[FOM] Arithmetical soundness of ZFC

Timothy Y. Chow tchow at alum.mit.edu
Fri May 22 21:27:29 EDT 2009

Nik Weaver wrote:
> However, it is a sociological fact that many philosophers of mathematics 
> assume that all that matters is whether ZFC is consistent.

I can vouch for this (anecdotally).  My impression, though, is that this 
is simply because the philosophers in question don't understand the 
mathematical distinction between "ZFC is consistent" and "ZFC proves only 
true arithmetical theorems," and not because they *do* appreciate the 
difference clearly yet insist that consistency is all that matters.

> 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.

You didn't say it explicitly, but I think you're implicitly appealing to 
that principle.  Above you say:

1. arithmetical soundness is a rare property;
2. therefore ZFC is probably not arithmetically sound.

But 2 does not follow from 1.  We cannot assign a probability to "ZFC is 
not arithmetically sound" on the basis of 1, unless we have some reason to 
regard ZFC as a *random* formal system.  We just don't know.

Anyway, since we're starting to go in circles, let me introduce a 
different argument.  Apparently you believe that "experience" with ZFC can 
legitimately give one confidence that ZFC is consistent.  I am wondering 
why this is.  Perhaps it's because we have looked hard for inconsistencies 
and failed to find them?  Well, if that is the case, then 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.  
Surely whatever principle allows us to extrapolate from our experience to 
"ZFC is consistent" also allows us to extrapolate from our experience to 
"ZFC is arithmetically sound"?


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