Does Singleton Set Meet Zermelo-Fraenkel Set Theory with the Axiom of Choice?

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1 Adv. Studies Theor. Phys., Vol. 5, 2011, no. 2, Does Singleton Set Meet Zerelo-Fraenkel Set Theory with the Axio of Choice? Koji Nagata Future University Hakodate ko i na@yahoo.co.jp Tadao Nakaura Keio University Science and Technology Abstract We show that a singleton set, i.e., {1} does not eet Zerelo- Fraenkel set theory with the axio of choice. Our discussion relies on the validity of Addition, Subtraction, Multiplication, and Division. Our result shows the current axioatic set theory has a contradiction even if we restrict ourselves to Zerelo-Fraenkel set theory, without the axio of choice. Matheatics Subject Classification: 03A10, Zerelo-Fraenkel set theory with the axio of choice, coonly abbreviated ZFC, is the standard for of axioatic set theory and as such is the ost coon foundation of atheatics. It has a single priitive ontological notion, that of a hereditary well-founded set, and a single ontological assuption, naely that all individuals in the universe of discourse are such sets. ZFC is a one-sorted theory in first-order logic. The signature has equality and a single priitive binary relation, set ebership, which is usually denoted. The forula a b eans that the set a is a eber of the set b which is also read, a is an eleent of b or a is in b ). Most of the ZFC axios state that particular sets exist. For exaple, the axio of pairing says that given any two sets a and b there is a new set {a, b} containing exactly a and b. Other axios describe properties of set ebership. A goal of the ZFC axios is that each axio should be true if interpreted as a stateent about the collection of all sets in the von Neuann universe also known as the cuulative hierarchy). The etaatheatics of ZFC has been extensively studied. Landark results in this area that is established the independence

2 58 K. Nagata and T. Nakaura of the continuu hypothesis fro ZFC, and of the axio of choice fro the reaining ZFC axios [1]. Mach literature concerning above topic can be seen in Refs. [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. Surprisingly, we show that a singleton set, i.e., {1} does not eet Zerelo- Fraenkel set theory with the axio of choice. Our discussion relies on the validity of Addition, Subtraction, Multiplication, and Division. Our result shows the current axioatic set theory has a contradiction even if we restrict ourselves to Zerelo-Fraenkel set theory, without the axio of choice. We use an established atheatical ethod presented in Refs. [18, 19, 20, 21, 22, 23, 24]. Assue all axios of Zerelo-Fraenkel set theory with the axio of choice is true. Let us start with a singleton set We treat here Addition. We have Thus we obtain 2. By using the obtained 2, we have Thus we obtain 3. By repeating this ethod, we have Thus we have the following set We assue Subtraction. We have Thus we obtain 0. By using the obtained 0, we have {1}. 1) 1+1=2. 2) 2+1=3. 3) 1, 2, 3,... 4) {1, 2,...}. 5) 1 1=0. 6) 0 1= 1. 7) Thus we obtain 1. We next treat Division. We have +1 =+. 8) ɛ +0 ɛ

3 Does singleton set eet Zerelo-Fraenkel set theory 59 Thus we obtain +. In what follows, eans +. Finally, we have the following set { 1, 0, 1, 2,..., + }. 9) Our ai is to show that the set 9) does not eet ZFC axios. We consider an expected value E. We assue E =0. 10) We derive the possible value of the product E E =: E 2 of the expected value E. It is We have E 2 =0. 11) E 2 ) ax =0. 12) The expected value E = 0) which is the average of the results of easureents is given by E = r l. 13) We assue that the possible values of the actually easured results r l are ±1. We have 1 E ) The sae expected value is given by l E = =1 r l. 15) We only change the labels as and l l. The possible values of the actually easured results r l are ±1. We have and {l l N r l =1} = {l l N r l =1} 16) {l l N r l = 1} = {l l N r l = 1}. 17) Here N = {1, 2,...,+ }. By using these facts we derive a necessary condition for the expected value given in 13). We derive the possible value of the product

