Worked solution: Gödel and Cohen: the continuum hypothesis is independent of ZFC (1963)

Georg Cantor laid the foundation of transfinite set theory by showing in 1874 that the real numbers $\mathbb{R}$ are uncountable, and in 1891 (via his diagonal argument) that for every set $X$, the power set $\mathcal{P}(X)$ has strictly greater cardinality than $X$: $|X| < |\mathcal{P}(X)|$. Applying this to the countable natural numbers $\mathbb{N}$ of cardinality $\aleph_0$ gives $\aleph_0 < |\mathcal{P}(\mathbb{N})| = |\mathbb{R}| = 2^{\aleph_0}$. The natural numbers represent the smallest infinite cardinality; the real numbers form a strictly larger one.

In 1878, Cantor formulated the Continuum Hypothesis ($\mathrm{CH}$): there is no intermediate cardinality strictly between $\aleph_0$ and $2^{\aleph_0}$. Every infinite subset of $\mathbb{R}$ is either in bijection with $\mathbb{N}$ (countable) or in bijection with $\mathbb{R}$ itself. In the aleph hierarchy of well-ordered cardinalities $\aleph_0 < \aleph_1 < \aleph_2 < \cdots$, where $\aleph_1$ is by definition the least uncountable cardinal, Cantor's theorem implies $\aleph_1 \le 2^{\aleph_0}$. The Continuum Hypothesis is therefore the assertion of exact equality: $2^{\aleph_0} = \aleph_1$.

David Hilbert placed the Continuum Hypothesis first among his famous 23 problems at the 1900 International Congress of Mathematicians in Paris, expressing complete confidence that it would be settled by the methods of ordinary mathematics. Over the next six decades, however, attempts to prove or disprove $\mathrm{CH}$ ran into an unprecedented barrier: the hypothesis turned out to be neither provable nor refutable from the standard axioms of mathematics. The next step defines precisely what that means.