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Selected article 32

The continuum hypothesis is a hypothesis, advanced by Georg Cantor, about the possible sizes of infinite sets. Cantor introduced the concept of cardinality to compare the sizes of infinite sets, and he showed that the set of integers is strictly smaller than the set of real numbers. The continuum hypothesis states the following:

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

Or mathematically speaking, noting that the cardinality for the integers |\mathbb{Z}| is \aleph_0 ("aleph-null") and the cardinality of the real numbers |\mathbb{R}| is 2^{\aleph_0}, the continuum hypothesis says:

\nexists \mathbb{A}: \aleph_0 < |\mathbb{A}| < 2^{\aleph_0}.

This is equivalent to:

2^{\aleph_0} = \aleph_1

The real numbers have also been called the continuum, hence the name.

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