Fundamental Zero Science
by
© Ph. D. & Dr. Sc. Lev Gelimson
Academic Institute for Creating Fundamental Sciences (Munich, Germany)
Mathematical Monograph
The “Collegium” All World Academy of Sciences Publishers
Munich (Germany)
2012
Keywords: Fundamental mathematics, megascience, revolution, megamathematics, mega-overmathematics, object, operation, fundamental zero science.
Introduction
It is usual to consider 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]) 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.
Therefore, in classical mathematics, there is no possibility to adequately consider the empty value and hence empty objects also as operands, arguments, or inputs, as well as outputs, results, or values.
Hence classical mathematics cannot (and does not want to) regard (adequately solve and even consider) very many typical urgent 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 [Encyclopaedia of Physics 1973]. This is also very important for chaos theory (Ilya Prigogine [1993, 1997]) and fractal theory (Benoît Mandelbrot [1975, 1977, 1982]).
It is urgent to regard the empty value and empty objects also as operands, arguments, or inputs, as well as outputs, results, or values.
Mega-overmathematics by Lev Gelimson [1987-2012] based on its uninumbers, quantielements, quantisets, and uniquantities with quantioperations and quantirelations provides universally and adequately modeling, expressing, measuring, evaluating, and estimating general objects. This all creates the basis for many further developing, extending, and applying mega-overmathematics fundamental sciences systems. Among them is, in particular, empty value theory.
Empty Value Theory
Definition. The empty value EV is the "result"/"output" of "performing"/"applying" no actions, e.g. operations, transformations, functions, functionals, operators, mappings, etc. (hence on/to no objects, e.g. operands, arguments, inputs, etc.).
Corollary. This "result" may not depend on any particular operation and must be universal.
Corollary. Both the empty sum ES and the empty product EP are particular cases of the empty value (output) EV .
Nota bene: Lev Gelimson [1995a] first explicitly introduced the empting (voiding) operation transforming any object to the empty (void) object (element) # (or the empty set ∅ so that # ∈ ∅ and # = ∅ simultaneously). Using the empty (void) object, e.g. operand, argument, input, etc., # (or ∅) excludes (drops) any action, e.g. operation, transformation, function, functional, operator, mapping, etc. on this object so that this object neutralizes any action. Then the "result"/"output" of "performing"/"applying" no actions, e.g. operations, transformations, functions, functionals, operators, mappings, etc. (hence on/to no objects, e.g. operands, arguments, inputs, etc.) equals namely the empty value # (or ∅), which is universal. It seems to be suitable to denote this universal empty value as this "result"/"output" namely via the empty set symbol ∅ because in this case, both the set of all the actions, e.g. operations, transformations, functions, functionals, operators, mappings, etc., and the set of all the objects, e.g. operands, arguments, inputs, etc., are empty.
Notation. Let ∅ denote this universal empty value so that EV = ∅ .
Corollary. The empty sum ES = ∅ and the empty product EP = ∅ .
Definition. Any object representable as the result/output of performing/applying any actions, e.g. operations, transformations, functions, functionals, operators, etc., on/to any objects, e.g. operands, arguments, inputs, etc., all taking the empty value ∅ only, is the empty value ∅ .
Examples:
any (possibly uncountable) sum
Σj∈J ∅ = ... + ∅ + ... + ∅ + ... = ∅
of the empty values ∅ , i.e. copies (exemplars) of the universal (and hence unique) empty value ∅ , with denoting usual addition by +
where (and also similarly further) the index j one-time takes all the values in any index set J and the linear notation is optional only as a simple graphical illustration;
the additive inverse
∅- = -∅ = ∅
of the empty value ∅ so that
∅ + (∅-) = (∅-) + ∅ = ∅ + (-∅) = (-∅) + ∅ = ∅ ;
any (possibly uncountable) algebraic sum
Σj∈J (±∅) = ... + ∅ + ... + ∅- + ... = ... + ∅ + ... + (-∅) + ... = ... + ∅ + ... - ∅ + ... = ∅ ;
any (possibly uncountable) product
Πj∈J ∅ = ... • ∅ • ... • ∅ • ... = ∅
with denoting usual multiplication by • (or ×);
the multiplicative inverse
∅/ = ∅
of the empty value ∅ so that
∅ • ∅/ = ∅/ • ∅ = ∅ × ∅/ = ∅/ × ∅ = ∅ ;
any (possibly uncountable) algebraic product
Πj∈J ∅(/) = ... • ∅ • ... • ∅/ • ... = ... • ∅ • ... / ∅ • ... = ∅
with denoting usual division by / , the parentheses in the index (/) showing that the index / is optional so that ∅(/) is the unified notation both for ∅ and for ∅/ like ±∅ is the unified notation both for ∅ and for -∅ ;
any, e.g. zero, positive or negative, rational or irrational, imaginary, etc., power of the empty value ∅ is the empty value ∅ .
