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| In [[mathematics]], the [[infinite set|infinite]] [[cardinal number]]s are represented by the [[Hebrew letter]] <math>\aleph</math> ([[Aleph (letter)|aleph]]) indexed with a subscript that runs over the [[ordinal number]]s (see [[aleph number]]). The second [[Hebrew alphabet|Hebrew letter]] <math>\beth</math> ([[bet (letter)|beth]]) is used in a related way, but does not necessarily index all of the numbers indexed by <math>\aleph</math>.
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| == Definition ==
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| To define the '''beth numbers''', start by letting
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| :<math>\beth_0=\aleph_0</math>
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| be the cardinality of any [[countably infinite]] [[set (mathematics)|set]]; for concreteness, take the set <math>\mathbb{N}</math> of [[natural number]]s to be a typical case. Denote by ''P''(''A'') the [[power set]] of ''A''; i.e., the set of all subsets of ''A''. Then define
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| :<math>\beth_{\alpha+1}=2^{\beth_{\alpha}},</math>
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| which is the cardinality of the power set of ''A'' if <math>\beth_{\alpha}</math> is the cardinality of ''A''.
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| Given this definition,
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| :<math>\beth_0,\ \beth_1,\ \beth_2,\ \beth_3,\ \dots</math>
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| are respectively the cardinalities of
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| :<math>\mathbb{N},\ P(\mathbb{N}),\ P(P(\mathbb{N})),\ P(P(P(\mathbb{N}))),\ \dots.</math>
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| so that the second beth number <math>\beth_1</math> is equal to <math>\mathfrak c</math>, the [[cardinality of the continuum]], and the third beth number <math>\beth_2</math> is the cardinality of the power set of the continuum.
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| Because of [[Cantor's theorem]] each set in the preceding sequence has cardinality strictly greater than the one preceding it. For infinite [[limit ordinal]]s λ the corresponding beth number is defined as the [[supremum]] of the beth numbers for all ordinals strictly smaller than λ:
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| :<math>\beth_{\lambda}=\sup\{ \beth_{\alpha}:\alpha<\lambda \}.</math>
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| One can also show that the [[von Neumann universe]]s <math>V_{\omega+\alpha} \!</math> have cardinality <math>\beth_{\alpha} \!</math>.
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| == Relation to the aleph numbers ==
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| Assuming the [[axiom of choice]], infinite cardinalities are [[total order|linearly ordered]]; no two cardinalities can fail to be comparable. Thus, since by definition no infinite cardinalities are between <math>\aleph_0</math> and <math>\aleph_1</math>, it follows that
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| :<math>\beth_1 \ge \aleph_1.</math> | |
| Repeating this argument (see [[transfinite induction]]) yields
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| <math>\beth_\alpha \ge \aleph_\alpha</math>
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| for all ordinals <math>\alpha</math>.
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| The [[continuum hypothesis]] is equivalent to
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| :<math>\beth_1=\aleph_1.</math> | |
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| The [[Continuum hypothesis#The generalized continuum hypothesis|generalized continuum hypothesis]] says the sequence of beth numbers thus defined is the same as the sequence of [[aleph number]]s, i.e.,
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| <math>\beth_\alpha = \aleph_\alpha</math>
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| for all ordinals <math>\alpha</math>.
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| == Specific cardinals ==
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| === Beth null ===
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| Since this is defined to be <math>\aleph_0</math> or [[aleph null]] then sets with cardinality <math>\beth_0</math> include:
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| *the [[natural number]]s '''N'''
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| *the [[rational number]]s '''Q'''
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| *the [[algebraic number]]s
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| *the [[computable number]]s and [[computable set]]s
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| *the set of [[finite set]]s of [[integer]]s
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| === Beth one ===
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| {{main|cardinality of the continuum}}
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| Sets with cardinality <math>\beth_1</math> include:
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| *the [[transcendental numbers]]
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| *the [[irrational number]]s
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| *the [[real number]]s '''R'''
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| *the [[complex number]]s '''C'''
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| *[[Euclidean space]] '''R'''<sup>''n''</sup>
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| *the [[power set]] of the [[natural number]]s (the set of all subsets of the natural numbers)
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| *the set of [[sequence]]s of integers (i.e. all functions '''N''' → '''Z''', often denoted '''Z'''<sup>'''N'''</sup>)
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| *the set of sequences of real numbers, '''R'''<sup>'''N'''</sup>
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| *the set of all [[continuous function]]s from '''R''' to '''R'''
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| *the set of finite subsets of real numbers
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| === Beth two ===
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| <math>\beth_2</math> (pronounced ''beth two'') is also referred to as '''2<sup>''c''</sup>''' (pronounced ''two to the power of c'').
