[FOM] Arithmetical soundness of ZFC

[Monroe Eskew]

On Sun, May 24, 2009 at 2:37 PM, Nik Weaver <nweaver at math.wustl.edu> wrote:
> Well, what would evidence of the arithetical soundness of ZFC look like?

Perhaps it would be an argument that it is unlikely for ZFC to be
unsound.  You said you think unsoundness is likely, simply because we
have no particular reason to think it is sound.  Here's a shot at a
reason:

Suppose ZFC is consistent but unsound.  Then it has a model.  Let A be
an arbitrary model of ZFC.  There is a unique object in A satisfying
the definition of "omega."  Let N_A be the collection of x such that A
satisfies "x is a member of omega."  Now whatever the natural numbers
really are, you should agree that we can embed them into N_A.  Send 0
to the empty set and inductively extend the map, sending n to the nth
successor of the empty set.  We can prove by induction that the
arithmetical operations are preserved.  If this map is onto, then we
have an isomorphism.  This implies that whatever ZFC proves about
omega must be true of the natural numbers.  Therefore if ZFC is
unsound, then all models of ZFC have nonstandard omega.  To me this
seems unlikely.  Probably, at least some of them have standard omega.
I see no particular reason that there is something lurking in the
axioms causing omega to always be nonstandard.

Best,
Monroe