[FOM] arithmetical soundness of ZFC

[Monroe Eskew]

On Sun, May 17, 2009 at 8:10 AM, Nik Weaver <nweaver at math.wustl.edu> wrote:
> I am not just playing devil's advocate.  I think it is
> quite likely that ZFC is consistent but not arithmetically
> sound.  I spell out this objection in more detail in
> Section 2 of my "indispensable" paper (announced in FOM
> post # 013607).

This point of view seems to require there to be a unique intended
model of arithmetic, to make sense of what "soundness" means here.
But here's an argument for why you shouldn't be too worried about the
arithmetical soundness of ZFC:

If you have a natural (transitive) model of ZFC, then it has as a
definable submodel, something isomorphic to the intended model of
natural numbers.  In general, if B is a definable submodel of A, and A
satisfies a theory T, then if T proves a sentence S that has its
quantifiers relativized to the formula defining B, then B satisfies
S', where S' is S without relativization  So given a natural model of
ZFC, if ZFC proves something about natural numbers, then it must be
true in the intended model of arithmetic.  To deny this conclusion
would require denying the possibility of a natural model of ZFC.  From
my perspective, the existence of such a model is a fairly weak claim,
and so its negation is fairly strong.  Hence denying the conclusion
requires a strong and, in my opinion, unintuitive claim.  It's not
quite as strong as saying ZFC is inconsistent, but it's close.

Best,
Monroe