Continuum hypothesis

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.

The name of the hypothesis comes from the term continuum for 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. It became one of the most studied problems in set theory, and establishing its truth or falsehood was the first of Hilbert's 23 problems presented in 1900. The answer to this problem is independent of ZFC. This means the axioms of ZFC can neither prove nor disprove the continuum hypothesis, meaning 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 generalized continuum hypothesis states that ${\displaystyle \aleph _{\alpha +1}=2^{\aleph _{\alpha }}}$ for every ordinal ${\displaystyle \alpha }$.