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Christian Thalmann scripsit:

> The whole building of math is built on the axioms of the
> real numbers, which can't be proven, but must be assumed
> to be true for the rest of math to work.  Everything else
> in math is proven logically on the basis of these axioms.

I think you are confusing the foundational status of the reals and
the integers here.  The real numbers have a complete and consistent
axiomatization; the integers (as shown by Goedel's Theorem) do not.
This is probably why analysis is so full of powerful methods and
results and advances progressively, whereas number theory is
wonky and irregular, and advances only when fundamental connections
with the rest of mathematics (typically analysis) are found.
(And why it's number theory that has my heart, for I love
wonky and complicated domains.)

--
Values of beeta will give rise to dom!          John Cowan
(5th/6th edition 'mv' said this if you tried    http://www.ccil.org/~cowan
to rename '.' or '..' entries; see              [log in to unmask]
http://cm.bell-labs.com/cm/cs/who/dmr/odd.html)