The second theorem says that the consistency of a system cannot be proved within that system (sufficiently strong, etc).

If a contradiction is discovered in set theory, either it will be fixed by a change to the axioms, or another set of foundational axioms altogether will become the standard foundation of mathematics. Perhaps the canonical example of this is naive set theory itself - paradoxes were discovered, but set theory was interesting enough that mathematicians looked for alternate definitions that did not contain the paradoxes, and they succeeded.

There are two things to note here. First, axiomatization was a useful tool in distinguishing various types of set theory and defining the ones that did not contain the (known) paradoxes. Second, the full formal treatment of these set theories came relatively late to mathematics, as did widespread use of formal, axiomatic methods in general. Rigorous axiomatic formalisms are very useful mathematical tools, but mathematics got along without them for a long time (not even Euclid counts, by modern standards) and can do so again if necessary. It is unlikely that this will happen, not because axiomatics can be proved to always work or because mathematics cannot get along without it, but because it is too useful a tool to abandon easily.

This was really Godel's point: mathematics is not identical with formalism. They stand and fall separately. This is not to say that mathematics could never collapse for any reason, only that it would take a lot more than finding a paradox at the center of ZFC to make it happen.

"This was really Godel's point: mathematics is not identical with formalism. They stand and fall separately. This is not to say that mathematics could never collapse for any reason, only that it would take a lot more than finding a paradox at the center of ZFC to make it happen."

That's significant because another great mathematician David Hilbert challenged mathematicians to come up with a complete and consistent set of axioms for all of mathematics. This was a great hope at one time that was shattered by Godel. Most people don't really talk about this old program anymore (except as a historical curiosity) because there is utterly no hope in it at all. The other implications that people try to draw from Godel's work is probably a consequence of the fact that his work sounds like it says so much more than it actually does when it's translated into normal English (and out of math-speak).

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.

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)

Ah, that makes a lot more sense now!

Consistency is overrated. General relativity and quantum mechanics are inconsistent with each other but at the same time they are extremely precise when applied to an appropriate domain (I remember watching a lecture where Feynman wrote down a number with 20 or so digits representing conformity of theory prediction with experiment).

To fill in some of the details: the so-called 'Second Incompleteness Theorem' states that recursively inumerable systems containing arithmetic are insufficient for proving their own consistency. It is possible to prove the consistency of arithmetic, but to do so you must use an even more complex system of axioms whose consistency can itself only be proved with an even more complex system, and so on.

So strictly speaking, the theorem doesn't rule out consistency proofs, but it does rule out proving the consistency of complex systems by simpler systems, meaning that the consistency of the consistency proofs can't be trusted any more than the consistency of the original system!

Even though they're not axiomatizable, natural numbers rest on reality - a contradiction would let us make two apples from one.

Banach-Tarski paradox: http://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox.

Which rests on the grossly-unphysical "Axiom of Choice". That doesn't disprove cousin_it's point. Though it's still an interesting link, if that's why you're posting it.

Almost all people use mathematics where "Axiom of Choice" is set to be true. It is grossly-unusual to consider systems without that axiom.

If you are not a mathematician chances are you've never encountered a system based on axioms without "Axiom of Choice" (for systems where it is applicable at all).

I agree it's "grossly-unphysical", but you can do hardly any math without it. Now try to do physics (or computer science) without math.

A view which Gödel, an ardent Platonist, would have no doubt agreed with.

Do they, though? Is there infinity in reality?

Infinity is not a number.

Infinity is not a natural number (as is in Peano arithmetic) but in many cases it can be treated as an ordinary number e.g., for some problems on a complex plane.

You are in a good company thinking that infinity cannot be ever treated as a number http://scienceblogs.com/goodmath/2008/10/infinity_is_not_a_n...

It is. Just depends which field. Surreals, superreals, hyperreals... People have been taking the reals and adding infinity all the time.

Also, there are the cardinal numbers, for which the interesting ones consist entirely of infinity.

I know, but there are more numbers than real things, or at least maybe they do. Perhaps numbers fail only in extremely large regions that we don't perceive in everyday life. (Unlikely I know, just nitpicking really).

Reality is a scary word.

Are there negative numbers? Zero? Imaginary unit?

There were times when people even didn't know a concept of zero.

A concept of infinity is really useful and it is used almost everywhere.