I don't quite understand this.
If it's independent of the axioms of set theory
then we are free to accept it as either true or false.
Sort of like
if I don't like playing chess tonight
I can always look for a bridge party.
... it's independent of the axioms of set theory and we're waiting for further insight into whether it's true or false.
You replied:
I don't quite understand this. If it's independent of the axioms of set theory then we are free to accept it as either true or false.
Sort of like if I don't like playing chess tonight I can always look for a bridge party.
This is an interesting question. There's clearly a sense in which one can do exactly as you say. If a sentence is independent of set theory, both the sentence and its negation are consistent with set theory, so you can develop two different versions of mathematics without hitting a contradiction in either one. (This is much like Euclidean and non-Euclidean geometries.)
But people generally want their mathematics to mean something, not just to be a formal game with symbols. (Consistency is just a formal, syntactic property.) The axioms of set theory are intended to be true of the mathematical universe as a whole, whatever that might consist of.
Plenty of axiomatizations are consistent but are of no particular interest. For example, by Gödel, assuming that ZF is consistent, the following is also a consistent theory: ZF together with the sentence "ZF is inconsistent". But this isn't really a very interesting theory, and no one would propose adopting it as a base for mathematics. Why? Because it's not satisfied by the actual mathematical universe (since ZF really is consistent).
Why do people take Zermelo-Fraenkel set theory (possibly with the Axiom of Choice, depending on your taste) as being a description of the general accepted principles of mathematics? ZF isn't set in stone, after all. It's just that people know what mathematical principles they accept, and ZF seems to embody them.
In fact, when Zermelo first compiled his axiomatization of set theory, he didn't include what is now known as the Replacement Axiom Schema. Fraenkel pointed this out, and it's reported that Zermelo agreed that it should be there and said that he just forgot to put it in.
The thing is that mathematicians have a good idea as to what the mathematical universe looks like, and they want to use an axiom system that correctly describes that universe and that lets them prove as much as possible. By Gödel's Incompleteness Theorem, no consistent axiomatization of mathematics can be complete, so there's always the possibility of realizing that there are mathematical principles that you would accept but that haven't been included in the theory, and then adding them in.
From a formal perspective, one can add a new axiom "ZF is consistent". (The consistency of ZF is widely accepted, although it's not provable within ZF.) One can then add the consistency of the new theory. And one can then add the consistency of that new theory, and so on. Each one of these axioms can be seen informally to be true, but cannot be proven from the axioms admitted before it.
But it's more interesting to look at one's intuition about the mathematical universe and use that to find new axioms that one can argue are true in the mathematical universe but that ZF doesn't prove. This would be very much like the situation with the Replacement Axiom Schema -- you may think that you have a good axiom system, but a little reflection indicates that the mathematical universe is richer in structure than the axiom system requires, so you decide to add a new axiom.
The basic intuition of set theory is that the universe of sets ought to be as large as possible and as rich as possible, since it's supposed to contain all conceivable mathematical objects. Several of the axioms of ZF can be thought of as expressing (some small part of) this idea.
Many axioms that go beyond ZF are known, and people have varying amounts of confidence in whether these axioms are true in the universe of sets. Most of these axioms are so-called large cardinal axioms; a large cardinal axiom requires the existence of sets larger than can be proven to exist without that axiom.
A simple example of a large cardinal axiom is the axiom of inaccessibility, which states that there exists a "strongly inaccessible" cardinal. [This is an uncountable cardinal kappa so large that: (1) the union of fewer than kappa-many sets of size less than kappa must have size less than kappa; and (2) the power set of any set of size less than kappa must have size less than kappa.] One gets the feeling that if the universe is to be as large as possible, then strongly inaccessible cardinals ought to exist. If they don't exist, it means you just didn't include everything you could have. It turns out that the existence of a strongly inaccessible cardinal proves the consistency of ZF. It follows that one can't prove, assuming just ZF, that strongly inaccessible cardinals exist. But one's intuition says that they should exist; ZF is just too weak to prove it.
And so on to larger and larger cardinals.... Mahlo cardinals, weakly compact cardinals, Ramsey cardinals, measurable cardinals, strongly compact cardinals, supercompact cardinals, huge cardinals, ..., the existence of these being progressively more powerful assumptions. Strongly inaccessible cardinals are, in fact, very small in the hierarchy of large cardinals.