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Continuum Hypothesis

Statement

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.

Solution

undecidable

Alternate names

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

History

Current evidence

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.

Associated prizes

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.

Classes

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

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