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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
*x**a**y*(<*x,y*>
*X*), then, for some *b*
*A*,
*x**a**y**b*(<*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

*jbell@julian.uwo.ca*
Return to Infinitary Logic

*First published: January 23, 2000 *

*Content last modified: September 20, 2000 *