Nota bene: Power exponents may be considered as means to provide suitable notation only rather than inevitable operands. We can avoid explicitly using power exponents via repeating the empty value ∅ , its additive ∅- = -∅ and multiplicative ∅/ inverses, and operations on them, as well as introducing, denoting, and considering intermediate results (outputs), etc. We may also simply consider any power of the empty value ∅ as the value of the power function whose only argument is the empty value ∅ and whose only parameter is the required (desired) power exponent and then explicitly use the last definition.
Examples:
∅5 = ∅ • ∅ • ∅ • ∅ • ∅ = ∅ ;
∅-3 = ∅/ • ∅/ • ∅/ = ∅ • ∅ • ∅ = ∅ ;
∅-3/5 = (∅/)1/5 • (∅/)1/5 • (∅/)1/5 = ∅1/5 • ∅1/5 • ∅1/5 = ∅ • ∅ • ∅ = ∅ ;
Let us prove that this universal empty value may not be any number known 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]).
Zero Theorem. u = 0.
Proof (via reductio ad absurdum). If, on the contrary, the inequality u ≠ 0 were true, then the empty value would be
EV = u ≠ 0
and, in particular, the empty sum ES would be
ES = u ≠ 0.
Add any number a to the empty sum ES with denoting usual addition by + . Then the relations
a = a + ES = a + u ≠ a
would hold implying the inequality a ≠ a , which is false.
Nonzero Theorem. u ≠ 0.
Proof (via reductio ad absurdum). If, on the contrary, the equality u = 0 were true, then the empty value would equal
EV = u = 0
and, in particular, the empty product EP would equal
EP = u = 0.
Multiply the empty product EP with any nonzero number a denoting usual multiplication by • (or ×). Then the equalities
a = a • EP = a • u = a • 0 = 0
would hold, which is false because
a ≠ 0.
Unit Theorem. u = 1.
Proof (via reductio ad absurdum). If, on the contrary, the inequality u ≠ 1 were true, then the empty value would be
EV = u ≠ 1
and, in particular, the empty product EP would be
EP = u ≠ 1.
Multiply the empty product EP with any nonzero number a . Then the relations
a = a • EP = a • u ≠ a
would hold implying the inequality a ≠ a , which is false.
Nonunit Theorem. u ≠ 1.
Proof (via reductio ad absurdum). If, on the contrary, the equality u = 1 were true, then the empty value would equal
EV = u = 1
and, in particular, the empty sum ES would equal
ES = u = 1.
Add any number a to the empty sum ES . Then the equalities
a = a + ES = a + u = a + 1
would hold, which is false.
Corollary. The universal empty value may not be any number known in classical mathematics. In particular, neither 0 nor 1 can provide such universality. Hence it is necessary to extra introduce the universal empty value.
In particular, for any c ∈ C (the complex numbers), we have
c + ∅ = ∅ + c = c ,
c • ∅ = ∅ • c = c
where + and × (or •) and are usual addition and multiplication, respectively.
For c = 0, we have
0 + ∅ = ∅ + 0 = 0,
0 • ∅ = ∅ • 0 = 0.
For c = 1, we have
1 + ∅ = ∅ + 1 = 1,
1 • ∅ = ∅ • 1 = 1.
Let us extra prove the following theorems also in this particular case to further compare the universal empty value ∅ with 0 and 1.
Nonzero Theorem. ∅ ≠ 0.
Proof (via reductio ad absurdum). If, on the contrary, the equality ∅ = 0 were true, then the equalities 1 • ∅ = ∅ • 1 = 1 would imply
1 • 0 = 0 • 1 = 1,
which is false.
Nonunit Theorem. ∅ ≠ 1.
