 2n | n ∈ N} = ω/|2| - 0/2 - 1/2 + 1/(2|2|) = ω/2 - 1/4;

for the sets of all odd natural numbers excluding 1,

Q{3, 5, 7, ...} = Q{1 + 2n | n ∈ N} = ω/|2| - 1/2 - 1/2 + 1/(2|2|) = ω/2 - 3/4;

for the sets of all even natural numbers excluding 2,

Q{4, 6, 8, ...} = Q{2 + 2n | n ∈ N} = ω/|2| - 2/2 - 1/2 + 1/(2|2|) = ω/2 - 5/4;

for the sets of all natural numbers giving remainder 1 if divided by 3,

Q{1, 4, 7, ...} = Q{-2 + 3n | n ∈ N} = ω/|3| - (-2)/3 - 1/2 + 1/(2|3|) = ω/3 + 1/3;

for the sets of all natural numbers giving remainder 2 if divided by 3,

Q{2, 5, 8, ...} = Q{-1 + 3n | n ∈ N} = ω/|3| - (-1)/3 - 1/2 + 1/(2|3|) = ω/3;

for the sets of all natural numbers divisible by 3,

Q{3, 6, 9, ...} = Q{0 + 3n | n ∈ N} = ω/|3| - 0/3 - 1/2 + 1/(2|3|) = ω/3 - 1/3;

for the sets of all natural numbers giving remainder 1 if divided by 3 excluding 1,

Q{4, 7, 10, ...} = Q{1 + 3n | n ∈ N} = ω/|3| - 1/3 - 1/2 + 1/(2|3|) = ω/3 - 2/3;

for the sets of all natural numbers giving remainder 2 if divided by 3 excluding 2,

Q{5, 8, 11, ...} = Q{2 + 3n | n ∈ N} = ω/|3| - 2/3 - 1/2 + 1/(2|3|) = ω/3 - 1;

for the sets of all natural numbers divisible by 3 excluding 3,

Q{6, 9, 12, ...} = Q{3 + 3n | n ∈ N} = ω/|3| - 3/3 - 1/2 + 1/(2|3|) = ω/3 - 4/3;

for the set H^- of all negative half-integers,

Q(H^-) = Q{-1/2, -3/2, -5/2, ...} = Q{1/2 + (-1)n | n ∈ N} = ω/|-1| - (1/2)/(-1) - 1/2 + 1/(2|-1|) = ω + 1/2;

for the set H⁺ of all positive half-integers,

Q(H⁺) = Q{1/2, 3/2, 5/2, ...} = Q{-1/2 + 1 × n | n ∈ N} = ω/|1| - (-1/2)/1 - 1/2 + 1/(2|1|) = ω + 1/2;

for the set H of all half-integers,

Q(H) = Q{... , -5/2, -3/2, -1/2, 1/2, 3/2, 5/2, ...} = Q(H^- +° H⁺) = ω + 1/2 + ω + 1/2 = 2ω + 1.

Remark 3.2.1.9. Definition 3.2.1.6

Q{a + bn | n ∈ N} = ω/|b| - a/b - 1/2 + 1/(2|b|),

seems to be the only natural linear definition due to expression

ω/|b| - a/b - 1/2 + 1/(2|b|).

The first term ω/|b| provides satisfying Axiom A7 and, due to using the absolute value, Axiom A4. Further this term together with the second term - a/b also satisfying Axiom A7 and Axiom A4 naturally makes sequence

Q{a_m + bn | n ∈ N} (m ∈ N)

in m an arithmetic progression provided that sequence

a_m (m ∈ N)

is an arithmetic progression. Adding - 1/2 + 1/(2|b|) is also necessary. Let us assume

Q{a + bn | n ∈ N} = ω/|b| - a/b + c

where c is a constant. Then for b ∈ Z⁺ = N ,

ω = Q{1, 2, ... , (b - 1), b , b + 1, ...} =

Q({1, b + 1, 2b + 1, ...} +° {2, b + 2, 2b + 2, ...} +° ... +° {b - 1, 2b - 1, 3b - 1, ...} +° {b , 2b , 3b , ...}) =

Q({-(b - 1) + bn | n ∈ N} +° {-(b - 2) + bn | n ∈ N} +° ... +° {-1 + bn | n ∈ N} +° {0 + bn | n ∈ N}) =

∑_a=-(b-1)⁰ Q{a + bn | n ∈ N} =

∑_a=-(b-1)⁰ [ω/|b| - a/b + c] =

bω/b + [(b - 1) + (b - 2) + ... + 1 + 0]/b + bc =

ω + b(b - 1)/(2b) + bc = ω - 1/2 + b/2 + bc

which has to equal ω . Then we obtain

- 1/2 + b/2 + bc = 0,

c = - 1/2 + 1/(2b).

Similarly, for b ∈ Z^- , we obtain

c = - 1/2 + 1/[2(-b)].

Unifying the last two formulae via using the absolute value, we finally obtain

c = - 1/2 + 1/(2|b|)

and

Q{a + bn | n ∈ N} = ω/|b| - a/b - 1/2 + 1/(2|b|),

quod erat demonstrandum.

Definition 3.2.1.10. If a real function f(N) has inverse f ^-1, then

Q{f(n) | n ∈ N} = (f ^-1(ω))₀

where ( )₀ is a correction such that

Q{-f(n) | n ∈ N} = Q{f(n) | n ∈ N}.

Examples 3.2.1.11.

Q{a + bn^k | n ∈ N} = (ω/|b| - a/b - 1/2 + 1/(2|b|))^1/k

where

a ∈ R°,

b , k ∈ R° \ {0},

k > 0,

ℵ/|b| - a/b - 1/2 + 1/(2|b|) > 0;

Q{a + bcⁿ | n ∈ N} = log_c (ω/|b| - a/b - 1/2 + 1/(2|b|))

where

a , b , c ∈ R°,

b ≠ 0,

c > 1,

ω/|b| - a/b - 1/2 + 1/(2|b|) > 0.

