To be a bit more specific: the cardinality of the reals is equal to the cardinality of the power set of the naturals. This has also been known since the time of

To be a bit more specific: the cardinality of the reals is equal to the cardinality of the power set of the naturals. This has also been known since the time of Cantor (read: late 19th century, at the dawn of set theory). CH states that no cardinal numbers exist between the cardinalities of the reals and the naturals[1]. Cantor believed CH to be true, but couldn’t prove it, and it was eventually (ca. 1960) proven to be independent of the usual (ZFC) axioms of set theory. [1] Though CH could of course be stated in infinitely many equivalent ways, by replacing “reals” and “naturals” in this statement with any other pair of sets whose cardinalities are equal to those of the reals and naturals, respectively. For (uncountably) infinitely many stupid examples, take any non-empty open subset of any connected, finite-dimensional (Hausdorff, second-countable) manifold with at least two points, and any infinite subset of the integers.