[FOM] Consistency strength order

pax0@seznam.cz pax0 at seznam.cz
Tue May 6 07:52:58 EDT 2008

Dear Hendrik,
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.
Hendric wrote:
>  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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