Remark 3.2.1.12. So the uninumbers and functions of them can be interpreted by the quantisets. Note that the existence of f ^-1 is not necessary for

Q{f(n) | n ∈ N}

to be defined. For example,

Q{a + bn + cn² | n ∈ N} =

[ω/|c| - a/c - b/(2c) + 1/(2|c|) + b²/(2c)²]^1/2 - b/(2c) =

[ω/|c| - a/c - b/(2c) + 1/(2|c|)]/{[ω/|c| - a/c - b/(2c) + 1/(2|c|) + b²/(2c)²]^1/2 + b/(2c)}

where

a , b , c ∈ R° ,

c ≠ 0,

giving Definition 3.2.1.6 as c → 0, the correcting sign of modulus jumping from c to b with b ≠ 0 instead of c ≠ 0.

Remark 3.2.1.13. The uniquantity of a quantiset is the uninumber of its elements if their own quantities are uninumbers of a common uninumber set. The uniquantities evaluated by the uninumbers enable us to solve many problems. Thus the same probability p_n = p of the random sampling of a certain n ∈ N is the solution

p = 1/Q(N) = 1/ω

to the equation

∑_n∈N p = 1

giving

Q(N)p = 1.

3.2.2. Uniquantities of Continual Quantisets

Definition 3.2.2.1. The following semiquantiintervals with ±ω are:

|-ω , =° ]-∞ , ;

, ω| =° , ∞[ ;

]-ω , =° -°_1/2(-ω) +° |-ω , ;

, ω[ =° , ω| -° _1/2ω ;

[-ω , =° _1/2(-ω) +° |-ω , ;

, ω] =° , ω| +° _1/2ω

where ±∞ are conventional symbols in

]-∞ , a), (a , ∞[ ⊂ R (a ∈ R),

]-∞ , ∞[ = R .

Corollary 3.2.2.2.

|-ω , ω| =° ]-∞ , ∞[ = R .

Theorem 3.2.2.3. For any a , b ∈ R :

Q]_qa , _rb[ = |b - a|Ω - 1,

Q|_qa , _rb[ = |b - a|Ω - 1 + q/2,

Q]_qa , _rb| = |b - a|Ω - 1 + r/2,

Q[_qa , _rb[ = |b - a|Ω - 1 + q ,

Q|_qa , _rb| = |b - a|Ω - 1 + (q + r)/2,

Q]_qa , _rb] = |b - a|Ω - 1 + r ,

Q[_qa , _rb| = |b - a|Ω - 1 + q + r/2,

Q|_qa , _rb] = |b - a|Ω - 1 + q/2 + r ,

Q[_qa , _rb] = |b - a|Ω - 1 + q + r .

Proof. We begin with the equality

Q|a, b| = |b - a|Ω .

Let first a , b ∈ R and a ≤ b. For the integer part [b - a] ∈ N ,

Q|a , a + [b - a]| = Q(|a , a + 1| +° |a + 1, a + 2| +°

... +° |a + [b - a] - 1, a + [b - a]|) = [b - a]Ω

(see Axioms A1, A5, and A7). Further we have

Q|a + [b - a], a + (b - a)| = (b - a - [b - a])Ω

when

b - a - [b - a] ∈ Q

[the rational numbers] since the partition of |0, 1| into n equal parts (for any n ∈ Z⁺) gives (see Axioms A1, A5, and A7)

Q|0, 1/n| = Q|1/n , 2/n| = ... =

Q|(n - 1)/n , 1| = Q|0, 1|/n = Ω/n

and when

b - a - [b - a] ∉ Q

since the set Q is everywhere dense in R (see Axiom A6).

For

a = -ω

and

b = ω

(see Axioms A1, A5, and A7),

Q(R) = Q|-ω , ω| = Q(|-ω , 0| +° |0, ω|) = Q|-ω , 0| + Q|0, ω|,

Q|-ω , 0| = Q(+°_n∈N |- n , - n + 1|) = ωΩ ,

Q|0, ω| = Q(+°_n∈N |n - 1, n|) = ωΩ ,

Q(R) = ωΩ + ωΩ = 2ωΩ .

We also have (see Axioms A1, A2, and A4, Definitions 2.3.4.5 and 2.3.4.6, and Corollary 2.3.4.7)

Q|-ω , b| = Q/-ω , b/ = Q(/-ω , 0/ +° /0, b/) =

ωΩ + bΩ = (ω + b)Ω ,

Q|a , ω| = Q/a , ω/ = Q(/0, ω/ -°/0, a/) =

ωΩ - aΩ = (ω - a)Ω .

For

a < -ω

or

b > ω ,

see Axioms A1 and A5 and Examples 1.1.1.9 (R is translational).

For a > b , see Corollary 2.3.4.3.

Hence,

Q|a , b| = |b - a|Ω

holds for any a , b ∈ R . Finally,

Q]_qa , _rb[ = Q]a , b[ = Q (|a , b| -° _1/2a -° _1/2b) =

|b - a|Ω - 1/2 - 1/2 = |b - a|Ω - 1,

Q|_qa , _rb[ = Q(_q/2a +° ]_qa , _rb[) = |b - a|Ω - 1 + q/2,

Q]_qa , _rb| = Q(]_qa , _rb[ +° _r/2b) = |b - a|Ω - 1 + r/2,

Q[_qa , _rb[ = Q(_qa +° ]_qa , _rb[) = |b - a|Ω - 1 + q ,

Q|_qa , _rb| = Q(_q/2a +° ]_qa , _rb[ +° _r/2b) =

|b - a|Ω - 1 + (q + r)/2,

Q]_qa , _rb] = Q(]_qa , _rb[ +° _rb) = |b - a|Ω - 1 + r ,

Q[_qa , _rb| = Q(_qa +° ]_qa , _rb[ +° _r/2b) =

|b - a|Ω - 1 + q + r/2 ,

Q|_qa , _rb] = Q(_q/2a +° ]_qa , _rb[ +° _rb) = |b - a|Ω - 1 + q/2 + r ,

Q[_qa , _rb] = Q(_qa +° ]_qa , _rb[ +° _rb) = |b - a|Ω - 1 + q + r .

