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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.

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Smullyan and Fitting say "proved or disproved from the present day axioms of set theory". The present day axioms are the generally accepted ones of ZFC. The axioms of set theory can change. One may add additional axioms, or perhaps change the underlying logic. There is not yet any way to do this that is generally accepted. – MikeC Apr 23 '12 at 4:41

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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!)

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What bothers me is the first sentence: "So far, no attempts have been the slightest bit successful in determining whether the continuum hypothesis is true or false!" If it's known not to be provable or disprovable, how can they attempt to determine its truth? – aprer Apr 23 '12 at 4:08
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It is conceivable that the negation of CH, or perhaps CH, will follow from a "generally accepted" new axiom for Set Theory. – André Nicolas Apr 23 '12 at 4:12
@aprer: in the physics analogy, we find a better theory of physics. In mathematics, we find "better" axioms for set theory (although what "better" means here is debatable, it is debatable whether this is a good way to think about set theory, and I am not qualified to discuss it). – Qiaochu Yuan Apr 23 '12 at 4:29
I see. So the "truth" here has a subtle philosophical meaning. Thanks. – aprer Apr 23 '12 at 4:36
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Finding a "generally accepted" new axiom that would settle CH is an active area of research. – Quinn Culver Apr 23 '12 at 20:10

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