[FOM] Wittgenstein Inspired Skepticism

[tchow]

Harvey Friedman wrote:

> ZFC is clearly the currently undisputed foundation of mathematics in
> the following specific sense. Let me start with an anecdote. Charles
> Fefferman once served (at least) a term as principal Editor of the
> Annals of Mathematics. We did have an occasion to talk in person about
> f.o.m. some and the matter came up of what he viewed the standards of
> publication were for the Annals of Mathematics. On the issue of
> correctness, he brought up the standard that the proof must be readily
> formalizable in ZFC, and any use of additional axioms needs to be put
> in as an hypothesis to an implication.

This anecdote says absolutely *nothing* about whether ZFC is the 
undisputed foundation of mathematics.

First of all, it says only that ZFC is a useful acronym to quote in 
bureaucratic contexts where some convention is needed.  Nobody actually 
verifies, in any meaningful sense, that submissions are formalizable in 
ZFC.  The only exception might be certain papers in logic and set theory 
that need to be particularly careful about which axioms are being used.  
Here, the choice of ZFC as opposed to some other set of axioms does make 
some difference, but it has nothing to do with ZFC's status as a 
foundation for mathematics; again, it's just because it is convenient to 
have some kind of convention.

Secondly, even an ultrafinitist could be happy with this bureaucratic 
policy, because if for some reason we decided to enforce it, we would 
only need to agree on the ability to establish *provability of the 
theorems from ZFC*.  No commitment to ZFC itself is needed.

Tim