Remark 3.2.2.4. If [possibly] a and/or b ∈ R° \ R , then

)_qa , _rb) = [_q’a’, _r’b’]

where

a’, b’ ∈ R ,

a’ ≅ a ,

b’ ≅ b ,

with the same quantities q’ and r’ of a’ and b’ in

)_qa , _rb)

and in

[_q’a’, _r’b’] ,

e.g.

)2 - ω^-1, 3 - Ω^-2( = [2, 3[

for any choice of

) , ( ∈ {] , | , [}

on the left-hand side.

Theorem 3.2.2.5. If addition and subtraction (the both without any restriction) are S-operations, the uniquantity of the S-monad of any S-uninumber is the same.

Proof. By Axiom A5, we have

Qa_S = Qb_S

for any

a , b , a’ ∈ S

with a’ ≅ a by putting

b’ = b + (a’ - a) ≅ b .

Note that

b’ - b = a’ - a ,

and the one-to-one mapping

a’ ↔ b’

is an isometry as

|b’₂ - b’₁| = |a’₂ - a’₁|.

Theorem 3.2.2.6. Under the hypotheses of Theorem 3.2.2.5, each of the positive and negative submonads of any S-uninumber has the same uniquantity.

Proof. By Axiom A5, for any

a ∈ S ,

a’ ∈ a⁺ ,

we have

Qa⁺ = Qa^-

by putting

a’’ = a - (a’ - a) = 2a - a’ ∈ a^-

because the one-to-one mapping

a’ ↔ a’’

is an isometry since

|a’’₂ - a’’₁| = |a’₂ - a’₁|.

Notation 3.2.2.7. For any a ∈ R° and for the positive and negative R°-submonads of a ,

g = Qa⁺ = Qa^- ,

then

Qa = 2g + 1

for the [unireal] R°-monad a of any R-uninumber a ∈ R°.

Corollary 3.2.2.8. For any a , b ∈ R°, the uniquantities of the following quantiintervals and semiquantiintervals are (see Notations 2.3.3.8 and 2.3.4.8, Definition 2.3.4.5, and Axiom A1):

Q)a ,° b( = Q)a , b( × (2g + 1),

Q|a ,° = Q|a ,° = Q]a ,° + g + 1/2,

Q]a ,° = Q]a ,° + g , Q[a ,° = Q]a ,° + g + 1,

Q(,° b[) = Q(,° b[) + g , Q(,° b]) = Q(,° b[) + g + 1,

Q(,° b|) = Q(,° b|) = Q(,° b[) + g + 1/2.

Remark 3.2.2.9. Just the semiquantiintervals introduced enable us to use such a brief notation and to obtain by their combining a number of results for the quantiintervals. In the further results, the passage from R to R° can proceed by the above.

Corollary 3.2.2.10. The uniquantity of a Cartesian power of a quantiset is its uniquantity raised to the same power n ∈ Z⁺ (see Axiom A3), say for a ≤ b :

Q]a , b[ⁿ = ((b - a)Ω - 1)ⁿ,

Q|a , b[ⁿ = Q]a , b|ⁿ = ((b - a)Ω - 1/2)ⁿ,

Q[a , b[ⁿ = Q|a , b|ⁿ = Q]a , b]ⁿ = (b - a)ⁿΩⁿ ,

Q[a , b|ⁿ = Q|a , b]ⁿ = ((b - a)Ω + 1/2)ⁿ,

Q[a , b]ⁿ = ((b - a)Ω + 1)ⁿ;

Q|-ω , b|ⁿ = (ω + b)ⁿΩⁿ if b ≥ -ω ,

Q|a , ω|ⁿ = (ω - a)ⁿΩⁿ if a ≤ ω ,

Q(Rⁿ) = 2ⁿωⁿΩⁿ .

Definition 3.2.2.11. For a simple arc, say AB , a simple quantiarc of the equivalent forms

_qA _rB =° [_qA _rB] (=° _qA +° ]AB[ +° _rB)

is the quantiunion of both

(a) the ordinary set ]AB[ of all arc intermediate points and

(b) its bounds A and B with their arbitrary quantities q and r , respectively,

say

a simple quantiarc

[AB]

including the both bounds A and B is called closed,

]AB[

without them open,

|AB| =° _1/2A +° ]AB[ +° _1/2B

symmetric half-open, or symmetric half-closed,

[AB[

or

]AB]

half-open, or half-closed,

|AB[ ,

]AB| ,

|AB] ,

or

[AB|

partially open, or partially closed.

If a simple arc AB has length L ≥ 0, the uniquantity of the symmetric half-open simple quantiarc |AB| is

Q|AB| = LΩ .

Definition 3.2.2.12. Open, some partially open [partially closed], in particular, [symmetric] half-open [half-closed], and closed simple quantiarcs of zero length are:

]AA[ =° _-1A ,

|AA[ =° ]AA| =° _-1/2A ,

[AA[ =° |AA| =° ]AA] =° ₀A =° ₀# =° ∅ ,

|AA] =° [AA| =° _1/2A ,

[AA] =° ₁A =° {A} .

Definition 3.2.2.13. A quantiarc is the quantiunion of a linearly ordered family of simple quantiarcs, each of them having the unique immediately preceding one [possibly except the first one] and the unique immediately following one [possibly except the last one] in the family, such that the quantisum of the quantities of the common bound of any two adjacent simple ones is 1, each point of self-intersection being counted so many times as it is passed.

