Provability and truth The following is quoted from Set Theory and the Continuum Problem by Raymond M. Smullyan and Melvin Fitting.

So far, no attempts have been the slightest bit successful in determining whether the continuum hypothesis is true or false! Another question not to be confused with the truth or falsity of the continuum hypothesis is whether it can be formally proved or disproved from the present day axioms of set theory. This question is completely settled.

I always thought that when a statement is not provable nor disprovable, then it is up to convention to decide whether it is true or false. How can we determine the truth of CH if we know it is not provable nor disprovable?
Please explain in layman's language.
 A: Here's an analogy: we don't know the physics of the universe, but we have various models of it, some of which might disagree. The "truth of Maxwell's equations" is a statement about the universe, whereas the "derivability of Maxwell's equations from quantum electrodynamics" is a statement about a particular model of physics (namely QED). 
Similarly, we don't know the "physics of sets," and from certain philosophical points of view there is no "physics of sets." What we do have are various models of the "physics of sets," some of which might disagree. The "truth of the continuum hypothesis" is a statement about the set-theoretic universe, whereas the "provability of the continuum hypothesis from the currently accepted axioms of set theory" is a statement about a particular model of set theory known as ZFC, or Zermelo-Fraenkel set theory with the axiom of choice. (The continuum hypothesis is known to be neither provable nor disprovable from these axioms!) 
A: My answer to this question also answers your question, I think. The key point: many set theorists don't think that these questions are matters of convention, but rather (to quote Drake), "in some sense sets do exist, as objects to be studied, and that set theory is just as much about fixed objects as is number theory."
Examples of this attitude can be found in Gödel's essay "What is Cantor's Continuum Problem?" and in the conclusion to Cohen's book Set Theory and the Continuum Hypothesis, also Drake's book Set Theory: an Introduction to Large Cardinals, from which the above quote is taken. The first sentence of Schindler's book Set Theory: Exploring Independence and Truth expresses it succinctly: "Set theory aims at proving interesting true statements about the mathematical universe."
Suppose $X$ is a statement in the language of Peano arithmetic that cannot be proved in PA, but can be proved in ZFC. (Examples: the Paris-Harrington theorem, or Goodstein's theorem, or the assertion that PA is consistent.) Most number theorists are happy to say that $X$ is "really true", not that its truth is a matter of convention.
Analogously, many set theorists think that $2^{\aleph_0}$ has an "actual value". Yes, the axiom system ZFC can't determine what it is, but that's evidence of the weakness of this axiom system. We need more axioms that are "true about the universe of actual sets", or so the thinking goes.
It's also worth noting that the axioms for ZFC weren't written down at random, but flow from an intuitive picture, known as the cumulative hierarchy. Perhaps this intuition can be pushed further, suggesting new axioms that will settle CH. Or so it is hoped.
I should note finally that Platonism is by no means universal among set theorists, or even mathematicians at large. Many would agree that the adoption of CH or an alternative is a matter of convention, or personal preference, or whatever, but not a question of "truth". For a more extensive discussion, see the entries Platonism in the Philosophy of Mathematics and The Continuum Hypothesisin the Stanford Encyclopedia of Philosophy, and the panel discussion "Does Mathematics Need New Axioms?" in the Bulletin of Symbolic Logic, 6(4) (Dec., 2000), pp. 401-446, with participants Solomon Feferman, Harvey M. Friedman, Penelope Maddy and John R. Steel.