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| Sets with cardinality <math>\beth_2</math> include:
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| * The [[power set]] of the set of [[real number]]s, so it is the number of [[subset]]s of the [[real line]], or the number of sets of real numbers
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| * The power set of the power set of the set of natural numbers
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| * The set of all [[function (mathematics)|functions]] from '''R''' to '''R''' ('''R'''<sup>'''R'''</sup>)
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| * The set of all functions from '''R'''<sup>''m''</sup> to '''R'''<sup>''n''</sup>
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| * The power set of the set of all functions from the set of natural numbers to itself, so it is the number of sets of sequences of natural numbers
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| * The [[Stone–Čech compactification]]s of '''R''', '''Q''', and '''N'''
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| === Beth omega ===
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| <math>\beth_\omega</math> (pronounced ''beth omega'') is the smallest uncountable [[strong limit cardinal]].
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| ==Generalization==
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| The more general symbol <math>\beth_\alpha(\kappa)</math>, for ordinals α and cardinals κ, is occasionally used. It is defined by:
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| :<math>\beth_0(\kappa)=\kappa,</math>
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| :<math>\beth_{\alpha+1}(\kappa)=2^{\beth_{\alpha}(\kappa)},</math>
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| :<math>\beth_{\lambda}(\kappa)=\sup\{ \beth_{\alpha}(\kappa):\alpha<\lambda \}</math> if λ is a limit ordinal.
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| So <math>\beth_{\alpha}=\beth_{\alpha}(\aleph_0).</math>
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| In ZF, for any cardinals κ and μ, there is an ordinal α such that:
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| :<math>\kappa \le \beth_{\alpha}(\mu).</math>
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| And in ZF, for any cardinal κ and ordinals α and β:
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| :<math>\beth_{\beta}(\beth_{\alpha}(\kappa)) = \beth_{\alpha+\beta}(\kappa).</math>
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| Consequently, in [[Zermelo–Fraenkel set theory]] absent [[ur-element]]s with or without the [[axiom of choice]], for any cardinals κ and μ, the equality
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| :<math>\beth_{\beta}(\kappa) = \beth_{\beta}(\mu)</math>
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| holds for all sufficiently large ordinals β (that is, there is an ordinal α such that the equality holds for every ordinal β ≥ α).
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| This also holds in Zermelo–Fraenkel set theory with ur-elements with or without the axiom of choice provided the ur-elements form a set which is equinumerous with a [[pure set]] (a set whose [[transitive set#Transitive closure|transitive closure]] contains no ur-elements). If the axiom of choice holds, then any set of ur-elements is equinumerous with a pure set.
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| ==References==
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| * T. E. Forster, ''Set Theory with a Universal Set: Exploring an Untyped Universe'', [[Oxford University Press]], 1995 — ''Beth number'' is defined on page 5.
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| * {{ cite book | last=Bell | first=John Lane | coauthors=Slomson, Alan B. | year=2006 | title=Models and Ultraproducts: An Introduction | edition=reprint of 1974 edition | origyear=1969 | publisher=[[Dover Publications]] | isbn=0-486-44979-3 }} See pages 6 and 204–205 for beth numbers.
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| * {{cite book
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| | last = Roitman
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| | first = Judith
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| | title = Introduction to Modern Set Theory
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| | date = 2011
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| | publisher = [[Virginia Commonwealth University]]
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| | isbn = 978-0-9824062-4-3 }} See page 109 for beth numbers.
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| [[Category:Cardinal numbers]]
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| [[Category:Infinity]]
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