Continuum hypothesis
The continuum hypothesis (CH) is the assertion that there are \(\aleph_1\) many real numbers, or, equivalently, that \(2^{\aleph_0} = \aleph_1\). This is formulated in the context of the axiom of choice, and \(\aleph_1\) is the smallest uncountable cardinal. It is equivalent to the following assertion: "for every \(A \subseteq \mathbb{N}\), either \(A\) and \(\mathbb{N}\) have the same size, or \(A\) and \(\mathbb{R}\) have the same size". In the context of the axiom of determinacy, it holds that, for every \(A \subseteq \mathbb{N}\), either \(A\) and \(\mathbb{N}\) have the same size, or \(A\) and \(\mathbb{R}\) have the same size, and yet \(2^{\aleph_0} \neq \aleph_1\) (in particular, the two are incomparable, since one needs choice to prove cardinals are linearly ordered).
The continuum hypothesis was originally posed by Georg Cantor after his proof of the diagonal argument. Cantor was unable to prove or disprove the continuum hypothesis, primarily because he was unable to find a well-order of the reals. It is now known that there is no definable well-order on the reals: therefore, this approach can't work. Solving the continuum hypothesis was the first on Hilbert's famous list of problems to be solved in the 20th century. After proving his first incompleteness theorem, Gödel had a suspicion that the continuum hypothesis may be independent of ZFC. He proved that the continuum hypothesis could not be disproved in ZFC, by showing that it was true in his inner model \(L\).[1] (Assuming ZFC is consistent to begin with, the alternative being that ZFC proves and refutes any statement by the principle of explosion.)
Cohen then proved it could not be proved in ZFC either, assuming ZFC is consistent: given a countable standard transitive model \(M\) of ZFC, he proved that there was a forcing extension \(M[G]\) which added \(\aleph_2^M\) reals, and therefore that the continuum hypothesis fails within \(M[G]\).[2] The powerful method of forcing could also be used to show the opposite: if \(M\) was a countable standard transitive model of ZFC, then there was a forcing extension \(M[G]\) that added a surjection from \(\aleph_1^M \to \mathfrak{c}^M\), and therefore the continuum hypothesis holds in \(M[G]\). As such, the continuum hypothesis is independent of ZFC, if ZFC is consistent. It is therefore often regarded as one of the biggest unsolved problems in set theory.