FOM: large cardinals and P vs NP

[Mitchell Steven Spector]

On Tue, Aug 07, 2001 at 10:59:06AM -0700, Robert M. Solovay wrote:
> 
> 
> On Sat, 4 Aug 2001, Mitchell Steven Spector wrote:
> 
> >    If the theory ZFC + C is omega-consistent, then A must be
> > true, even if A is not arithmetical but just Sigma-1-1.
> > (Alternatively, if A is a Pi-0-1 sentence of number theory,
> > then the consistency of the theory ZFC + C is sufficient to
> > prove that A is true.)
> > 
> 	This is not correct. The statement "T is omega-consistent" is
> arithmetic. 
> 
> 	For a specific example, take "ZFC + 'ZFC is *not*
> omega-consitent'.
> 
> 	This theory is omega-consistent, but proves the arithmetically
> false statement 'ZFC is omega-inconsistent'.
> 
> 	I think you are confusing omega-consistent with "has an
> omega-model".
> 
> 	--Bob Solovay

You're right, of course -- I meant "has an omega-model."
Thanks for the correction.

I still would be interested in an example of a sentence
phi that appears at first to be a large cardinal axiom,
but that turns out on further investigation to have the
following properties:

(1) phi is consistent with ZFC, and

(2) ZFC + phi proves an arithmetically false statement.

[Presumably statement (1), the consistency of phi with ZFC,
would have to be proven relative to the consistency of some
"normal" large cardinal axiom, since otherwise phi wouldn't
have looked like a large cardinal axiom to begin with.]

Mitchell