TITLE: Can we prove that axioms do not contradict?
QUESTION [5 upvotes]: We construct many structures by chosing a set of axioms and deriving everything else from them. As far as I remember we never proved in our lectures that those axioms do not contradict. So:
Is it always possible to prove that a set of axioms does not contain a contradiction? (So if not could we perhaps find a contradiction in a structure that we definde by axioms someday?) Or is it proven that doing so is impossible?

REPLY [9 votes]: Proved from what?
Even logic is formalized in a mathematical fashion, so it has a mathematical backbone. We need to first assume that first-order logic does not include a contradiction. Then we need to decide on which system we are going to use to develop our mathematics. Arithmetical systems will suffice for syntactical manipulations of strings, and set theoretical systems will suffice for semantical proofs using models.
Whatever meta-theory, in which we formalized the logic and the proofs and whatnot, you are working in, if it is sufficiently strong, it will not prove its own consistency. This means that along with the assumption that first-order logic does not contain a contradiction, we implicitly assume that our meta-theory is consistent, whatever it might be. And the more we need, the more consistency we will have to prove.
Goedel proved that a theory which is both consistent, recursively enumerable, and interprets arithmetics, cannot prove its own consistency (and is therefore not complete). This means that we really have to assume the consistency of our meta-theory (which is usually a theory satisfying these conditions).
Of course, we can prove the consistency of arithmetic, using set theory, and the consistency of set theory using an even stronger set theory, and so on. But that's not what you're looking for in an answer. The answer, simply, is that at some point, you have to take some things for granted. The question is how much you are willing to take for granted, how much logic and foundational theories you are familiar with, to transform an argument form one system to another.