Mathematics

Is there a correct formulation of set theory? For example, it's been proven by Gödel and Paul Cohen that the continuum hypothesis can neither be proven nor disproven in ZFC. Should we take from this that there exists a hitherto undiscovered formulation of set theory that can conclusively establish whether the continuum hypothesis is true or false?

Comments

Logically wiser heads (such as Peter Smith) may want to weigh in, but I think we can at least say this much. As you note, the axioms of Zermelo-Frankel set theory don't answer the question. We can add either the Axiom of Choice or its denial and to the basic ZFC axioms. As long as the basic axioms are consistent, the result will be consistent too. The same goes for the Continuum Hypothesis. But by itself this doesn't give us any reason one way or another to think that there's some more agreeable formulation without this ambiguity. And even if we came up with a particularly elegant formulation of set theory that didn't leave these axioms dangling, it's not clear that this would settle the question of truth conclusively.

Behind this lies a harder question: what would it mean in the first place for the matter to be settled conclusively? The problem is that just what truth amounts to here is not a settled matter, and not a matter that formulations of axioms by themselves can settle.

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