Let us return to the continuum function on regular cardinals and concentrate on the simplest case, the size of 2ℵ0. One of Cantor's original approaches to CH was by investigating “simple” sets of real numbers (see Hallett (1984), pp. 3–5 and §2.3(b)).  One of the first results in this direction is the Cantor-Bendixson theorem that every infinite closed set is either countable or contains a perfect subset, in which case it has the same cardinality as the set of reals. In other words, CH holds (in this formulation) when one restricts one's attention to closed sets of reals. In general, questions about “definable” sets of reals are more tractable than questions about arbitrary sets of reals and this suggests looking at definable versions of the continuum hypothesis. 