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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?

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    $\begingroup$ Note: neither direction of the independence of CH was proved by Cantor, so even as a simlification this does not make sense. $\endgroup$ Jul 12 '16 at 9:31
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    $\begingroup$ Du Sautoy is being informal: he means that Cantor at times thought he had a proof, only to discover it to be flawed, or lacking somehow. And maybe at times he thought he had a proof of the opposite, only to realize that it too was incorrect. This is a bit of a historical oversimplification, but it is all that was meant in the podcast. $\endgroup$ Jul 12 '16 at 12:11
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    $\begingroup$ To pile on the other comments, that is the problem of popular mathematics. It is aimed for the masses, and is therefore generally being oversimplified. History is no difference here. Not that this is necessarily a bad thing; but it's important to point out the oversimplifications when you're making them. Whether or not Du Sautoy did that, I don't know, I haven't listened to his podcast. $\endgroup$
    – Asaf Karagila
    Jul 12 '16 at 17:56
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    $\begingroup$ Also, it is not the case that "both answers are correct"; the two answers contradict each other, so you'd need a very strange notion of correctness to make both of them correct. It is true that each answer is consistent with the usual axioms of set theory (ZFC), but that's very different from saying that the answers are correct. $\endgroup$ Jul 13 '16 at 6:11
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    $\begingroup$ Let me point out that what @Andrés wrote is a very common thing. If you have a conjecture, you try and prove it; if you haven't succeeded for a while, you try to disprove it; if you haven't succeeded in a while, you sometimes feel that perhaps it is true after, so you return to trying to prove it; and thus the vicious circle continues, until someone proves, disproves, or shows that the conjecture is neither provable nor disprovable. $\endgroup$
    – Asaf Karagila
    Jul 14 '16 at 7:54
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From Wikipedia:

Cantor believed the continuum hypothesis to be true and tried for many years to prove it, in vain. It became the first question on David Hilbert's list of important open questions that was presented at the International Congress of Mathematicians in 1900.

Kurt Gödel showed in 1940 that the continuum hypothesis (CH for short) cannot be disproved from the standard Zermelo–Fraenkel set theory with Choice (ZFC). Paul Cohen showed in 1963 that CH cannot be proven from those same axioms either [by using a method called "Forcing" to create a model of ZFC where CH was false]. Hence, CH is independent of ZFC.

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