FOM: an application of reverse math to algebraic geometry

[Stephen G Simpson]

In my FOM posting of 14 Sep 1999 18:18:11, I referred to Friedman's
FOM posting of 17 May 1999 11:07:03.  I mentioned that Friedman had
used my 1988 reverse mathematics analysis of the Hilbert Basis Theorem
(``every ideal in a polynomial ring is finitely generated'') to obtain
upper and lower bounds for a certain interesting function that arises
in algebraic geometry.

I should have pointed out that Friedman's result can be stated very
simply and elegantly in a few lines.  I am not adding anything new
here.  I am merely restating Friedman's result.

Define j(n) to be the maximum length j of a descending chain of
algebraic sets X_1, X_2, ..., X_j in n-dimensional Euclidean space,
such that the degree of each X_i is exactly i.  

The finiteness of j(n) is a consequence of the Hilbert Basis Theorem.

Friedman's result is that j(n) grows like the Ackermann function.

I think this result of Friedman qualifies as a very nice application
of reverse mathematics to algebraic geometry.  I admit it does not
answer any question which has been posed by algebraic geometers.  But
I find that algebraic geometers readily understand it and get
interested in it.

Instead of algebraic sets in n-dimensional Euclidean space, we could
consider algebraic sets in n-dimensional complex space, and the same
result holds.  This is useful, because algebraic geometers tend to
find n-dimensional complex space more appealing and easier to deal
with than n-dimensional Euclidean space.

Nevertheless, I want to stress once again that reverse mathematics
does not need such applications in order to indirectly justify its
existence.  That is because reverse mathematics directly addresses
basic foundational questions which are of obvious general intellectual
interest, questions such as ``What are the appropriate axioms for core
mathematics?''.

-- Steve

PS 

I want to thank Arnon Avron for his FOM posting of 15 Sep 1999
14:20:01 in defense of the f.o.m. enterprise.  Also Jeff Ketland
15 Sep 1999 19:51:53 and Joe Shipman 15 Sep 1999 17:11:24.

A small correction.  Avron said

 > In his very agressive posting from September 9, B. Soare made the
 > following unbelievable claim:
 >
 >   "It is time for Reverse Mathematics to add something *new* to
 >   mathematics, or to get off the pot."

but actually Soare did not make this claim in any of his numerous,
aggressive FOM postings.  The truth is that Soare made it in one of
his angry, hostile, spam messages.