[FOM] Arithmetical soundness of ZFC (platonic)

[Timothy Y. Chow]

Nik Weaver wrote:
> From what I've read on the FOM list, I get the impression that
> people basically fall into two camps.  Some want to carefully
> build foundations up from the bottom, starting with principles
> in which we have complete confidence and demanding a thorough
> justification of any proposed extension.
[...]
> The other group takes an "anything goes" attitude, the idea being
> that we should look for the most powerful axioms we can find, and
> if something turns out to be inconsistent we give it up.  That's
> fine, but my feeling is that if that is your approach then don't
> claim that what you're doing is arithmetically sound, because if
> your axioms weren't sound you wouldn't have any reliable way of
> knowing this.

Perhaps someone who is more familiar with the history of ZFC can correct 
me if I'm wrong, but my impression is that the genesis of ZFC falls into 
neither of these alleged camps.  We got to ZFC from ZF and to ZF from Z, 
and the extensions were motivated neither by the desire to find the most 
powerful axioms nor the desire to be totally safe from error.  A large 
part of the motivation was to capture actual mathematical practice as 
simply and elegantly as possible.

Nowadays we take it for granted that ZFC suffices for virtually all 
mathematics, and we tend to forget what an achievement it was to unify
the disparate areas of mathematics under a single, simple conceptual 
framework.  When you're not yet sure that such a unification is even 
possible, your main concern is to produce a "proof of concept."  It
would be a case of premature optimization to strive for the most 
parsimonious solution before finding any solution at all.

So it seems to me that the confidence in ZFC comes down to this: ZFC is a 
pretty good approximation to actual mathematical practice, and most of us 
feel that actual mathematical practice is sound (with the occasional 
unsoundness being relatively quickly detected and eliminated).  Though 
most mathematics uses much less than ZFC, some of it uses more (e.g., 
Grothendieck and Voevodsky, to mention just two top mathematicians, felt 
free to go slightly beyond ZFC in their pursuit of real mathematical 
problems).  The general feeling is that if there were something wrong with 
ZFC then we would have noticed something funny by now.

Such a trusting attitude may not be philosophically justified, of course.  
However, I think that Weaver's "anything goes" argument attacks a straw 
man, since ZFC did not arise that way.

Tim