By A.A. Fraenkel, Y. Bar-Hillel
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Extra resources for Foundations of Set Theory
Mostowski 51a and 53. Mostowski 55 (p. 20) gives a modification, due to Tarski of -4xiom V which makes the system finitizable; thus an essential weakening of the system is achieved. The modification consists in admitting in Axiom V only predicates in which the quantifiers are restricted in a certain way. 8 ) Quine 37 and 53. See the proof in Hailperin 44 which is rather surprising since Quine’s original class axiom schema is impredicative. Cf. below § 7. “CONSTRUCTIVE” AXIOMS OF GENERAL SET THEORY 43 The null-set will be denoted by ‘0’1).
42) is here postponcd until 3 5. Instead of dogmatically introducing the various axioms we shall precede them by inl’ormal remarks apt to point out the purpose of each axiom and to investigate its consequences. I n accordance with the character of the axiomatic method in general the axioms have essentially been chosen by means of an a posteriori analysis of the methods of Cantor’s “genetic” theory, with restrictions meant (at least) to avoid antinomies. The simple operation of uniting two different sets to a set 1) is expressed by the AXIOM(11) OF PAIRING 2 ) .
The sets UC and UA prove to contain the same members, hence to be equal. In a similar way the general associativity of the union operation can be shown. UA contains the members of three different sets u, b, c. Proceeding further in the same or in a similar way - provided we have at our disposal more than three different sets - we may obtain more and more comprehensive sets. But in spite of the considerable strength of Axiom 111, a glance at Cantor’s theory shows that the axioms of pairing and of sum-set do not give us sufficient liberty in forming new sets, even though fairly strong assumptions about the existence of sets to start with be made.