In mathematics, specifically set theory, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states:

There is no set whose cardinality is strictly between that of the integers and the real numbers.

In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to the following equation in aleph numbers: ${\displaystyle 2^{\aleph _{0}}=\aleph _{1}}$, or even shorter with beth numbers: ${\displaystyle \beth _{1}=\aleph _{1}}$.

The continuum hypothesis was advanced by Georg Cantor in 1878, and establishing its truth or falsehood is the first of Hilbert's 23 problems presented in 1900. The answer to this problem is independent of ZFC, so that either the continuum hypothesis or its negation can be added as an axiom to ZFC set theory, with the resulting theory being consistent if and only if ZFC is consistent. This independence was proved in 1963 by Paul Cohen, complementing earlier work by Kurt Gödel in 1940.

The name of the hypothesis comes from the term continuum for the real numbers.

Cantor believed the continuum hypothesis to be true and for many years tried in vain to prove it. It became the first on David Hilbert's list of important open questions that was presented at the International Congress of Mathematicians in the year 1900 in Paris. Axiomatic set theory was at that point not yet formulated. Kurt Gödel proved in 1940 that the negation of the continuum hypothesis, i.e., the existence of a set with intermediate cardinality, could not be proved in standard set theory. The second half of the independence of the continuum hypothesis – i.e., unprovability of the nonexistence of an intermediate-sized set – was proved in 1963 by Paul Cohen.

Two sets are said to have the same cardinality or cardinal number if there exists a bijection (a one-to-one correspondence) between them. Intuitively, for two sets ${\displaystyle S}$ and ${\displaystyle T}$ to have the same cardinality means that it is possible to "pair off" elements of ${\displaystyle S}$ with elements of ${\displaystyle T}$ in such a fashion that every element of ${\displaystyle S}$ is paired off with exactly one element of ${\displaystyle T}$ and vice versa. Hence, the set ${\displaystyle \{{\text{banana}},{\text{apple}},{\text{pear}}\}}$ has the same cardinality as ${\displaystyle \{{\text{yellow}},{\text{red}},{\text{green}}\}}$ despite the sets themselves containing different elements.

With infinite sets such as the set of integers or rational numbers, the existence of a bijection between two sets becomes more difficult to demonstrate. The rational numbers ${\displaystyle \mathbb {Q} }$ seemingly form a counterexample to the continuum hypothesis: the integers form a proper subset of the rationals, which themselves form a proper subset of the reals, so intuitively, there are more rational numbers than integers and more real numbers than rational numbers. However, this intuitive analysis is flawed since it does not take into account the fact that all three sets are infinite. Perhaps more importantly, it in fact conflates the concept of "size" of the set ${\displaystyle \mathbb {Q} }$ with the order or topological structure placed on it. In fact, it turns out the rational numbers can actually be placed in one-to-one correspondence with the integers, and therefore the set of rational numbers is the same size (cardinality) as the set of integers: they are both countable sets.