Definition 3.2.2.14. A directed quantiarc, say

)_qA _rB)

[the italic brackets, A the origin, B the endpoint, q and r any quantities of these bounds], is its generating quantiarc, say

)_qA _rB)

[the relative orientation of each bracket with reference to the adjacent bound is preserved], with the same point quantities if either it contains no point other than the bounds or the direction from A to B is considered positive else with the opposite ones:

)_qA _rB)⁺ =° )_qA _rB)

[ )_qA _rB)^- =° _-1)_qA _rB) ,

respectively].

Corollary 3.2.2.15. The rearrangement of the bounds with their own quantities by preserving the relative orientation of each bracket with reference to the adjacent bound of a quantiarc does not affect it, that of the distinct bounds of a directed quantiarc implies its additive inversion:

)_qA _rB) =° (_rB _qA( ,

say

[_qA _rB| =° |_rB _qA] ;

)_qA _rB) =° _-1(_rB _qA( ,

say

[_qA _rB/ =° _-1/_rB _qA] .

Definition 3.2.2.16. A (directed) quanticontour [closed quantipath] is a (directed, respectively) quantiarc whose bounds coincide, the quantisum of their quantities being -1 for the negatively directed quantiarc [and quanticontour] and otherwise 1.

Corollary 3.2.2.17. The uniquantity of a quanticontour or a positively directed one of length L is LΩ , of a negatively directed one -LΩ . The uniquantity of a quantiarc

[_qA _rB]

or a positively directed one

[_qA _rB]⁺

of length L is

Q[_qA _rB] = Q[_qA _rB]⁺ = LΩ - 1 + q + r ,

of a negatively directed one

Q[_qA _rB]^- = - (LΩ - 1 + q + r) = - LΩ + 1 - q - r .

Examples 3.2.2.18.

Q[(0, 0) ; (3, 4)] = 5Ω + 1

(the closed straight line interval),

Q{(x, y) | (x - x₀)² + (y - y₀)² = r²} = 2πrΩ

(the circle with radius r > 0 and center at point (x₀ , y₀)).

Definition 3.2.2.19. A (symmetric) half-open point set is an algebraic quantiunion of Cartesian products of some (symmetric, respectively) half-open straight line intervals.

Definition 3.2.2.20. The uniquantity of a simply connected point set P of measure

µ(P) = f(_j∈J µ_j(a_j))

that is a function of the determining parameters µ_j(a_j), each of them being interpreted by some greatest straight line interval a_j whose both bounds lie on the boundary of the set, is

Q(P) = f(_j∈J Q(a_j)) ,

say for such an open, symmetric half-open, or closed set, respectively,

Q]P[ = f(_j∈J (a_jΩ - 1)) ,

Q|P| = f(_j∈J (a_jΩ)) ,

Q[P] = f(_j∈J (a_jΩ + 1)) .

Examples 3.2.2.21. Let [P] be a disc

(x - x₀)² + (y - y₀)² ≤ r² (r > 0).

There is the unique acceptable determining parameter, namely the diameter d = 2r (but not the radius r). We have

µ(P) = π/4 × d²;

Q]P[ = π/4 × (2rΩ - 1)²,

Q|P| = πr²Ω² ,

Q[P] = π/4 × (2rΩ + 1)².

See also Corollary 3.2.2.10 (the unique parameter is the edge).

For the interior ]P[ of an ellipsoid

(x - x₀)²/a² + (y - y₀)²/b² + (z - z₀)²/c² < 1

(a , b , c ∈ R⁺;

x , y , z , x₀ , y₀ , z₀ ∈ R) ,

the only three such parameters are the axes

2a , 2b , 2c

(but not the semiaxes

a , b , c) .

From

µ(P) = π/6 × (2a)(2b)(2c)

we obtain

Q]P[ = π/6 × (2aΩ - 1)(2bΩ - 1)(2cΩ - 1) ,

Q|P| = 4π/3 × abcΩ³ ,

Q[P] = π/6 × (2aΩ + 1)(2bΩ + 1)(2cΩ + 1) .

Definition 3.2.2.22. The uniquantity of an open, half-open, or closed multiply connected point set [whose boundary consists of an outer quanticontour and inner ones] is the difference between the uniquantity of the open, half-open, or closed point set determined by the outer quanticontour, and the sum of the uniquantities of all closed, half-open, or open point sets, respectively, determined by all the inner quanticontours as their outer boundaries.

Example 3.2.2.23. Let [P] be an annulus

a² ≤ (x - x₀)² + (y - y₀)² ≤ b² (0 < a < b) .

The unique outer contour

(x - x₀)² + (y - y₀)² = b²

and the unique inner contour

(x - x₀)² + (y - y₀)² = a²

determine the point sets [discs]

[P_e] = {(x , y) | (x - x₀)² + (y - y₀)² ≤ b²} ,

[P_i] = {(x , y) | (x - x₀)² + (y - y₀)² ≤ a²} ,

respectively, as their outer contours. For the following open, half-open, and closed point sets,

Q]P[ = Q]P_e[ - Q[P_i] = π/4 × (2bΩ - 1)² - π/4 × (2aΩ + 1)² =

π(b² - a²)Ω² - π(b + a)Ω ,

Q|P| = Q|P_e| - Q|P_i| = π/4 × (2bΩ)² - π/4 × (2aΩ)² =

π(b² - a²)Ω² ,

Q[P] = Q[P_e] - Q]P_i[ = π/4 × (2bΩ + 1)² - π/4 × (2aΩ - 1)² =

π(b² - a²)Ω² + π(b + a)Ω .

