The continuum hypothesis states that there is no infinite set with a cardinal number between that of the "small" infinite set of integers and the "large" infinite set of real numbers.

undecidable

Cantor's problem of the cardinal number of the continuum | Hilbert's first problem

formulation date | 1877 (149 years ago) formulator | Georg Cantor status | ambiguous proof date | 1963 (86 years later) (63 years ago) provers | Kurt Gödel | Paul Joseph Cohen additional people involved | David Hilbert

Proved by Gödel and Cohen to be undecidable within Zermelo-Frankel set theory with or without the axiom of choice, but there is no consensus on whether this is a solution to the problem.

prize awarded for solution | Paul Cohen received the Fields Medal in 1966 for showing that if set theory is consistent, then no contradiction would arise if the negation of the continuum hypothesis was added to set theory.

Hilbert's original 10 problems | Hilbert's 23 problems | mathematical hypotheses | undecidable mathematics problems