[FOM] Consistency strength order

Robert M. Solovay solovay at Math.Berkeley.EDU
Mon May 5 15:27:15 EDT 2008

On Sat, 3 May 2008, hendrik at topoi.pooq.com wrote:

> 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 But the following (old unpublished) example of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.

 	If phi is taken as "there is an inaccssible cadinal" then ZFC + 
phi has new arithmetical consequences (not proved in ZFC). For example 
the assertion  "Con(ZFC)" is provable in "ZFC + phi" but not in ZFC.

> In 
either case, I would appreciate an explanation of what you meant. >
> -- hendrik boom

 	My statement seems perfectly clear to me. What is it that you 
don't understand.

 	--Bob Solovay

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