Definition 3.2.2.24. Let

P be a quantiset of elements of

Rⁿ (n ∈ Z⁺)

with S-uninumber quantities;

_modP the quantiset of the same elements with the moduli of their quantities in P ;

D’ a n-dimensional “cube’’ obtained by a rotation of

D = |-d , d|ⁿ (d ∈ R⁺)

with respect to the origin (_n0);

P_D’(_modP_D’)

the quantiset of the same elements, the quantity of each of them being the product of those in P (_modP, respectively) and in D’.

If, for any large enough d and for any D’ ∈ M minimizing

Q(_modP_D’)

globally, the uniquantity of P_D’ is a common function of d, say f(d), then the uniquantity of the quantiset P is

Q(P) = f(ω) .

Examples 3.2.2.25. (1) The uniquantity of the ray

P = [a , ω| × ∏_k=2ⁿ{a_k}

in Rⁿ is defined as

Q(P) = (ω - a)Ω + 1/2

depending on its origin. We have

_modP = P

and, for any large enough d , M is obtained by all the rotations of D around the x₁-axis;

f(d) = (d - a)Ω + 1/2.

(2) The uniquantity of a straight line (plane) in R³ is defined as

2ωΩ

(4ω²Ω² ,

respectively). In these cases,

_modP = P

and, for any large enough d , M consists of all the D’ with some edges parallel to the line (perpendicular to the plane);

f(d) = 2dΩ

(f(d) = 4d²Ω ,

respectively).

(3) The uniquantity of the set of the points strictly between any two parallel lines (planes) with their distance a ∈ R⁺ in R³ is defined as

2ωΩ(aΩ - 1)

(4ω²Ω²(aΩ - 1),

respectively). Again

_modP = P

and, for any large enough d , M unifies all the D’ with some edges parallel to the lines (perpendicular to the planes);

f(d) = 2dΩ(aΩ - 1)

(f(d) = 4d²Ω²(aΩ - 1) ,

respectively).

Remark 3.2.2.26. Q(P) is the same for bounded P and evidently bifurcates for unbounded P if, together with Definition 3.2.2.24, the simplified one with

f(d) = Q(P_D)

is considered leading to Q(P) depending on the orientation of P in Rⁿ. Then the uniquantity of a straight line P , e.g. in R²,

Q{(x , y) | ax + by + c = 0} ,

where

a , b , c , x , y ∈ R ,

a² + b² > 0,

is:

Q(P) = 2ωΩ(a² + b²)^1/2/max{|a|, |b|} if |a| ≠ |b|,

Q(P) = √2 × (2ω - |c|/|a|)Ω if |a| = |b| and c ≠ 0,

Q(P) = 2√2 × ωΩ - 1/2 if |a| = |b| and c = 0.

In fact, for |a| ≠ |b| and any large enough d > 0, P intersects two opposite sides of the square

|- d , d|²,

the length of P_d is

2d/cos α

where

α = arctan min{|b/a|, |a/b|},

1/cos α = (1 + tan² α)^1/2 =

(1 + min{b²/a², a²/b²})^1/2 = (a² + b²)^1/2/max{|a|, |b|}.

The bounds A and B of P_d do not coincide with any vertex of the square and have the same quantity

1 × 1/2 = 1/2.

Hence, the uniquantity of this symmetric half-open quantiinterval is

f(d) = Q(P_d) = Q|AB| = 2dΩ(a² + b²)^1/2/max{|a|, |b|}.

For |a| = |b|, c ≠ 0, and any large enough d > 0, P intersects two adjacent sides of the square at the points C , D outside of its vertices, the interval

P_d = |CD|

has length

√2 × (2d - |c/a|)

and uniquantity

f(d) = Q(P_d) = Q|CD| = √2 × (2d - |c|/|a|)Ω .

For |a| = |b|, c = 0, and any d > 0, P_d is a diagonal of the square with length

2√2 × d

and each bound quantity

1 × (1/2 × 1/2) = 1/4,

therefore

f(d) = Q(P_d) = 2√2 × dΩ - 1 + 1/4 + 1/4 = 2√2 × dΩ - 1/2.

Definition 3.2.2.27. The [whole] uniquantity, internal and external ones of the boundary B of a point set P are

Q(B) = Q[P] - Q]P[ ,

Q_i(B) = Q|P| - Q]P[ ,

Q_e(B) = Q[P] - Q|P| ,

respectively.

Corollary 3.2.2.28. The uniquantity of a closed point set is the sum of the uniquantities of its interior and boundary. The whole uniquantity of the boundary of a point set is the sum of the internal and external uniquantities of the boundary.

Example 3.2.2.29. Let [P] be a disc (see Examples 3.2.2.21). The whole, internal, and external uniquantities of its circle are

Q(B) = 2πrΩ ,

Q_i(B) = π(rΩ - 1/4) ,

Q_e(B) = π(rΩ + 1/4) .

Example 3.2.2.30. Let P be an n-dimensional "cube" (see Corollary 3.2.2.10). The whole, internal, and external uniquantitites of its boundary are

Q(B) = ((b - a)Ω + 1)ⁿ - ((b - a)Ω - 1)ⁿ,

Q_i(B) = (b - a)ⁿΩⁿ - ((b - a)Ω - 1)ⁿ,

Q_e(B) = ((b - a)Ω + 1)ⁿ - (b - a)ⁿΩⁿ ,

respectively.

Example 3.2.2.31. Let C be the Cantor ternary set [1, 9] built beginning with the closed segment [0, 1] by sequentially countably removing all the middle thirds as open segments from all the set parts at each step of building. C is the limit, or countable intersection, of all its iterative "approximations from above":

C₀ = [0, 1],

C₁ = [0, 1] \ (1/3, 2/3),

C₂ = [0, 1] \ (1/3, 2/3) \ (1/9, 2/9) \ (7/9, 8/9),

C₃ = [0, 1] \ (1/3, 2/3) \ (1/9, 2/9) \ (7/9, 8/9) \

(1/27, 2/27) \ (7/27, 8/27) \ (19/27, 20/27) \ (25/27, 26/27),

..............................................................................................................

