[FOM] strong hypotheses and the theory of N

[Harvey Friedman]

Here are two relevant observed facts.

1. Any two natural formal systems that interpret EFA = exponential  
function arithmetic, are comparable under interpretability.
2. Any two natural formal systems of set theory satisfying minimal  
requirements, are comparable under interpretations that preserve the  
ordered ring of natural number parts.

Much stronger forms of 2 are in fact observed.

Item 2 ensures that the provable arithmetic sentences are comparable.

Harvey Friedman

On Mar 14, 2010, at 5:19 AM, Monroe Eskew 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.