How do we know that we'll never prove a contradiction in Math

This really isn't an answer to the question posed, but a response that is much too long for a comment.

You (Jim Thio) state that, "[S]et theory is pretty much upgraded to kingdom come to prevent this so called contradiction." I would argue that almost the opposite happened.

Cantor's original definition of a set was as followed (translated from the original German by someone other than myself):

By a set we mean any collection $M$ into a whole of definite, distinct objects $m$ (which are called the elements of $M$) of our perception or of our thought.

This definition (and the implicit set construction principles) are about as wide ranging and general as possible. Any sort of collection of objects you can think has mathematical existence as a set. Cartesian products are sets. Unions of sets are sets. There is a set of all ordinals. And finally there is a set of all sets, and the various "subsets" of this are sets.

What Russell's Paradox (and others, such as Berry's Paradox) showed is that far from being able to posit the existence of sets with abandon, one has to be careful about set existence/construction principles. The various axiomatizations of set theory are then a description of the sorts of sets that one can posit into existence (this includes the operations we can perform on sets to yield another set).

So while modern axiomatizations are undoubtedly more complicated than the naive conception of a set, what it comes down to is a restriction of the sorts of collections that may rightly be called sets. On the bright side, these axiomatizations imply that virtually any collection that you meet in the various areas of mathematics are sets. We may then ignore the particulars of the axiomatizations when constructing, say, the set of extreme points of a convex subset of a topological vector space, or the set of all smooth functions $\mathbb{R} \to \mathbb{R}$.

As a final note (and something actually connected to the specific question asked), I recommend that you seek out a paper by the late George Boolos entitled "Gödel's Second Incompleteness Theorem Explained in Words of One Syllable". The first page gives a very brief and readable account of why -- unless our conception of mathematics becomes drastically different -- we cannot know that we are free of contradictions.

The contradiction you mention is not limited to set theory: consider the statement "this statement is false". If it is true then it is false, and if it is false then it is true.

Does this imply that language is contradictory? No. What it does is to show that you cannot expect every sentence to have a truth value.

In a similar way Russell's Paradox, as your contradiction is known, only shows that you cannot go happily saying "let $D$ be a set such that this and that" and always expect it to make sense. "Naive" set theory was indeed naive.

Regarding the strength of math as a whole, the lack of formal proofs of consistency is not ideal; but there are so many connections between diverse areas and even between abstract math and real-world experience that it is very unlikely that a hidden contradiction would be there lurking to come up to the light and make the whole building crumble.

To be more precise than LLLLL, we cannot prove in ZFC (for example) that ZFC itself will not result in a contradiction; or if we could, this would show that ZFC is inconsistent. This is Gödel's Incompleteness Theorem.

But say we have a logic system where an is axiom encoded to look like "ZFC is consistent", then this logic system will "prove" that ZFC is consistent. But is this new logic system consistent...? You begin to see the problem.