What is the continuum hypothesis in set theory?

The continuum hypothesis is a mathematical proposition about the possible sizes of infinite sets, stating that no set has a cardinality strictly between the integers and the real numbers. Advanced by Georg Cantor in 1878, it became the first of Hilbert's famous problems. It was later proven to be independent of the standard axioms of Zermelo-Fraenkel set theory with the axiom of choice.

What we know

• The continuum hypothesis states that there is no set whose cardinality is strictly between that of the integers and the real numbers. [1, 2]
• Georg Cantor advanced the continuum hypothesis in 1878. [2]
• Establishing the truth or falsehood of the hypothesis was presented as the first of Hilbert's problems in 1900. [1, 2]
• The answer to the continuum hypothesis is independent of Zermelo-Fraenkel set theory with the axiom of choice, meaning its axioms can neither prove nor disprove it. [2, 5]

Where to go next

• whoWho was Georg Cantor?
• the bigger pictureWhat are Hilbert's problems?
• how it workedHow did Paul Cohen prove independence?
• compared withWhat is the generalized continuum hypothesis?
• how we knowWhat is the axiom of choice?
• an unexpected connectionHow did a Fields Medal result from set theory?Paul Cohen was awarded a Fields Medal for proving the independence of the continuum hypothesis and the axiom of choice from Zermelo-Fraenkel set theory.

Sources

• [1]continuum hypothesis (Wikidata Q208416) · CC0 1.0
• [2]Continuum hypothesis · CC BY-SA 4.0
• [3]Second continuum hypothesis · CC BY-SA 4.0
• [4]Weak continuum hypothesis · CC BY-SA 4.0
• [5]Paul Cohen · CC BY-SA 4.0