do we assume "logic"? Mathematics has no foundations on reality, it stands on its own. But to construct it we have to assume a number of axioms, like if I wanted to create a "science" where the only rules are all circles are red and all squares are blue, then from that we can build theories and prove them from the assumed axioms, like a particular square has to be blue because all squares are blue, right? However, if the defining rules of my "science" are contradictory, like all circles are red and all circles are blue, then my "science" is "ill defined". But what indicates that? If the cornerstones of mathematics have to be "logical", not in the sense of mathematical logic or propositional logic per se but in the sense that they have to be "well defined", does that mean that we are assuming a set of implicit primordial rules that the rest of mathematics have to abide to? If even the most basic laws of mathematics have to be "logical", does that mean that we are assuming a set of rules that dictate whether something is logical or not? And could these rules be "logical" if they define logic? In other words, can they abide by their own rules? Is that even possible?
The short answer to your question is ... yes, most working mathematicians do their work with under the assumption that ZFC (the "math world" in which 99% of them work) is consistent (there are no contradictions) and complete (everything that is true is also provable and vice versa). You should look into topics such as Godel's Incompleteness Theorem, metamathematics, and philosophy of math for the beginnings of the long answer to your concerns.
However, just as we can work in many different sets of axioms, getting different mathematical "stuctures" so we can work in different "logics", ZFC or not. And just as we have to specify what axioms we are using, we have to specify what logic.
ZFC has been proven to be incomplete. The continuum hypothesis for example cannot be proven. Most mathematicians do believe that ZFC is consistent. But this can never be proven. In fact, whether ZFC is consistent is totally irrelevant to most. If it were inconsistent, we would find a new axiom system and formulate all of our mathematics in there. Most mathematicians wouldn't even notice anything changed.
This is assuming Con(ZFC) of course :tongue: Assuming Con(ZFC) then yes ZFC cannot prove its own consistency. This is technically different from not being provable altogether since you could hypothetically move to another set theoretic universe where you can prove consistency. Results of this nature are actually of some interest. For example PA is unable to prove its own consistency yet we have results like this: http://en.wikipedia.org/wiki/Gentzen's_consistency_proof