# [FOM] Consistency strength order

Robert M. Solovay solovay at Math.Berkeley.EDU
Fri May 2 21:19:19 EDT 2008

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 But the following (old unpublished) example of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.

--Bob Solovay

On Wed, 30 Apr 2008, pax0 at seznam.cz wrote:

> Can someone of Dear FOMers make the notion of  "Consistency strength order" sufficiently clear to me?
> For first order theories S, T (in the same language) we define that T has higher consistency strength then S:
> S \le_{Cons} T iff P_S \subseteq P_T, where P_S and P_T are Pi^0_1 consequences of S or T, respectively.
> Second definition could be
> S  \le_{Cons} T iff U |-- Con(T) --> Con(S), where U is some weak base theory.
> What is the relation between these two definitions, and how should the right definition read?
> Thank you, J.P.
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