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From: hensm@csc2.essex.ac.uk (Henson M C)
Newsgroups: comp.theory
Subject: Re: A set F of functions from F to F ??
Summary: axiom of foundation in ZF, Aczel's non-well founded sets
Message-ID: <4648@servax0.essex.ac.uk>
Date: 23 Jan 91 14:05:07 GMT
References: <16470@gremlin.nrtc.northrop.com>
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Reply-To: hensm@essex.ac.uk (Henson M C)
Organization: University of Essex, Colchester, UK
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 In ZF one has the axiom of foundation which rules out sets which have
 themselves as members. Levy's text [Springer Perspectives in Mathematical
 logic series: "Basic Set Theory" ] is a good presentation.

 In intuitionistic set theories like IZF this axiom gets replaced by
 epsilon-induction which is classically equivalent to foundation. This
 is done because foundation implies excluded middle which is not universally
 valid in intuitionistic theories. Beeson's text ["Foundations of Constructive
 Mathematics", Springer] is a good presentation.

 Peter Aczel has written a book on non-well founded sets which is very nice
 and published in the CLSI monograph series.