The real numbers are UNCOUNTABLE [III.11], but do they form the “smallest” uncountable set? Equivalently, is it the case that if A is any set of real numbers, then either A is countable or there is a bijection between A and the set of all real numbers? The continuum hypothesis (or CH) is the assertion that this is indeed true. The notions of countability and uncountability were invented by CANTOR [VI.54], who was also the first to formulate CH. He tried hard to prove or disprove it, as did many others after him, but nobody succeeded.
Gradually, mathematicians came to entertain the idea that CH might be “independent” of normal mathematics: that is, independent of the usual ZFC AXIOMS [IV.22 §3.1] of set theory. This would mean that it could be neither proved nor disproved from the ZFC axioms.
The first result in this direction was due to GÖDEL [VI.92], who showed that CH could not be disproved from the usual axioms. In other words, one could not reach a contradiction by assuming CH. To do this, he showed that inside every MODEL OF SET THEORY [IV.22 §3.2] there is a model in which CH holds. This model is called the “constructible universe.” Roughly speaking, it consists just of those sets that “have to exist” if the axioms are true. So, in this model, the set of reals is as small as it could possibly be. The “smallest uncountable size” is usually denoted ℵ1, and in Gödel’s construction the reals appear in ℵ1 stages, with only countably many reals appearing at each stage. From this one can deduce that the number of reals is ℵ1, which is precisely the assertion of CH.
The other direction had to wait thirty years, until Paul Cohen invented the method of forcing. How would we make CH false? Starting from some model of set theory (in which CH might well hold), we would like to “add” some reals to it. Indeed, we would like to add enough that there are now more than ℵ1 of them. But how do we “add” a real? We need to ensure that what we end up with is still a model of set theory, which is hard enough, but also that when we add new reals we do not alter the value of ℵ1 (since otherwise the statement “the number of reals is ℵ1” may still be true in the new model). This is an extremely complicated task, both conceptually and technically. See SET THEORY [IV.22] for more details about how it is carried out.