Proof (via reductio ad absurdum). If, on the contrary, the equality ∅ = 1 were true, then the equalities
0 + ∅ = ∅ + 0 = 0,
1 + ∅ = ∅ + 1 = 1
would imply the equalities
0 + 1 = 1 + 0 = 0,
1 + 1 = 1 + 1 = 1,
respectively, which both are false.
Additive Neutral Element Nonuniqueness Theorem. Along with the usual (number) additive neutral element 0, there is another extra (nonnumber) additive neutral element.
Proof. The universal empty value ∅ differs from 0 and is an extra (nonnumber) additive neutral element because for any c ∈ C (the complex numbers), we have
c + ∅ = ∅ + c = c .
Multiplicative Neutral Element Nonuniqueness Theorem. Along with the usual (number) multiplicative neutral element 1, there is another extra (nonnumber) multiplicative neutral element.
Proof. The universal empty value ∅ differs from 1 and is an extra (nonnumber) multiplicative neutral element because for any c ∈ C (the complex numbers), we have
c • ∅ = ∅ • c = c .
Fundamental zero science in mega-overmathematics by Lev Gelimson [1987-2012] is universal and very efficient.
References
[Bourbaki 1949] Nicolas Bourbaki. Elements de mathematique. Hermann, Paris, 1949 etc.
[Bronstein Semendjajew 1989] I. N. Bronstein, K. A. Semendjajew. Taschenbuch der Mathematik. Frankfurt/M., 1989
[Encyclopaedia of Mathematics 1988] Encyclopaedia of Mathematics / Managing editor M. Hazewinkel. Volumes 1 to 10. Kluwer Academic Publ., Dordrecht, 1988-1994
[Encyclopaedia of Physics 1973] Encyclopaedia of Physics / Chief Editor S. Flugge. Springer, Berlin etc., 1973 etc.
[Gelimson 1987] Lev Gelimson. The Stress State and Strength of Transparent Elements in High-Pressure Portholes (Side-Lights) [In Russian]. Ph. D. dissertation. Institute for Strength Problems, Academy of Sciences of Ukraine, Kiev, 1987
[Gelimson 1992] Lev Gelimson. Generalization of Analytic Methods for Solving Strength Problems [In Russian]. Drukar Publishers, Sumy, 1992
[Gelimson 1993a] Lev Gelimson. General Strength Theory. Drukar Publishers, Sumy, 1993
[Gelimson 1993b] Lev Gelimson. Generalized Methods for Solving Functional Equations and Their Sets [In Russian]. International Scientific and Technical Conference "Glass Technology and Quality". Mathematical Methods in Applying Glass Materials (1993), Theses of Reports, p. 106-108
[Gelimson 1994a] Lev Gelimson. Generalization of Analytic Methods for Solving Strength Problems for Typical Structure Elements in High-Pressure Engineering [In Russian]. Dr. Sc. dissertation. Institute for Strength Problems, National Academy of Sciences of Ukraine, Kiev, 1994
[Gelimson 1994b] Lev Gelimson. The method of least normalized powers and the method of equalizing errors to solve functional equations. Transactions of the Ukraine Glass Institute, 1 (1994), 209-214
[Gelimson 1994c] Lev Gelimson. General Estimation Theory. Transactions of the Ukrainian Glass Institute 1 (1994), p. 214-221 (both this article and a further mathematical monograph have been also translated from English into Japanese)
[Gelimson 1995a] Lev Gelimson. Basic New Mathematics. Drukar Publishers, Sumy, 1995
[Gelimson 1995b] Lev Gelimson. New mathematics as new scientific thinking language [in Russian]. International Scientific and Technical Conference "Energy and Resource Saving Technologies in Glass Production", General Problems, Theses of Reports, 1995, p. 67-68
[Gelimson 1995c] Lev Gelimson. General objects, operations, sets, and numbers [in Russian]. International Scientific and Technical Conference "Energy and Resource Saving Technologies in Glass Production", General Problems, Theses of Reports, 1995, p. 68-70
[Gelimson 1995d] Lev Gelimson. General systems, states, and processes [in Russian]. International Scientific and Technical Conference "Energy and Resource Saving Technologies in Glass Production", General Problems, Theses of Reports, 1995, p. 71-72
[Gelimson 1995e] Lev Gelimson. General estimations and approximations [in Russian]. International Scientific and Technical Conference "Energy and Resource Saving Technologies in Glass Production", General Problems, Theses of Reports, 1995, p. 72-74
