From: jbaez@zermelo.mit.edu (John C. Baez)
Subject: Re: You know, the integers (was: Re: Stupid question about FLT)
Organization: MIT Department of Mathematics, Cambridge, MA
Date: Tue, 21 Jul 92 16:35:33 GMT

In article <1992Jul21.132554.152734@ns1.cc.lehigh.edu> fc03@ns1.cc.lehigh.edu (Frederick W. Chapman) writes:

>I can't speak for anyone else, but I find the notion that the
>consistency of ZFC has not yet been established to be the most
>singularly disturbing mathematical news to ever reach my ears, given
>that ZFC is intended to serve as a foundation for the rest of
>mathematics.

The carefully guarded secret about these "consistency proofs" is that
for you to believe them you must have faith in the consistency of the
system in which you are doing the proof.  Suppose you could prove
Con(ZF) in ZF.  Well, if you are smart and know Goedel's theorem you
know you're in deep trouble at this point since ZF must be inconsistent.
But even if you weren't so smart you could realize that it's still quite
possible for ZF to be inconsistent, in which case one could prove
EVERYTHING in ZF, e.g., Con(ZF).  

Now consider the propositional calculus.  There's a consistency proof
for it.  However, I claim that this proof isn't worth much if what you
are seeking is reassurance that your ideas on logic aren't screwed up.
For the reasoning used in the proof of consistency of the propositional
calculus, while usually left informal, is no more "self-evidently
consistent" than the propositional calculus itself.  It's possible that
our ideas on logic are somehow very deeply confused and that one day
someone will come up with a proof of P & not(P) in the propositional
calculus.  Of course, in this case our ideas on logic would be revealed
to be a bunch of baloney - including our proof of the consistency of the
propositional calculus.

For a while I was planning on writing a short story entitled "The
inconsistency of the propositional calculus", that would imagine this
happening.  

I used to crave certainty and this sort of thing bugged me.  Now I'm
fairly used to it -- as well as the fact (which once seemed disturbing)
that no r.e. set of axioms about the integers will have a unique model
up to elementary equivalence, and NO set of axioms about the integers
will have a unique model up to isomorphism.   It irritates Torkel when I
say that this means we're not quite sure what the "real" integers ARE.
I think the sense in which I mean this was most clearly indicated by my
game show in which one tried to figure out which mathematician was using
the real integers and which two were fakes.  (What's the name of that
show, anyway?  Truth or Consequences?)  

Please don't think I'm losing any sleep over this.  I like to think of
it this way.  I have a notion - "the integers" - and I try to capture
this notion in some axioms and prove some things using the axioms.  All
I get is what I pay for, I should not expect the axioms to answer all possible
questions, I should not expect the axioms to be categorical, and I
should not expect to get some sort of guarantee that the axioms are
consistent, other than the empirical fact that nobody has found an
inconsistency yet.  This may seem tough but it's certainly no worse than
the rest of life.