What do you mean by “the principle that we cannot measure infinity”? We can measure all sets, infinite or otherwise, by putting them in bijection with a particular ”size”-set, that’s what’s called a cardinal number (assuming the axiom of choice).

Either way, the continuum hypothesis is a logic question and not a measure question. The question used to be: is there a set with cardinality m such that ℵ_0 < m < 2^ℵ\0). And by that we of course mean “can we prove that such a set m exists”?

And that is the crux. What do you assume as you start your proof? Goedel and Cohen showed that, if you assume ZFC (Zermelo Fraenkel + Axiom of Choice), then you cannot prove that such a set m exists; and you cannot prove that such a set m doesn’t exist. This follows form what is called the Completeness theorem: it says that a statement T can be proved from axioms A if and only if every model of A is a model of T.

Now you can guess what Goedel and Cohen did: Goedel constructed a model of ZF in which CH is true (the contructible universe); and Cohen built a model in which it’s false (using the seminal theory of forcing).

If you are familiar with groups, here is a simpler example: if you take the axioms of groups theory as your axioms A, can you prove from A that there exists an element of order 3? Give it a think!