Is there really a size of infinity strictly between the integers and the real numbers, and if not, does that fact hold absolutely or only relative to which further axioms mathematicians choose to accept?

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.