Re:Does Gödel Matter?
by Tichy on Monday December 22, 2008 @10:15AM
(#406424)
Hm, I thought set theory already contains arithmetic, so wouldn't Goedel's proof show that such axioms without paradoxes can't be found? I thought that is why he is so famout - for years it had been the goal of mathematics to find the ultimate axioms, and then Goedel showed that it can't be done, ever.
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The second theorem tells us that we can't prove the consistency of set theory without using an even more powerful system. This doesn't mean that set theory is inconsistent, it just means that we can't prove its consistency in any meaningful way. So yes, Gödel indeed shows that there are no 'ultimate axioms' that contain everything of interest in mathematics.
Although it does lead to a mathematically uninteresting paradox: if you let A be the axioms of set theory and you add an axiom P which states that A proves x and not x, (i.e. set theory is inconsistent) then A' = A and P is still consistent.
For clearity: A' = (A and P)