[FOM] Truth and set theory

Thomas Forster T.Forster at dpmms.cam.ac.uk
Wed Dec 5 02:15:37 EST 2007

Richard Kaye has done some interesting work on matters closely related
to this.  It turns out that the status of transitive containment (TCo: 
``every set has a transitive superset'') in ZF- is very sensitive to 
whether or not one adds the *negation* of infinity. I *think* (if i 
remember correctly) that Kaye and his co-author showed that transitive 
containment is a theorem of ZF- + not-infinity but is not a theorem of 
ZF-.  This may matter: after all, TCo is important in proving at least
some versions of the recursion theorem and one obviously needs recursion
in truth-definitions.  One has to tread very carefully!

If the Wang paper you allude to is the TAMS paper (``Truth definitions
and consistency proofs'') that I have in mind then you should read it:
it's actually very well written and quite clear.  Are you quite sure
that it is NBG- that he gives the truth-definition in?  Not the version
with the impredicative class existence scheme..?


On Tue, 4 Dec 2007, praatika at mappi.helsinki.fi wrote:

> Dear FOMers
> Let us denote ZF (NBG) withouth the axiom of infinity as ZF- (NBG-). Z is
> the original Zermelo set theory, without the axiom of replacement.
> Now apparently Wang (1952) showed that it is possible to give a (materially
> adequate) truth-definition for the infinitary Z in finitistic set theory
> NBG-. This is interesting enough.
> But is there some reason why one could not also give a truth definition for
> ZF, again in NBG- ? (Z and ZF have, after all, the same language)
> Is anyone here familiar with this stuff?
> Best, Panu
> Panu Raatikainen
> Ph.D., Academy Research Fellow,
> Docent in Theoretical Philosophy
> Department of Philosophy
> University of Helsinki
> Finland
> E-mail: panu.raatikainen at helsinki.fi
> http://www.mv.helsinki.fi/home/praatika/
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