[FOM] Arithmetical soundness of ZFC

Nik Weaver nweaver at math.wustl.edu
Fri May 22 01:21:34 EDT 2009

Laureano Luna wrote, in response to my comment that "it is
fairly common for philosophical defenses of set theory to
... assume that the only issue [about ZFC] is the question
of consistency":

> I wonder how this could be the case after Godel.
> Incompleteness entails the existence of many consistent
> but mutually incompatible arithmetical extensions of any
> arithmetical system. I'd say this compels to distinguish
> soundness from consistency.

You're quite right --- in fact Godel himself apparently
made the argument that the formalist goal of proving
infinitary mathematics consistent was not adequate, as
it still might not be arithmetically sound.  (At least,
Smorynski quotes him to this effect, but without giving
a primary reference.  The quote is reproduced near the
bottom of page 6 of my "indispensable" paper.)

However, it is a sociological fact that many philosophers
of mathematics assume that all that matters is whether
ZFC is consistent.  I could give plenty of examples if
you insist (though I don't think there's any real point
to doing this).  Fred Muller's paper is hardly unique in
this regard.


Nik Weaver
Math Dept.
Washington University
St. Louis, MO 63130 USA

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