[FOM] Arithmetical soundness of ZFC

Nik Weaver nweaver at math.wustl.edu
Fri May 22 01:59:36 EDT 2009

Tim Chow wrote:

> ZFC is highly structured, and so it is not clear that
> we have any reason to believe that it behaves like a
> random consistent extension of PA.

That's true.  Well, I think we are not so far apart at this
point.  You say that a random set of axioms is unlikely to
be arithmetically sound, but ZFC is structured, not random,
so "all we can say is that we don't know".  I say that we
have no particular reason to believe that ZFC is structured
in a way that would make it arithmetically sound, and as
this is an extremely special and rare property, it probably
is not. Perhaps we needn't argue this much further.  But
feel free if you like.

I do take issue with your attribution to me of the view
that "we expect (the arithmetical part of) ZFC to behave
like a random consistent extension of PA."  I don't think
I said that.


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

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