Ph. D. & Dr. Sc. Lev Gelimson's Uniarithmetics, Quantianalysis, and Quantialgebra: Uninumbers, Quantielements, Quantisets, and Uniquantities with Quantioperations and Quantirelations (Essential)

Uniarithmetics, Quantianalysis, and Quantialgebra: Uninumbers, Quantielements, Quantisets, and Uniquantities with Quantioperations and Quantirelations (Essential)

© Ph. D. & Dr. Sc. Lev Gelimson

Academic Institute for Creating Fundamental Sciences (Munich, Germany)

Mathematical Journal

of the "Collegium" All World Academy of Sciences

Munich (Germany)

11 (2011), 26

UDC 501:510

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. In the introduced overmathematics based on 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 the infinite cardinal numbers 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 Math. Subj. Classification: primary 00A71; second. 03E10, 03E72, 08B99, 26E30, 28A75.

Keywords: Overmathematics, uninumber, quantiset, uniquantity, uniarithmetics, quantialgebra, quantianalysis.

Introduction

The very fundamentals of classical mathematics [1] have evident lacks and shortcomings.

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. It is urgent to exactly express (in some suitable extension of R) all infinite and infinitesimal quantities, e.g., such a p for any countable or uncountable set, as well as distributions and distribution functions on any sets of infinite measures.

2. The Cantor sets [1] with either unit or zero quantities of their possible elements may contain any object as an element either once or not at all with ignoring its true quantity. The same holds for the Cantor set relations and operations. That is why those set operations are only restrictedly invertible. In the Cantor sets, the simplest equations X ∪ A = B and X ∩ A = B in X are solvable by A ⊆ B and A ⊇ B only, respectively [uniquely by A = ∅ (the empty set) and A = B = U (a universal set), respectively]. The equations X ∪ A = B and X = B \ A in the Cantor sets are equivalent by A = ∅ only. In a fuzzy set, the membership function of each element may also lie strictly between these ultimate values 1 and 0 in the case of uncertainty only. Element repetitions are taken into account in multisets with any cardinal numbers as multiplicities and in ordered sets (tuples, sequences, vectors, permutations, arrangements, etc.) [1]. They and unordered combinations with repetitions cannot express many collections, e.g., that of half an apple and a quarter pear. For any quantities with measurement units, e.g., "5 L (liter) fuel", there is no suitable mathematical model and no known operation, say between "5 L" and "fuel" (not: "5 L" × "fuel" or "fuel" × "5 L"). Note that multiplication is the evident operation between the number "5" and the measurement unit "L". The Cantor set relations and operations only restrictedly reversible with absorption contradict the conservation law of nature because of ignoring element quantities and hinder constructing any universal degrees of quantity.

3. The cardinality is sensitive to finite unions of disjoint finite sets only and gives the same continuum cardinality C for clearly very distinct point sets in a Cartesian coordinate system between two parallel lines or planes differently distant from one another.

4. The measures are 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 [1].

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

6. The operations are considered to be at most countable.

I. Uniarithmetics

The uninumbers [2] naturally extend and refine the scale of R via supplementing the real numbers with the infinite cardinal numbers by preserving the properties of the usual operations and relations. The uninumbers exactly express all infinite and infinitesimal quantities [e.g., that p = 1/(ℵ + 1/2)] for any countable or uncountable set, as well as distributions and distribution functions on sets of infinite measure with modeling them in Lobachevskian geometry.

II. Quantianalysis

1. The also uncountable quantioperations and quantirelations [2] including quantification and quantity determination exactly and perfectly sensitively express, evaluate, and estimate even infinitesimal distinctions in infinite objects.

2. The integral quantisets [2] of quantielements with any elements quantities

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

(+° quantiaddition) exactly and perfectly sensitively express any unordered collection of objects and are evaluated by the uninumbers.

3. The fractional quantisets [2] are introduced as quantisets fractions of the form A/B.

4. The uniquantities [2] are possibly uncountable quantisums of their element quantities

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

extend the numbers of elements and point (zero-dimensional) measures, and build exact and perfectly sensitive universal quantimeasures also of any infinities and infinitesimals.

III. Quantialgebra

1. The quantielements can form [2]:

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.

2. The integral quantisets form [2]:

together with quantiaddition and quantimultiplication – a ring with unit and an algebra,

together with quantiunification and quantiintersection – a so-called extremely Boolean algebra.

3. The fractional quantisets form [2]:

together with quantiaddition and quantimultiplication – a field and an algebra,

together with quantiunification and quantiintersection – a distributive algebra.

These fundamental concepts provide principally new possibilities for modeling general objects and systems, as well as for solving very many urgent scientific and life problems.

References

[1] Encyclopaedia of Mathematics / Managing editor M. Hazewinkel. Volumes 1 to 10. Kluwer Academic Publ., Dordrecht, 1988-1994.

[2] Lev Gelimson. Elastic Mathematics. General Strength Theory. The "Collegium" All World Academy of Sciences Publishers, Munich (Germany), 2004, 496 pp.