Godel (1940) and Cohen (1963) proved that the continuum hypothesis can be neither proved nor disproved from the standard ZFC axioms of set theory. That result closes the question of what ZFC alone can say, but it opens a harder one: whether CH has a determinate truth value at all, and if so what further axiom would reveal it. Mathematicians who accept that set theory describes a single true universe of sets, Godel among them, hold this is a real unanswered question rather than a dead end; committed formalists tend to hold there is nothing further to ask.
What would resolve this Either a broadly accepted new axiom for set theory, beyond ZFC, that settles CH one way or the other and gains the kind of consensus ZFC itself enjoys, or a philosophical argument persuasive enough to convince most set theorists that the independence result is the end of the matter rather than the start of a harder question.
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