In article <email@example.com>, "Gene Ward Smith" <firstname.lastname@example.org> wrote:
> Gerry Myerson wrote: > > > We do hear that all of mathematics can be expressed in ZFC (or at > > any rate all the mathematics that isn't specifically designed to be > > done outside ZFC), and this makes the independence results of set > > theory a bit more worrisome than those of geometry. > > Isn't this a contradiction? The independence results show that > mathematics can't all be reduced to ZFC. Of course, the independence > results themselves involve inner models and forcing models, but it > seems to me the obvious way to interpret it all is that there are > reasonable ways to add axioms to ZFC.
Let me try to get at what the worry is.
How do we know the Wiles-Taylor solution of the Fermat problem is correct? If you push a mathematician hard enough on this, or any similar question, never accepting an answer as final until you come down to bedrock, the eventual answer will be that it can all be formalized and validated in ZFC. But ZFC doesn't even decide a simple question like, is there an infinite subset of the reals that can be put into one-one correspondence with neither the integers nor the reals? If ZFC doesn't capture such a basic part of our mathematical world, what reason is there to think it's a good framework for our mathematical work? and what good does it do to say that you can carry out the Wiles-Taylor proof in it?
The independence of the parallel postulate doesn't make anyone worry, because no one says you can carry out the proof of Fermat in Euclidean geometry. The independence of the continuum hypothesis says we really don't understand sets, and that's a worry.
-- Gerry Myerson (email@example.com) (i -> u for email)