Their uniquantities are

Q(C₀) = Ω + 1,

Q(C₁) = Ω + 1 - 2⁰(Ω/3¹ - 1) = (1 - 2⁰/3¹)Ω + 1 + 2⁰,

Q(C₂) = Ω + 1 - 2⁰(Ω/3¹ - 1) - 2¹(Ω/3² - 1) = (1 - 2⁰/3¹ - 2¹/3²)Ω + 1 + 2⁰ + 2¹,

Q(C₃) = Ω + 1 - 2⁰(Ω/3¹ - 1) - 2¹(Ω/3² - 1) - 2²(Ω/3³ - 1) =

(1 - 2⁰/3¹ - 2¹/3² - 2²/3³)Ω + 1 + 2⁰ + 2¹ + 2²,

.......................................................................................................................................

Q(C_n) = Ω + 1 - 2⁰(Ω/3¹ - 1) - 2¹(Ω/3² - 1) - 2²(Ω/3³ - 1) - ... - 2^n-1(Ω/3ⁿ - 1) =

(1 - 2⁰/3¹ - 2¹/3² - 2²/3³ - ... - 2^n-1/3ⁿ)Ω + 1 + 2⁰ + 2¹ + 2² + ... + 2^n-1 =

(1 - 2⁰/3¹ (1 - 2ⁿ/3ⁿ)/(1 - 2/3))Ω + 1 + 2⁰ (2ⁿ - 1)/(2 - 1) =

(2/3)ⁿΩ + 2ⁿ (n ∈ N) ,

.......................................................................................................................................

Using

Q(N) = ω ,

we may consider that the uniquantity of C itself is

Q(C) = (2/3)^ωΩ + 2^ω .

This result is very natural, eloquent, interesting, typical, and important. It corresponds to intuition and obviously shows the universality and perfect sensitivity of uniquantities in contrast to the cardinality and measures. The cardinality of C is

card(C) = Ω

[1, 9] which is the same as by any continual set. Among them are [0, 1], the whole three-dimensional space, any countably-dimensional space, etc. Each known measure can be sensitive to a certain dimensionality only. For C which is a linear set, the linear measure µ only could be sensitive. But it is not the case because

µ(C) = 1 - 2⁰/3¹ - 2¹/3² - 2²/3³ - ... = 1 - 2⁰/3¹ (1/(1 - 2/3)) = 1 - 1 = 0

[1, 9] which is the same as by any null set (in the sense of a measure-zero set) and not only by the empty set. Hence this example clearly shows that the cardinality and measures are not sensitive enough and do not correspond to intuition.

Remark 3.2.2.32. Only at most countable sums [series] are usually regarded. Due to the uniquantities evaluated by the uninumbers, arbitrary [also uncountable] sums can be considered, for instance:

∑_x∈R Q{x} = ∑_x∈R 1 = Q(R) = 2ωΩ .

Remark 3.2.2.33. In probability theory, many probability distributions and cumulative distribution functions (as the probabilities that the variables are less than or equal to the arguments of these functions) on sets of infinite measure can be evaluated and interpreted only by the uninumbers and uniquantities and in non-Euclidean geometry.

Example 3.2.2.34. The uniform probability distribution on ]-ω , ω] and its cumulative distribution function are

f(x) = 1/Q]-ω , ω] = 1/(2ωΩ),

F(x) = Q]-ω, x]/Q]-ω , ω] = (ω + x)/(2ω),

respectively.

This F(x) is interpreted by the straight line

F(x) = 1/2 + x/tan Π(1/2)

in the Lobachevskian geometry where Π(y) is the parallelism angle at the point (0, y) with respect to the x-axis.

Example 3.2.2.35. The uniform probability distribution on ]-ω , a] (a ∈ R) and its cumulative distribution function are

f(x) = 1/Q]-ω , a] = 1/[(ω + a)Ω] if x ≤ a ,

f(x) = 0 if x > a ,

F(x) = Q]-ω , x]/Q]-ω , a] = (ω + x)/(ω + a) if x ≤ a ,

F(x) = 1 if x > a ,

respectively.

This F(x) by x ≤ a is interpreted by the straight line

F(x) = 1 + (x - a)/tan Π(1)

in the Lobachevskian geometry where Π(y) is the parallelism angle at the point (a, y) with respect to the x-axis.

Example 3.2.2.36. The uniform probability distribution on ]b , ω] (b ∈ R) and its cumulative distribution function are

f(x) = 0 if x ≤ b ,

f(x) = 1/Q]b , ω] = 1/[(ω - b)Ω] if x > b ,

F(x) = 0 if x ≤ b ,

F(x) = Q]b , x]/Q]b , ω] = (x - b)/(ω - b) if x > b ,

respectively.

This F(x) by x > b is interpreted by the straight line

F(x) = (x - b)/tan Π(1)

in the Lobachevskian geometry where Π(y) is the parallelism angle at the point (a, y) with respect to the x-axis.

Example 3.2.2.37. The uniform probability distribution on ]-ω , a] +° ]b , ω] (a ∈ R , b ∈ R , a < b) and its cumulative distribution function are

f(x) = 1/Q{]-ω , a] +° ]b , ω]} = 1/[(2ω + a - b)Ω] if x ≤ a ,

f(x) = 0 if a < x ≤ b ,

f(x) = 1/Q{]-ω , a] +° ]b , ω]} = 1/[(2ω + a - b)Ω] if x > b ,

F(x) = Q]-ω , x]/Q{]-ω , a] +° ]b , ω]} = (ω + x)/(2ω + a - b) if x ≤ a ,

F(x) = Q]-ω , a]/Q{]-ω , a] +° ]b , ω]} = (ω + a)/(2ω + a - b) if a < x ≤ b ,

F(x) = Q{]-ω , a] +° ]b , x]}/Q{]-ω , a] +° ]b , ω]} =

(ω + a - b + x)/(2ω + a - b) if x > b ,

respectively.

