Does Gödel's Theorem Matter to Mathematics?
Gina Bari Kolata · Science · 1982
Publisher Summary This chapter focuses on Kurt Godel's theorem contribution to mathematics. The recent discovery of two natural but undecidable statements indicates that Godel's theorem is more than just a logician's trick. Godel shook the world of mathematics by showing that there are statements in every logical system whose truth or falsehood simply cannot be determined by staying within the system. The Paris–Harrington theorem is a slight variation of Ramsey's theorem. According to Ramsey's theorem, if there is an infinite set and a color, say red or blue, is assigned arbitrarily to each pair of members of the set, then an infinite subset can be found, all of whose pairs are red or all of whose pairs are blue. Paris and Harrington used model theory, a standard method of mathematical logic, to show that their theorem is undecidable in Peano arithmetic. The more natural but undecidable theorems that are found, of course, the more willing mathematicians are to believe that Godel's theorem might apply to important results.