[FOM] Consistency strength order
pax0 at seznam.cz
Tue May 6 07:52:58 EDT 2008
I do not know how Robert Solovay's phi sentence looks like,
but your observation is trivially false:
if phi were "there is an inaccessible" then 1) would fail: one particular arithmetical
consequence in T would be Con(ZFC), but this Godel number is not a consequence of ZFC (unless ZFC is inconsistent, of course).
I would like too see what Robert's phi says.
> > Assume that "ZFC + "there is an inaccessible cardinal" is
> > consistent. (Call this theory ZFCI for short.)
> > Then there is a theory T, obtained by adjoining a single sentence
> > phi to ZFC such that:
> > 1) T has the same arithmetical consequences as ZFC.
> > 2) It is finitistically (that is, in "primitive recursive
> > arithmetic") provable that "Con(T) iff Con(ZFCI)".
> > The proof has much in common with the proofs of my old results
> > on the provability logic of Peano Arithmetic.
> Either this is trivial, or I completely misunderstand it.
> Can phi not just be "there is an inaccessible cardinal"? In that case,
> T would be ZFCI.
> In either case, I would appreciate an explanation of what you meant.
> -- hendrik boom
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> FOM at cs.nyu.edu
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