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On the axiom of extensionality, Part II1

Published online by Cambridge University Press:  12 March 2014

R. O. Gandy
R. O. Gandy*
Affiliation:
The University, Leeds, England
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Extract

In this paper it is shown that, if a certain form of Gödel-Bernays set theory which does not include the axiom of extensionality is consistent, then so is the whole system of set theory. The general line of argument is similar to that used in Part I. § 1 describes a reformulation of set theory in which the class-existence axioms are replaced by the use of abstracts. In § 2 some standard theorems, including the theory of the ancestral, are proved without using the axiom of extensionality. In § 3 the appropriate inner model is defined, and the validity in it of the most of the axioms is demonstrated. § 4 deals with the remaining axioms (of infinity and of choice).

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 24 , Issue 4 , December 1959 , pp. 287 - 300
Copyright
Copyright © Association for Symbolic Logic 1952

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Footnotes

1

Part I appeared in this Journal, vol. 21 (1956), pp. 36–48.

References

2 The consistency of the continuum hypothesis and of the generalized continuum-hypothesis with the axioms of set theory, Princeton (Princeton University Press), 1940, 66 pp. Second printing 1951, 74 pp.

3 A system of axiomatic set theory, this Journal, part I, vol. 2 (1937), pp. 65–77, and part II, vol. 6 (1941), pp. 1–17, contain what is relevant to this paper.

4 For simplicity we have used particular letters lor the bound variables in these definitions; in applications a change of bound variable may be necessary to avoid collisions with the free variables of α, β, γ.

5 In the extensional system C′.4 could be replaced by but in the non-extensional system C′.4 serves to ensure that is an extensional property of classes.

6 This essential step for the proof of 2.11 is omitted by Gödel (op. cit. p. 19).

7 For the sake of definiteness, notions and operations have been defined with arbitrary terms as arguments; but the required properties of and are only provable when the arguments are sets

8 Note that B does not necessarily lie in the model.

9 Cf. proof of 8.51 in Gödel, op. cit., p. 32.