Lou van den Dries
vddries at math.uiuc.edu
Sun Oct 26 21:04:07 EST 1997
A couple of quick remarks:
I formulated three statements 1., 2. and 3. I clearly did not assert
that 3. is known at present (it isn't, as far as I know.)
I hope this answers John Baldwin's question. The reason I formulated
1., 2. and 3. is because Harvey asked if results such as Faltings'
could be stated in a kind of headline form. I didn't do that either,
but i did show that, for example Siegel's theorem, which is in the
same tradition of number theory as Falting's, has a *consequence*
that can be formulated in the same headline style as (a consequence of)
the MRDP-theorem. This was statement 2.
The MRDP-theorem has certainly the virtue of being provable by only
elementary number theory and some basic facts on recursive and recursively
enumerable sets, as Martin Davis showed in his very nice article
in the American Math. Monthly (which I read with great interest as
an undergraduate). I don't think Siegel's theorem, let alone Faltings',
have that character. But these theorems do establish deep, unexpected,
and yes, fundamental links between arithmetic, geometry, and topology.
But yes, you do first have to learn a little about these topics
(and about number theory) before you can appreciate what these
theorems say (and not just some headline consequences of it).
I fail to understand what has been suggested by Steve (I am not sure
about Harvey) that number theorists have not given enough attention
to MRDP and similar issues of undecidability or independence. Much
of their work is so far "below" where MRDP is likely to be relevant
that they work almost in a different mental universe.
Hmm, this last statement is of course an exaggeration. When very
general issues are at stake, as in the Lang conjectures, I suppose they
do keep MRDP-like phenomena in mind. -Lou van den Dries-
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