If x ≤ a , then F(x) is interpreted by the straight line

F(x) = (ω + a)/(2ω + a - b) + (x - a)/tan Π[(ω + a)/(2ω + a - b)]

in the Lobachevskian geometry where Π(y) is the parallelism angle at the point (a, y) with respect to the x-axis.

If x > b , then F(x) is interpreted by the straight line

F(x) = (ω + a)/(2ω + a - b) + (x - b)/tan Π[(ω - b)/(2ω + a - b)]

in the Lobachevskian geometry where Π(y) is the parallelism angle at the point (b, y) with respect to the straight line y = 1.

Remark 3.2.2.38. Such a possible event as the choice of a certain real number r ∈ R is usually considered to have the same zero probability as an impossible event. If for any r ∈ R , that probability is the same, then it equals

1/Q(R) = 1/(2ωΩ) > 0.

Remark 3.2.2.39. The uniquantities evaluated by the uninumbers are natural and precise degrees of quantity valid also by intersecting sets of any mixed dimensions, e.g.

Q((0, 0, 0) +° {0} × |-1/2, 1/2] × {0} +° [3, +∞[² × {4} +° ]-∞ , -1|³) =

1 + (Ω + 1/2) + ((ω - 3)Ω + 1/2)² + (- 1 + ω)³Ω³ =

7/4 + (ω - 2)Ω + (ω - 3)²Ω² + (ω - 1)³Ω³ .

Conclusions

The sets, multisets, and fuzzy sets are extended by the quantisets with any quantity of each element, whose uniquantities are the possibly uncountable quantisums of their element quantities and extend the numbers of elements and point [zero-dimensional] measures. The uninumbers extending the numbers by including the infinite cardinal numbers evaluate the quantisets, their elements, element quantities, and uniquantities and are interpreted by them. The quantisets, their uniquantities, and uninumbers with their possibly uncountable quantioperations and quantirelations exactly and perfectly sensitively express, estimate, and evaluate even infinitesimal distinctions in arbitrarily infinite objects. In particular, each individual element of any countable or continual point set is precisely taken into account, the differently possible events have naturally discriminated positive probabilities, and distributions and distribution functions on sets of infinite measure are expressed and evaluated and with non-Euclidean interpretation. This realizes Bolzano’s dream on discovering the mysteries and paradoxes of the infinity with effective operations at it like those on the usual numbers. Such a quantianalysis is applicable to many urgent problems in pure and applied mathematics, mechanics, strength science, physics, and other sciences.

Any quantities of elements save space by notation and are natural in quantitative sets, or quantisets, including quantielements, integral and fractional quantisets. They are introduced along with appropriate quantification, quantity determination, and other quantioperations and quantirelations. They naturally generalize Cantor's sets, fuzzy sets, number and set operations and relations.

The quantielements can form:

together with quantiaddition and quantimultiplication –

a commutative additive group with zero and additive inverse, a commutative multiplicative group with unit and multiplicative inverse, and a commutative field,

together with quantiunification and quantiintersection – a so-called extremely Boolean algebra.

The integral (fractional) quantisets form:

together with quantiaddition and quantimultiplication – a ring with unit and an algebra (a field and an algebra, respectively),

together with quantiunification and quantiintersection – a so-called extremely Boolean algebra (a distributive algebra, respectively).

These fundamental concepts provide many new possibilities for representating objects and solving many urgent scientific and life problems.

The System of Revolutions in Fundamental Mathematics

The system of revolutions in fundamental mathematics includes:

the foundation subsystem of revolutions, in particular:

1) unifying the relations of belonging, inclusion, and part-whole;

2) justifying and efficiently using contradictions;

3) general (nonlogical) quantification;

the quantiobject subsystem of revolutions, in particular:

1) quantielements;

2) integer and fractional quantisets;

3) mereological quantiaggregates (quanticontents);

4) quantisystems;

the unioperation subsystem of revolutions, in particular:

1) alternative negativity-preserving multiplication;

2) base sign preserving exponentiation;

3) avoiding division by zero if necessary and/or useful;

4) the use of division by zero;

5) the universal empty, emptifying, void, voiding, neutral, and operation-neutralizing element (operand);

6) unioperations with noninteger numbers of operands or with uncountable sets of operands;

the uninumber subsystem of revolutions, in particular:

1) the system of canonical sets whose uniquantities are infinite cardinal numbers;

2) universally applicable explicit infinities and infinitesimals namely real but not potential;

3) the system of the canonical infinities (the infinite cardinal numbers as the canonical positive infinities namely real but not potential);

4) the system of the canonical infinitesimals (the infinite cardinal numbers reciprocals as the canonical positive infinitesimals namely real but not potential);

5) the first discovered nonnumber overinfinity-reciprocal nature and essence of zero;

6) the first invented and discovered universally applicable explicit overinfinities and overinfinitesimals namely real but not potential;

7) the system of the canonical overinfinities (the signed zeroes reciprocals as the canonical overinfinities namely real but not potential);

8) the system of the canonical overinfinitesimals (the canonical overinfinities reciprocals as the canonical overinfinitesimals namely real but not potential);

9) perfectly discriminating infinities or overinfinities even by infinitesimal or overinfinitesimal distinctions and differences;

10) precisely representing each uninumber as sums of purely overinfinite, purely infinite, purely finite, purely infinitesimal, and purely overinfinitesimal uninumber terms;

11) exactly expressing each purely overinfinite uninumber via canonic overinfinities;

12) exactly expressing each purely infinite uninumber via canonic infinities;

13) exactly expressing each purely infinitesimal uninumber via canonic infinitesimals;

14) exactly expressing each purely overinfinitesimal uninumber via canonic overinfinitesimals;

15) a finite uniscale of the uninumbers including sums of overinfinite, infinite, finite, infinitesimal, and overinfinitesimal terms in any combination;

the unimeasure subsystem of revolutions, in particular:

1) uniquantities as unimeasures,

2) the perfect additivity and sensitivity of such unimeasures,

3) providing universal conservation laws with no absorption for any (also finite, infinite, infinitesimal, overinfinite, and overinfinitesimal) uninumbers.

