FOM: Cardinals in ZFC fragments
Adrian-Richard-David.Mathias at helios.univ-reunion.fr
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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