Other than ZFC, what other axiomatic systems are useful?

[ainster31]

Other than ZFC, what other axiomatic systems have been shown to be useful to the real world?

I'm a layman that is reading through the Wikipedia articles and it seems like the axioms are becoming more Philosophy than Math. We can use a different axiomatic system other than ZFC, but are there any that are actually useful?

 

[jedishrfu]

I would have to say the Peano Axioms were quite influential too.

http://en.wikipedia.org/wiki/Peano_arithmetic

In either case, though Godel's theorem still looms over any system of logic stating that there will always be some statements that while true can't be proved.

http://en.wikipedia.org/wiki/Gödel's_incompleteness_theorems