Re: Set Theory: Should you believe?

In article <1152843662.206607.94160@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"Gene Ward Smith" <genewardsmith@xxxxxxxxx> 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 (gerry@xxxxxxxxxxxxxxx) (i -> u for email)
.

Relevant Pages

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