[FOM] Number theorist's interest in bounds
Harvey Friedman
friedman at math.ohio-state.edu
Sat Apr 8 23:35:32 EDT 2006
I just received a response from one of the three unnamed number theorists.
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"Some "effectivity" results are more interesting than others. Here's
one that I consider interesting. I don't speak for anybody else, but
I would expect that many other people would find it interesting too.
An effective version of Faltings' theorem that contained a bound on
the height (size of numerators and denominators) of the solutions
would give us an effective algorithm for finding all rational
points. That is a fundamental problem, and I would consider such a
thing very worthwhile.
On the other hand, an effective version of Faltings' theorem that
only provided a bound on the number of rational points would be less
interesting, because it would not (by itself) give us such an
algorithm. But such an effective version might still be interesting:
for example if that bound depended only on the genus of the curve,
that would be new and important information."
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The situation in the second paragraph above is precisely the situation of a
Pi03 sentence with a classical proof that is made constructive and in fact
into a Pi01 sentence.
Harvey Friedman
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