> But one of those two must be true, you just can't prove it. Three alternative accounts: * Both the CH and ¬CH mathematical universes really exist, so we jus

> But one of those two must be true, you just can't prove it. Three alternative accounts: * Both the CH and ¬CH mathematical universes really exist, so we just have to choose which one we're more interested in at a given time. Like one might say there are the "reall numbers" and the "realle numbers", both valid and interesting constructions which humanity was just slow to recognize the distinctions between (because they were initially less relevant to our interests and our day-to-day lives). * We are actually ultimately thinking about one or the other of them, or are in some sense in one or the other mathematical universe, but we don't know enough about our intuition about the reals to be able to specify or explain which one. (Maybe we need other properties whose obviousness or relevance humanity is not smart enough to notice?) * Some finitist or ultrafinitist approach is actually right: the real numbers are a formalism that, while reasonably motivated by historical attempts to "complete" mathematics in various ways, doesn't correspond to anything Platonically real or to anything intellectually relevant to humanity. (In this account, there is potentially no answer to the question because the real numbers don't exist at all. Neither CH nor ¬CH refers to a mathematical reality, just to games about formalisms.)