[Gelimson 1995f] Lev Gelimson. General problems and methods of solving them [in Russian]. International Scientific and Technical Conference "Energy and Resource Saving Technologies in Glass Production", General Problems, Theses of Reports, 1995, p. 74-76
[Gelimson 1995g] Lev Gelimson. New phenomena and general laws of nature and science [in Russian]. International Scientific and Technical Conference "Energy and Resource Saving Technologies in Glass Production", General Problems, Theses of Reports, 1995, p. 76-78
[Gelimson 1996] Lev Gelimson. General Implantation Theory in the New Mathematics. Second International Conference "Modification of Properties of Surface Layers of Non-Semiconducting Materials Using Particle Beams" (MPSL'96). Sumy, Ukraine, June 3-7, 1996. Session 3: Modelling of Processes of Ion, Electron Penetration, Profiles of Elastic-Plastic Waves Under Beam Treatment. Theses of Reports
[Gelimson 1997a] Lev Gelimson. Hyperanalisis: Hypernumbers, Hyperoperations, Hypersets and Hyperquantities. Collegium International Academy of Sciences Publishers, 1997
[Gelimson 1997b] Lev Gelimson. Mengen mit beliebiger Quantität von jedem Element. Collegium International Academy of Sciences Publishers, 1997
[Gelimson 2001a] Lev Gelimson. Elastic Mathematics: Theoretical Fundamentals. The "Collegium" All World Academy of Sciences Publishers, Munich (Germany), 2001
[Gelimson 2001b] Lev Gelimson. Elastic Mathematics: Principles, Theories, Methods, and Applications. The "Collegium" All World Academy of Sciences Publishers, Munich (Germany), 2001
[Gelimson 2001c] Lev Gelimson. General Estimation Theory. The "Collegium" All World Academy of Sciences Publishers, Munich (Germany), 2001
[Gelimson 2001d] Lev Gelimson. Hyperanalisis: Hypernumbers, Hyperoperations, Hypersets and Hyperquantities. The "Collegium" All World Academy of Sciences Publishers, Munich (Germany), 2001
[Gelimson 2001e] Lev Gelimson. Mengen mit beliebiger Quantität von jedem Element. The "Collegium" All World Academy of Sciences Publishers, Munich (Germany), 2001
[Gelimson 2001f] Lev Gelimson. Objektorientierte Mathematik in der Messtechnik. The "Collegium" All World Academy of Sciences Publishers, Munich (Germany), 2001
[Gelimson 2001g] Lev Gelimson. Measurement Theory in Physical Mathematics. The "Collegium" All World Academy of Sciences Publishers, Munich (Germany), 2001. Also published by Vuara along with a number of references to Lev Gelimson's scientific works.
[Gelimson 2003a] Lev Gelimson. Quantianalysis: Uninumbers, Quantioperations, Quantisets, and Multiquantities (now Uniquantities). Abhandlungen der WIGB (Wissenschaftlichen Gesellschaft zu Berlin), 3 (2003), Berlin, 15-21
[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
[Gelimson 2012] Lev Gelimson. Fundamental Mega-Overmathematics as Revolutions in Fundamental Mathematics: Uniarithmetics, Quantialgebra, and Quantianalysis: Uninumbers, Quantielements, Quantisets, and Uniquantities with Quantioperations and Quantirelations. The "Collegium" All World Academy of Sciences Publishers, Munich (Germany), 2012
[Kaufmann Denis-Papin Faure 1964] Arnold Kaufmann, Maurice Denis-Papin, Robert Faure. Mathématiques nouvelles. – Dunod (impr. Jouve), Paris, 1964
[Korn 1968] G. A. Korn, T. M. Korn. Mathematical Handbook for Scientists and Engineers. McGraw-Hill, N.Y. etc., 1968[Mandelbrot 1975] Benoît Mandelbrot. Les objets fractals. Flammarion, Paris, 1975, 1984
[Mandelbrot 1977] Benoît Mandelbrot. Fractals: Form, Chance and Dimension. Freeman, New York, 1977
[Mandelbrot 1982] Benoît Mandelbrot. The Fractal Geometry of Nature. Freeman, New York, 1982
[Prigogine 1993] Ilya Prigogine. Chaotic Dynamics and Transport in Fluids and Plasmas: Research Trends in Physics Series. American Institute of Physics, New York, 1993
[Prigogine 1997] Ilya Prigogine. End of Certainty. The Free Press, 1997