I have been listening to the podcasts of A Brief History of Mathematics on BBC Radio 4. In Episode 7 (titled Georg Cantor), Prof. Marcus du Sautoy says the following (listen from the timestamp 9:18 onwards here):
But the question that really vexed Cantor concerned the nature of the infinite set of decimal numbers. Yes, it's bigger than the set of whole numbers, but could there be a set in between—strictly bigger than the set of whole numbers (or the coins I had) and strictly smaller than the set of decimal numbers (or kumquats)? One day he proved there was, the next he proved the opposite. And the reason Cantor was having so much trouble answering his own question was that some decades later it was shown that both answers are correct—a revelation that threw many areas of mathematics into crises.
This confused me because if what Prof. du Sautoy says is true, then that would mean we have arrived at a contradiction in the axioms we started with.
I have read that the Continuum Hypothesis can neither be proved nor disproved within the axioms of ZFC. I understand that this is very different from both proving and disproving CH, or any proposition for that matter. So, did Prof. du Sautoy say this only to simplify for the sake of his listeners, or did Cantor actually manage to both prove and disprove the Continuum Hypothesis, after which (perhaps) the foundations of mathematics were examined more closely?