This is a somewhat different situation. Gödel's theorem gives you independent statements but they are highly contrived (containing expressions encoding provabil

This is a somewhat different situation. Gödel's theorem gives you independent statements but they are highly contrived (containing expressions encoding provability and consistency in an ad-hoc fashion). This is talking about the cardinalities of very basic sets, and it was (is? haha) extremely unclear to what extent statements about them are independent, if at all. I would personally be shocked if you can prove independence of CH directly from the incompleteness theorem, though I would love to be wrong! It required a lot more delicate machinery to be developed (Cohen's forcing method). Numbers certainly exist in a useful sense - they model real properties of the universe. But I agree that higher set theories may be moving away from that...