Applications of Multivalued Logic to Set Theory and Calculus
Abstract
The objectives of this paper are to make set theory a step farther from catastrophic contradictions such as the Russell paradox, which blow away some collections of objects such as the set of all sets, which intuitively should exist since their members exist, and make use of multivalued logic in developing non-standard models in set theory and calculus to make work easier. The set theoretic contradictions are derived through two-valued logic, which ends with sentences of the form “P and not P,” which always have the value “false.” By introducing other truth values, such sentences can end up with a non-false value if both “P” and “not P” are not false. Non-standard objects are obtained through collections of ordinary objects indicating that none of the elements are meant, whereas what is more desirable is indicated. For example, a positive number smaller than all positive real numbers can be indicated by a set of positive real numbers with values ever approaching but never truly attaining “true” as they approach zero. In this way, infinite sets can be indicated by multivalued collections of finite sets, and both infinite and infinitesimal numbers are reintroduced to calculus, improving the scope of integration and differentiation. This paper also presents a new way to accommodate truth values for set theory where the objects can be far more numerous than real numbers between zero and one representing values between “false” and “true.”
Keywords: multivalued logic, the Zermelo-Fraenkel set theory, equipollence classes, truth values, multivalued derivatives and integrals, sums of arbitrary series.
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SALIM R. A., & ERNASTUTI. Some Results in Calculus and Set Theory through the Directed Set Method. Journal of Hunan University Natural Sciences, 2023, 50(10): 195-204. https://doi.org/10.55463/issn.1674-2974.50.10.19
ZAMANSKY A. On recent applications of paraconsistent logic: an exploratory literature review. Journal of Applied Non-Classical Logics, 2019, 29(4): 382-391. https://doi.org/10.1080/11663081.2019.1656393
SULEIMENOV I. E., VITULYOVA Y. S., KABDUSHEV S. B., and BAKIROV A. S. Improving the Efficiency of Using Multivalued Logic Tools. Scientific Reports, 2023, 13: 1108. https://doi.org/10.1038/s41598-023-28272-1
FRAENKEL A. A., BAR-HILLEL Y., and LEVY A. Foundations of Set Theory. Elsevier, Amsterdam, 1973.
RUBIN J. E., & RUBIN H. Equivalents of the Axiom of Choice, II. Elsevier, Amsterdam, 1985.
HOWARD P., & RUBIN J. E. Consequences of the Axiom of Choice. American Mathematical Society, Providence, Rhode Island, 1998.
SALIM R. A., & ERNASTUTI. Definability and Strength of Fourier Series Transform Obtained by Employing Parametric Integration. Journal of Hunan University Natural Sciences, 2023, 50(8): 160-167. https://doi.org/10.55463/issn.1674-2974.50.8.13
SALIN H. On Russell’s Paradox and Attempted Resolutions. Bachelor's thesis, Department of Historical, Philosophical and Religious Studies, Umea University, Umea, 2023. https://www.researchgate.net/publication/371595938_On_Russell's_Paradox_and_Attempted_Resolutions
BARTLE R. G., & SHERBERT D. R. Introduction to Real Analysis. 4th ed. John Wiley & Sons, Hoboken, New Jersey, 2011.
MUNKRES J. R. Analysis on Manifolds. Addison Wesley Publishing Company, Redwood City, California, 1991.
GONZALEZ-VELASCO E. A. The Lebesgue Integral as a Riemann Integral. International Journal of Mathematics and Mathematical Sciences, 1987, 10(4): 693-706. https://doi.org/10.1155/S0161171287000802
GRABINSKI M., & KLINKOVA G. Like a Sum Is Generalized into an Integral, a Product May Be Generalized into an Inteduct. Applied Mathematics, 2023, 14: 279-289. https://doi.org/10.4236/am.2023.145017
DACK T.-J. S., & TRESSL M. Elementary Theory of Surreal Numbers. Master's thesis, University of Manchester, Manchester, 2018. http://dx.doi.org/10.13140/RG.2.2.35697.53602
VAN DER BERG I., & JANELIDZE Z. An Axiomatic Approach to the Ordinal Number System. Stellenbosch University, Matieland, 2021.
GITIK M. All Uncountable Cardinals Can Be Singular. Israel Journal of Mathematics, 1980, 35: 61-88. https://doi.org/10.1007/BF02760939
KUPERBERG V. Sums of Singular Series with Large Sets and the Tail of the Distribution of Primes. The Quarterly Journal of Mathematics, 2023, 74(4): 1457–1479. https://doi.org/10.1093/qmath/haad030
CHAGAS J. Q., TENREIRO MACHADO J. A., and LOPES A. M. Overview in Summabilities: Summation Methods for Divergent Series, Ramanujan Summation and Fractional Finite Sums. Mathematics, 2021, 9(22): 2963. https://doi.org/10.3390/math9222963
OZAWA M. From Boolean Valued Analysis to Quantum Set Theory. Mathematics, 2021, 9(4): 397. https://doi.org/10.3390/math9040397
RAMIREZ J. P. A New Set Theory for Analysis. Axioms, 2019, 8(1): 31. https://doi.org/10.3390/axioms8010031
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