4 60 K. Nagata and T. Nakaura E 2 of the expected value E given in 13). We have E 2 = = = r l The above inequality is saturated since and ) l =1 r l ) l =1 r l r l ) l =1 r l r l ) l =1 =1. 18) {l l N r l =1} = {l l N r l =1} 19) {l l N r l = 1} = {l l N r l = 1}. 20) We derive a proposition concerning the expected value given in 13) under the assuption that the possible values of the actually easured results are ±1 that is E 2 1. We derive the following proposition E 2 ) ax =1. 21) We do not assign the truth value 1 for the two propositions 12) and 21) siultaneously. We are in a contradiction. We do not treat all the other axios of Zerelo-Fraenkel set theory with that the axio of choice are true if we accept the existence of the singleton set {1}. Of course, our discussion relies on the validity of Addition, Subtraction, Multiplication, and Division. In conclusions we have shown that a singleton set, i.e., {1} does not eet Zerelo-Fraenkel set theory with the axio of choice. Our discussion has relied on the validity of Addition, Subtraction, Multiplication, and Division. Our results have shown that the current axioatic set theory has a contradiction even if we restrict our thoughts to Zerelo-Fraenkel set theory, without the axio of choice. Interestingly our discussion iplies that the faous truth-false set {0, 1} does not eet the ZFC axios. Therefore, we have to distinguish the current foralis of atheatics fro the truth-false arguentation, e.g., coputer science. In suary, coputer science is not always atheatics. It is an opinion of the authors, and generally this proble is open.

5 Does singleton set eet Zerelo-Fraenkel set theory 61 References [1] Zerelo-Fraenkel set theory - Wikipedia, the free encyclopedia. [2] Alexander Abian, The Theory of Sets and Transfinite Arithetic. W B Saunders. [3] and LaMacchia, Sauel, 1978, On the Consistency and Independence of Soe Set-Theoretical Axios, Notre Dae Journal of Foral Logic 19: [4] Keith Devlin, ). The Joy of Sets. Springer. [5] Abraha Fraenkel, Yehoshua Bar-Hillel, and Azriel Levy, ). Foundations of Set Theory. North Holland. Fraenkel s final word on ZF and ZFC. [6] Hatcher, Willia, ). The Logical Foundations of Matheatics. Pergaon. [7] Thoas Jech, Set Theory: The Third Millenniu Edition, Revised and Expanded. Springer. ISBN [8] Kenneth Kunen, Set Theory: An Introduction to Independence Proofs. Elsevier. ISBN [9] Richard Montague, 1961, Seantic closure and non-finite axioatizability in Infinistic Methods. London: Pergaon: [10] Patrick Suppes, ). Axioatic Set Theory. Dover reprint. Perhaps the best exposition of ZFC before the independence of AC and the Continuu hypothesis, and the eergence of large cardinals. Includes any theores. [11] Gaisi Takeuti and Zaring, W M, Introduction to Axioatic Set Theory. Springer Verlag. [12] Alfred Tarski, 1939, On well-ordered subsets of any set,, Fundaenta Matheaticae 32: [13] Tiles, Mary, ). The Philosophy of Set Theory. Dover reprint. Weak on etatheory; the author is not a atheatician. [14] Tourlakis, George, Lectures in Logic and Set Theory, Vol. 2. Cabridge Univ. Press.

6 62 K. Nagata and T. Nakaura [15] Jean van Heijenoort, Fro Frege to Godel: A Source Book in Matheatical Logic, Harvard Univ. Press. Includes annotated English translations of the classic articles by Zerelo, Fraenkel, and Skole bearing on ZFC. [16] Zerelo, Ernst 1908), Untersuchungen uber die Grundlagen der Mengenlehre I, Matheatische Annalen 65: , doi: /bf English translation in *Heijenoort, Jean van 1967), Investigations in the foundations of set theory, Fro Frege to Godel: A Source Book in Matheatical Logic, , Source Books in the History of the Sciences, Harvard Univ. Press, pp , ISBN [17] Zerelo, Ernst 1930), Uber Grenzzablen und Mengenbereiche, Fundaenta Matheaticae 16: 29-47, ISSN [18] K. Nagata, Eur. Phys. J. D 56, ). [19] K. Nagata and T. Nakaura, Int. J. Theor. Phys. 48, ). [20] K. Nagata and T. Nakaura, Int. J. Theor. Phys. 49, ). [21] K. Nagata, Int. J. Theor. Phys. 48, ). [22] K. Nagata and T. Nakaura, Adv. Studies Theor. Phys. 4, ). [23] K. Nagata and T. Nakaura, arxiv: [24] K. Nagata and T. Nakaura, Advances and Applications in Statistical Sciences, Volue 3, Issue 1, 2010), Page 195. Received: June, 2010