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Stanford Encyclopedia of Philosophy
Supplement to Infinitary Logic

Definition of the Concept of Admissible Set

A nonempty transitive set A is said to be admissible when the following conditions are satisfied:
(i) if a, b A, then {a, b} A and A A;

(ii) if a A and X A is 0 on A, then X a A;

(iii) if a A, X A is 0 on A, and xay(<x,y> X), then, for some b A, xayb(<x,y> X).

Condition (ii) -- the 0-separation scheme -- is a restricted version of Zermelo's axiom of separation. Condition (iii) -- a similarly weakened version of the axiom of replacement -- may be called the 0-replacement scheme.

It is quite easy to see that if A is a transitive set such that <A, | A> is a model of ZFC, then A is admissible. More generally, the result continues to hold when the power set axiom is omitted from ZFC, so that both H() and H(1) are admissible. However, since the latter is uncountable, the Barwise compactness theorem fails to apply to it.

Copyright © 2000 by
John L. Bell

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First published: January 23, 2000
Content last modified: September 20, 2000