[FOM] Arithmetical soundness of ZFC

Nik Weaver nweaver at math.wustl.edu
Sun May 24 14:00:22 EDT 2009

Tim Chow wrote:

> Apparently you believe that "experience" with ZFC can
> legitimately give one confidence that ZFC is consistent
> ... I would argue that we have analogous grounds for
> believing that ZFC is arithmetically sound.  We've looked
> hard for false theorems of ZFC and haven't found any.

How would we know?

If you mean we haven't found any theorems of ZFC that
contradict theorems of Peano arithmetic, then you're
talking about consistency, since every theorem of PA
is also a theorem of ZFC.  But we agree that ZFC is
probably consistent.

(Or maybe we don't.  I mean I can't assign a numerical
probability to this statement, so based on your reaction
to my view that ZFC is more likely than not to be unsound,
perhaps you would argue that all we can say about ZFC's
consistency is "We just don't know"?)

Anyway, the question is whether ZFC might be consistent
but not arithmetically sound.


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