FOM: Cardinals in ZFC fragments

Adrian-Richard-David Mathias Adrian-Richard-David.Mathias at
Thu Aug 8 16:12:07 EDT 2002

>       (ii) ZF without foundation lets you prove that the transitive closure
> of a set (the set containing its members, its members's members, and so on)
> exists.  (I think this is useful in working with Aczel's Anti-Founded set
> theory.)  So, take as measure of the "size" of a set, the cardinality of
> its transitive closure.

Um, what does "cardinality" mean ? 

>      (iii) **CONJECTURE** At least in Aczel's ZF-AFA (though ??maybe not in
> plain ZF with neither foundation nor anti-foundation??) I think (hope?) you
> can prove that for any set there is a SET of all those sets whose
> transitive closures are no larger than that of the given set.  Call this
> thing-- if it exists-- the "pseudorank" of the given set.

One context in which the conjecture would be false: call a set a 
Quine atom if it equals its own singleton. 
Suppose that there is a proper class of Quine atoms 
(easily consistent with ZF-Foundation by using permutations of 
the universe as in Scott's paper on Quine's individuals). 
Each Quine atom is a transitive set 
of cardinality 1.  [In the case of singletons I have no problem defining 
cardinality.] So the conjecture fails. 

[AFA, of course, implies that there is at most one Quine atom.]  

A R D Mathias

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