[FOM] Consistency strength order

[hendrik@topoi.pooq.com]

On Fri, May 02, 2008 at 06:19:19PM -0700, Robert M. Solovay wrote:
> I can't answer your question as to what the right consistency strength 
> notion is, in gneeral. But the following (old unpublished) example of
> mine shows that the two notions can diverge. (The theory in question is 
> rather artificial.)
> 
>  	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