Regarding the ending point on Godel. If we have proven that it is impossible to disprove the continuum hypothesis, then would we be safe in assuming it to be tr

Regarding the ending point on Godel. If we have proven that it is impossible to disprove the continuum hypothesis, then would we be safe in assuming it to be true? If such an assumption were to lead to contradictions, we would have a proof by contradiction that the hypothesis is false. Of course, the same argument could be made for assuming the hypothesis is false. Therefore, doesn't this demonstrate that their are 2 constructions of math we can use: one where it is true, and the other where it is false. And that both of these constructions are internally consistent (assuming that the math with neither assumption is itself consistent). At this point it just becomes a question of which version is more useful/interesting.