[FOM] strong hypotheses and the theory of N

[Monroe Eskew]

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