Adv. Studies Theor. Phys., Vol. 5, 2011, no. 2, 57 - 62

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

Koji Nagata

Future University Hakodate ko mi [email protected]

Tadao Nakamura

Keio University Science and Technology

Abstract We show that a singleton set, i.e., {1} does not meet Zermelo- Fraenkel set theory with the axiom of choice. Our discussion relies on the validity of Addition, Subtraction, Multiplication, and Division. Our result shows the current axiomatic set theory has a contradiction even if we restrict ourselves to Zermelo-Fraenkel set theory, without the axiom of choice.

Mathematics Subject Classification: 03A10, 11-00

Zermelo-Fraenkel set theory with the axiom of choice, commonly abbre- viated ZFC, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics. It has a single primitive ontological notion, that of a hereditary well-founded set, and a single ontological assump- tion, namely 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 primitive binary relation, set membership, which is usually de- noted ∈. The formula a ∈ b means that the set a is a member of the set b (which is also read, “a is an element of b”or“a is in b”). Most of the ZFC axioms state that particular sets exist. For example, the axiom of pairing says that given any two sets a and b there is a new set {a, b} containing exactly a and b. Other axioms describe properties of set membership. A goal of the ZFC axioms is that each axiom should be true if interpreted as a statement about the collection of all sets in the von Neumann universe (also known as the cumulative hierarchy). The metamathematics of ZFC has been extensively studied. Landmark results in this area that is established the independence 58 K. Nagata and T. Nakamura of the continuum hypothesis from ZFC, and of the axiom of choice from the remaining ZFC axioms [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 meet Zermelo- Fraenkel set theory with the axiom of choice. Our discussion relies on the validity of Addition, Subtraction, Multiplication, and Division. Our result shows the current axiomatic set theory has a contradiction even if we restrict ourselves to Zermelo-Fraenkel set theory, without the axiom of choice. We use an established mathematical method presented in Refs. [18, 19, 20, 21, 22, 23, 24]. Assume all axioms of Zermelo-Fraenkel set theory with the axiom of choice is true. Let us start with a singleton set

{1}. (1)

We treat here Addition. We have

1+1=2. (2)

Thus we obtain 2. By using the obtained 2, we have

2+1=3. (3)

Thus we obtain 3. By repeating this method, we have

1, 2, 3,... (4)

Thus we have the following set

{1, 2,...}. (5)

We assume Subtraction. We have

1 − 1=0. (6)

Thus we obtain 0. By using the obtained 0, we have

0 − 1=−1. (7)

Thus we obtain −1. We next treat Division. We have +1 lim =+∞. (8) →+0 Does singleton set meet Zermelo-Fraenkel set theory 59

Thus we obtain +∞. In what follows, ∞ means +∞. Finally, we have the following set

{−1, 0, 1, 2,... ,+∞}. (9)

Our aim is to show that the set (9) does not meet ZFC axioms. We consider an expected value E. We assume

E =0. (10)

We derive the possible value of the product E × E =: E2 of the expected value E.Itis

E2 =0. (11)

We have

2 (E )max =0. (12)

The expected value (E = 0) which is the average of the results of measurements is given by

m rl E = lim l=1 . (13) m→∞ m

We assume that the possible values of the actually measured results rl are ±1. We have

−1 ≤ E ≤ +1. (14)

The same expected value is given by

m  rl E = lim l =1 . (15) m→∞ m

We only change the labels as m → m and l → l. The possible values of the actually measured results rl are ±1. We have

{l|l ∈ N ∧ rl =1} = {l |l ∈ N ∧ rl =1} (16) and

{l|l ∈ N ∧ rl = −1} = {l |l ∈ N ∧ rl = −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 60 K. Nagata and T. Nakamura

E2 of the expected value E given in (13). We have E2   m m rl  rl = lim l=1 × lim l =1 m→∞ m m→∞ m   m m l=1 l=1 = lim · lim rlrl m→∞ m m→∞ m   m m l=1 l=1 ≤ lim · lim |rlrl | m→∞ m m→∞ m   m m  = lim l=1 · lim l =1 =1. (18) m→∞ m m→∞ m

The above inequality is saturated since {l|l ∈ N ∧ rl =1} = {l |l ∈ N ∧ rl =1} (19) and {l|l ∈ N ∧ rl = −1} = {l |l ∈ N ∧ rl = −1}. (20) We derive a proposition concerning the expected value given in (13) under the assumption that the possible values of the actually measured results are ±1 that is E2 ≤ 1. We derive the following proposition 2 (E )max =1. (21) We do not assign the truth value “1” for the two propositions (12) and (21) simultaneously. We are in a contradiction. We do not treat all the other axioms of Zermelo-Fraenkel set theory with that the axiom 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 meet Zermelo-Fraenkel set theory with the axiom of choice. Our discussion has relied on the validity of Addition, Subtraction, Multiplication, and Division. Our results have shown that the current axiomatic set theory has a contradiction even if we restrict our thoughts to Zermelo-Fraenkel set theory, without the axiom of choice. Interestingly our discussion implies that the famous truth-false set {0, 1} does not meet the ZFC axioms. Therefore, we have to distinguish the current formalism of mathematics from the truth-false argumentation, e.g., computer science. In summary, computer science is not always mathematics. It is an opinion of the authors, and generally this problem is open. Does singleton set meet Zermelo-Fraenkel set theory 61

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[24] K. Nagata and T. Nakamura, Advances and Applications in Statistical Sciences, Volume 3, Issue 1, (2010), Page 195.

Received: June, 2010