Beth number

In mathematics, the Hebrew letter leph (aleph) with various subscripts represents various infinite cardinal numbers (see aleph number). The second Hebrew letter eth (beth) is also used. To define the beth numbers, start by letting

the cardinality of the power set of A if is the cardinality of A.

Then

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:eth_0, eth_1, eth_2, eth_3, dots

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are respectively the cardinalities of

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:mathbb{N}, P(mathbb{N}), P(P(mathbb{N})), P(P(P(mathbb{N}))), dots.

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Each set in this sequence has cardinality strictly greater than the one preceding it, because of Cantor's theorem. Note that the 1st beth number eth_1 is equal to c, the cardinality of the continuum, and the 2nd beth number eth_2 is the power set of c.

Related Topics:
Cantor's theorem - 1st beth number - Cardinality of the continuum - 2nd beth number

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For infinite limit ordinals κ, we define

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:eth_kappa=sup{,eth_lambda:lambda

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If we assume the axiom of choice, then infinite cardinalities are linearly ordered; no two cardinalities can fail to be comparable, and so, since no infinite cardinalities are between leph_0 and leph_1, the celebrated continuum hypothesis can be stated in this notation by saying

Related Topics:
Axiom of choice - Continuum hypothesis

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:eth_1=leph_1.

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The generalized continuum hypothesis says the sequence of beth numbers thus defined is the same as the sequence of aleph numbers.

Related Topics:
Generalized continuum hypothesis - Aleph number

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