Continuum hypothesis: Difference between revisions

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 [[Countability|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).