Definition:Hilbert 23/1

Hilbert $23$: Problem $1$

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Continuum Hypothesis

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

Symbolically, the continuum hypothesis asserts:

$\aleph_1 = \mathfrak c$

where:

$\mathfrak c$ denotes the cardinality of the continuum

$\aleph_1$ denotes Aleph One.

Historical Note

The Hilbert 23 were delivered by David Hilbert in a famous address at Paris in $1900$.

He considered them to be the oustanding challenges to mathematicians in the future.

There was originally going to be a $24$th problem, on a criterion for simplicity and general methods in proof theory, but Hilbert decided not to include it, as it was (like numbers $4$, $6$, $16$ and $23$) too vague to ever be described as "solved".

Sources

• 1902: David Hilbert: Mathematical Problems (Bull. Amer. Math. Soc. Vol. 8, no. 10: pp. 437 – 479)

(translated by Mary Winston Newson from "Mathematische Probleme")