How is a system of axioms different from a system of beliefs?
Other ways to put it: Is there any faith required in the adoption of a system of axioms? How is a given system of axioms accepted or rejected if not based on blind faith?
To paraphrase Robert Mastragostino's comment, a system of axioms doesn't make any assertions that you can accept or reject as true or false; it only specifies the rules of a certain kind of game to play.
It's worth clarifying that a modern mathematician's attitude towards mathematical words is very different from that of a non-mathematician's attitude towards ordinary words (and possibly also very different from a classical mathematician's attitude towards mathematical words). A mathematical word with a precise definition means precisely what it was defined to mean. It's not possible to claim that such a definition is wrong; at best, you can only claim that a definition doesn't capture what it was intended to capture.
Thus a modern interpretation of, say, Euclid's axioms is that they describe the rules of a certain kind of game. Some of the pieces that we play with are called points, some of the pieces are called lines, and so forth, and the pieces obey certain rules. Euclid's axioms are not, from this point of view, asserting anything about the geometry of the world in which we actually live, so one can't accept or reject them on that basis. One can, at best, claim that they don't capture the geometry of the world in which we actually live. But people play unrealistic games all the time.
I think this is an important point which is not communicated well to non-mathematicians about how mathematics works. For a non-mathematician it is easy to say things like "but $i$ can't possibly be a number" or "but $\infty$ can't possibly be a number," and to a mathematician what those statements actually mean is that $i$ and $\infty$ aren't parts of the game Real Numbers, but there are all sorts of other wonderful games we can play using these new pieces, like Complex Numbers and Projective Geometry...
I want to emphasize that I am not using the word "game" in support of a purely formalist viewpoint on mathematics, but I think some formalism is an appropriate answer to this question as a way of clarifying what exactly it is that a mathematical axiom is asserting. Some people use the word "game" in this context to emphasize that mathematics is "meaningless". The word "meaningless" here has to be interpreted carefully; it is not meant in the colloquial sense (or at least I would not mean it this way). It means that the syntax of mathematics can be separated from its semantics, and that it is often less confusing to do so. But anyone who believes that games are meaningless in the colloquial sense has clearly never played a game...
The role of axioms is to describe a mathematical universe. Some settings, some objects. Axioms are there only to tell us what we know on such universe.
Indeed we have to believe that ZFC is consistent (or assume an even stronger theory, and believe that one is consistent, or assume... wait, I'm getting recursive here). But the role of ZFC is just to tell us how sets are behaved in a certain mathematical settings.
The grand beauty of mathematics is that we are able to extract so much merely from these rules which describe what properties sets should have.
Whether or not you should accept an axiomatic theory or not is up to you. The usual test is to see whether or not the properties described by the axioms make sense and seem to describe the idea behind the object in a reasonable manner.
We want to know that if a set exists, then its power set exists. Therefore the axiom of power set is reasonable. We want to know that two sets are equal if and only if they have the same elements, which is a very very reasonable requirement from sets, membership and equality. Therefore the axiom of extensionality makes sense.
What you should do when you attempt to decide whether or not you accept some axioms is to try and understand the idea these axioms try to formalize. If they convince you that the formalization is "good enough" then you should believe that the axioms are consistent and use them. Otherwise you should look for an alternative.
Related:
- I talked about the difference between an idea and its mathematical implementation in this answer of mine which might be relevant to this discussion as well (to some extent).
While I agree that faith is not required to accept the rules of certain game in order to play that game, I am pretty sure that, if the rules of the game, and the consequences of playing by those rules, were entirely arbitrary, a lot of mathematicians wouldn't be able to earn a living by doing mathematics.
For ex. according to some recent stats, mathematician is one of the most employable occupations in the U.S., mostly in finance industry. If that doesn't indicate that the rules have some validity in the real world, I don't know what does?!
Would physicists use mathematics if it were completely incapable to describe what they are theorizing and measuring?
There is evidence that axioms may be true, a lot of evidence.
EDIT: I have found the following article by R.W.Hamming very illuminating: The Unreasonable Effectiveness of Mathematics
Some quotations:
The Postulates of Mathematics Were Not on the Stone Tablets that Moses Brought Down from Mt. Sinai.
also
The idea that theorems follow from the postulates does not correspond to simple observation. If the Pythagorean theorem were found to not follow from the postulates, we would again search for a way to alter the postulates until it was true. Euclid's postulates came from the Pythagorean theorem, not the other way. For over thirty years I have been making the remark that if you came into my office and showed me a proof that Cauchy's theorem was false I would be very interested, but I believe that in the final analysis we would alter the assumptions until the theorem was true.
Amusingly, he warns in the introduction:
I am well aware that much of what I say, especially about the nature of mathematics, will annoy many mathematicians. My experimental approach is quite foreign to their mentality and preconceived beliefs
One can believe a set of clearly contradictory statements (indeed, people often do). If it can be proven that a set of statements entails a contradiction, though, it certainly cannot be taken as an axiomatic system.