One of the main reasons I study mathematics is because of the certainty of it. Given certain assumptions, one can deduce whether or not something is true.
Recently, I've just discovered that the foundations of mathematics is not self-consistent. There is a possibility that there may exist a contradiction in the system. But if there is a possibility that a contradiction may exist, then why should we be sure that the theorems we prove from the axioms are true?
Also, when we're talking about consistency in systems here, we're talking about set theory right? For example, for the field axioms, that system of axioms must be consistent, no?
This is really bothering me. All help would be appreciated!
I think you need to study this matter more methodically. It seems to me that you're going from bits and pieces of information about this. To get a clear grasp on these matters I would first learn the predicate calculus, then a certain amount of set theory, then some basic mathematical logic.