Continuum hypothesis

In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. Georg 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:

Related Topics:
Mathematics - Hypothesis - Infinite - Set - Georg Cantor - Cardinality - Integer - Real number

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

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Or mathematically speaking, noting that the cardinality for the integers |mathbb{Z}| is leph_0 ("aleph-null") and the cardinality of the real numbers |mathbb{R}| is 2^{leph_0}, the continuum hypothesis says:

Related Topics:
Cardinality - Aleph-null - Cardinality of the real numbers

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: otexists mathbb{A}: leph_0 < |mathbb{A}| < 2^{leph_0}.

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This implies:

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:|mathbb{R}| = leph_1

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The real numbers have also been called the continuum, hence the name. There is also a generalization of the continuum hypothesis called the generalized continuum hypothesis saying:

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: For all ordinals lpha: 2^{leph_lpha} = leph_{lpha+1}

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