Do the Two Operations Addition and Multiplication Commute with Each Other?

Koji Nagata¹, Tadao Nakamura²

¹Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon, Korea.
²Department of Information and Computer Science, Keio University, Yokohama, Japan.

DOI: 10.4236/oalib.1101803                

Abstract

We study about the metamathematics of Zermelo-Fraenkel set theory with the axiom of choice. We use the validity of Addition and Multiplication. We provide an example that the two operations Addition and Multiplication do not commute with each other. All analyses are performed in a finite set of natural numbers.

Keywords

Set Theory, Formalism

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Zermelo-Fraenkel Set Theory—Wikipedia, the Free Encyclopedia.
[2]	Abian, A. (1965) The Theory of Sets and Transfinite Arithmetic. W B Saunders.
[3]	Abian, A. and LaMacchia, S. (1978) On the Consistency and Independence of Some Set-Theoretical Axioms. Notre Dame Journal of Formal Logic, 19, 155-158. http://dx.doi.org/10.1305/ndjfl/1093888220
[4]	Devlin, K. (1984) The Joy of Sets. Springer.
[5]	Fraenkel, A., Bar-Hillel, Y. and Levy, A. (1958) Foundations of Set Theory. Fraenkel’s Final Word on ZF and ZFC, North Holland.
[6]	Hatcher, W. (1968) The Logical Foundations of Mathematics. Pergamon.
[7]	Jech, T. (2003) Set Theory: The Third Millennium Edition, Revised and Expanded. Springer.
[8]	Kunen, K. (1980) Set Theory: An Introduction to Independence Proofs. Elsevier.
[9]	Montague, R. (1961) “Semantic Closure and Non-Finite Axiomatizability” in Infinistic Methods. Pergamon, London, 45-69.
[10]	Suppes, P. (1960) Axiomatic Set Theory. Dover Reprint. Perhaps the Best Exposition of ZFC before the Independence of AC and the Continuum Hypothesis, and the Emergence of Large Cardinals. Includes Many Theorems.
[11]	Takeuti, G. and Zaring, W.M. (1971) Introduction to Axiomatic Set Theory. Springer Verlag.
[12]	Tarski, A. (1939) On Well-Ordered Subsets of Any Set. Fundamenta Mathematicae, 32, 176-183.
[13]	Tiles, M. (1989) The Philosophy of Set Theory. Dover Reprint. Weak on Metatheory; the Author Is Not a Mathematician.
[14]	Tourlakis, G. (2003) Lectures in Logic and Set Theory. Vol. 2, Cambridge University Press, Cambridge.
[15]	van Heijenoort, J. (1967) From Frege to Godel: A Source Book in Mathematical Logic, 1879-1931. Harvard Univ. Press. Includes Annotated English Translations of the Classic Articles by Zermelo, Fraenkel, and Skolem Bearing on ZFC.
[16]	Zermelo, E. (1908) Untersuchungen uber die Grundlagen der Mengenlehre I. Mathematische Annalen, 65, 261-281. http://dx.doi.org/10.1007/BF01449999
[17]	van Heijenoort, J. (1967) Investigations in the Foundations of Set Theory. From Frege to Godel: A Source Book in Mathematical Logic, 1879-1931. Source Books in the History of the Sciences, Harvard University Press, 199-215.
[18]	Zermelo, E. (1930) Uber Grenzzablen und Mengenbereiche. Fundamenta Mathematicae, 16, 29-47.