# [FOM] strong hypotheses and the theory of N

Monroe Eskew meskew at math.uci.edu
Sun Mar 14 05:27:53 EDT 2010

Clarification:  To rule out trivial counterexamples, I want \phi(T) to
be of the form (Con(T) & \psi(T)).  Obviously the Con(T) part is not
going to be decidable but I would like the \psi part to be decidable.
In other words I am looking for a criterion that says if this T is
consistent at all, then it will not say anything false about natural
numbers.

On Sun, Mar 14, 2010 at 1:19 AM, Monroe Eskew <meskew at math.uci.edu> wrote:
> It would seem a reasonable requirement that all strong hypotheses
> which set theorists explore or use should all agree on the theory of
> natural numbers.  So then how do we know whether whatever large
> cardinal, forcing axiom, determinacy statement, etc. we're looking at
> will not say anything about omega that a different such hypothesis
> contradicts?  Is there some computable property of theories \phi(T)
> that we can check in advance, to make sure that all T satisfying \phi
> have pairwise consistent theories about naturals?  Here I want of
> course \phi to be useful in practice so that the standard strong
> hypotheses in set theory like large cardinals are able to satisfy it.
>
> Monroe
>