From: kramsay@aol.com (KRamsay)
Newsgroups: sci.math
Subject: Re: Fermat, Gödel, Wiles
Date: 30 Sep 1998 21:05:52 GMT

In article <36115b06.25041509@news.prosurfr.com>,
jsavard@tenMAPSONeerf.edmonton.ab.ca (John Savard) writes:
|The proof that undecidable statements exist is based on statements
|having a very special form, such as "This statement cannot be proved
|using the axioms of classical number theory". There hasn't been any
|evidence that any mathematical statements of any real importance could
|belong to this category, so this possibility is not too worrisome.

Some progress has been made toward finding "ordinary" looking
statements which are independent of particular axiom systems. One
famous example is the Paris-Harrington statement, which is independent
of Peano arithmetic (the first-order theory of arithmetic). For each 
integer k>0 and integer n>0 there is an integer N>n with the property 
that if to every subset of {n,...,N} is assigned an integer in 
{0,...,k-1} (which we think of as a "color"), then there is a subset S 
of {n,...,N} which is "monochromatic" in the sense that any two 
subsets of S having the same number of elements are assigned the same 
color.

Keith Ramsay     "Thou Shalt not hunt statistical significance with
kramsay@aol.com   a shotgun." --Michael Driscoll's 1st commandment