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[Gelimson 2003b] Lev Gelimson. General Problem Theory. Abhandlungen der WIGB (Wissenschaftlichen Gesellschaft zu Berlin), 3 (2003), Berlin, 26-32

[Gelimson 2003c] Lev Gelimson. General Strength Theory. Abhandlungen der WIGB (Wissenschaftlichen Gesellschaft zu Berlin), 3 (2003), Berlin, 56-62

[Gelimson 2003d] Lev Gelimson. General Analytic Methods. Abhandlungen der WIGB (Wissenschaftlichen Gesellschaft zu Berlin), 3 (2003), Berlin, 260-261

[Gelimson 2003e] Lev Gelimson. Quantisets Algebra. Abhandlungen der WIGB (Wissenschaftlichen Gesellschaft zu Berlin), 3 (2003), Berlin, 262-263

[Gelimson 2003f] Lev Gelimson. Elastic Mathematics. Abhandlungen der WIGB (Wissenschaftlichen Gesellschaft zu Berlin), 3 (2003), Berlin, 264-265

[Gelimson 2004a] Lev Gelimson. Elastic Mathematics. General Strength Theory. The "Collegium" All World Academy of Sciences Publishers, Munich (Germany), 2004, 496 pp.

[Gelimson 2004b] Lev Gelimson. General Problem Theory. The Second International Science Conference "Contemporary methods of coding in electronic systems", Sumy, 26-27 October 2004

[Gelimson 2004c] Lev Gelimson. Quantisets Algebra. The Second International Science Conference “Contemporary methods of coding in electronic systems”, Sumy, 26-27 October 2004

[Gelimson 2005a] Lev Gelimson. Providing helicopter fatigue strength: Flight conditions [Megamathematics]. In: Structural Integrity of Advanced Aircraft and Life Extension for Current Fleets – Lessons Learned in 50 Years After the Comet Accidents, Proceedings of the 23rd ICAF Symposium, Vol. II, Dalle Donne, C. (Ed.), Hamburg, 2005, p. 405-416

[Gelimson 2005b] Lev Gelimson. Providing Helicopter Fatigue Strength: Unit Loads [Fundamental Mechanical and Strength Sciences]. In: Structural Integrity of Advanced Aircraft and Life Extension for Current Fleets – Lessons Learned in 50 Years After the Comet Accidents, Proceedings of the 23rd ICAF Symposium, Dalle Donne, C. (Ed.), 2005, Hamburg, Vol. II, 589-600

[Gelimson 2006a] Lev Gelimson. Quantisets and Their Quantirelations. The Third International Science Conference "Contemporary methods of coding in electronic systems", Sumy, 24-25 October 2006

[Gelimson 2006b] Lev Gelimson. Quantiintervals and Semiquantiintervals. The Third International Science Conference "Contemporary methods of coding in electronic systems", Sumy, 24-25 October 2006

[Gelimson 2006c] Lev Gelimson. Multiquantities (now Uniquantities). The Third International Science Conference "Contemporary methods of coding in electronic systems", Sumy, 24-25 October 2006

[Gelimson 2006d] Lev Gelimson. Sets with Any Quantity of Each Element. The "Collegium" All World Academy of Sciences Publishers, Munich (Germany), 2006

[Gelimson 2009a] Lev Gelimson. Overmathematics: Fundamental Principles, Theories, Methods, and Laws of Science. The "Collegium" All World Academy of Sciences Publishers, Munich (Germany), 2009

[Gelimson 2009b] Lev Gelimson. Overmathematics: Principles, Theories, Methods, and Applications. The "Collegium" All World Academy of Sciences Publishers, Munich (Germany), 2009

[Gelimson 2010] Lev Gelimson. Uniarithmetics, Quantialgebra, and Quantianalysis: Uninumbers, Quantielements, Quantisets, and Uniquantities with Quantioperations and Quantirelations. The "Collegium" All World Academy of Sciences Publishers, Munich (Germany), 2010

[Gelimson 2011a] Lev Gelimson. Uniarithmetics, Quantianalysis, and Quantialgebra: Uninumbers, Quantielements, Quantisets, and Uniquantities with Quantioperations and Quantirelations (Essential). Mathematical Journal of the "Collegium" All World Academy of Sciences, Munich (Germany), 11 (2011), 26

[Gelimson 2011b] Lev Gelimson. Science Unimathematical Test Fundamental Metasciences Systems. Monograph. The “Collegium” All World Academy of Sciences, Munich (Germany), 2011

[Gelimson 2011c] Lev Gelimson. Overmathematics Essence. Mathematical Journal of the "Collegium" All World Academy of Sciences, Munich (Germany), 11 (2011), 25

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[Wikibooks Hyperanalysis-MeasurementTheory] http://en.wikibooks.org/wiki/User:Vuara/Hyperanalysis-MeasurementTheory . Scientist Vuara has published in the English Wikipedia Lev Gelimson's Monograph "Measurement Theory in Physical Mathematics" (unfortunately with unproperly showing some symbols due to using NOT the Internet Explorer Web Browser without adding its character sets) along with the references of some academicians and professors to Lev Gelimson's monographs, articles, and other scientific publications, later "Measurement Theory in Elastic Mathematics", see http://scie.freehostia.com/MeaEnTxt.pdf

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[Wikipedia Infinity] Wikipedia entry: http://en.wikipedia.org/wiki/Infinity

[Wikipedia Iterated_logarithm] http://en.wikipedia.org/wiki/Iterated_logarithm

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