Editing Beth number

{{short description|Infinite Cardinal number}}
In [[mathematics]], particularly in [[set theory]], the '''beth numbers''' are a certain sequence of [[infinite set|infinite]] [[cardinal number]]s (also known as [[transfinite number]]s), conventionally written <math>\beth_0, \beth_1, \beth_2, \beth_3, \dots</math>, where <math>\beth</math> is the [[Hebrew alphabet|Hebrew letter]] [[bet (letter)|beth]]. The beth numbers are related to the [[aleph number]]s (<math>\aleph_0, \aleph_1, \dots</math>), but unless the [[generalized continuum hypothesis]] is true, there are numbers indexed by <math>\aleph</math> that are not indexed by <math>\beth</math> or <math>\gimel</math>. See: [[Gimel function]]

== Definition ==
Beth numbers are defined by [[transfinite recursion]]:

* <math>\beth_0 = \aleph_0,</math>
* <math>\beth_{\alpha+1} = 2^{\beth_\alpha},</math>
* <math>\beth_\lambda = \sup\Bigl\{ \beth_\alpha : \alpha < \lambda \Bigr\},</math>

where <math>\alpha</math> is an ordinal and <math>\lambda</math> is a [[limit ordinal]].<ref>{{cite book |last=Jech |first=Thomas |year=2002 |title=Set Theory |edition=3rd |quote = Millennium ed, rev. and expanded. Corrected 4th printing 2006. |location= |publisher=Springer |page=55 |isbn=978-3-540-44085-7 }}</ref>

The cardinal <math>\beth_0 = \aleph_0</math> is the cardinality of any [[countably infinite]] [[set (mathematics)|set]] such as the set <math>\mathbb{N}</math> of [[natural number]]s, so that <math>\beth_0 = |\mathbb{N}|</math>.

Let <math>\alpha</math> be an [[ordinal number|ordinal]], and <math>A_\alpha</math> be a set with cardinality <math>\beth_\alpha = |A_\alpha|</math>. Then, 

* <math>\mathcal{P}(A_\alpha)</math> denotes the [[power set]] of <math>A_\alpha</math> (i.e., the set of all subsets of <math>A_\alpha</math>), 

* the set <math>2^{A_\alpha} \subset \mathcal{P}(A_\alpha \times 2)</math> denotes the set of all functions from <math>A_\alpha</math> to <math>\{0, 1\}</math>,

* the cardinal <math>2^{\beth_\alpha}</math> is the result of [[cardinal exponentiation]], and

* <math>\beth_{\alpha+1} = 2^{\beth_\alpha} = \left| 2^{A_\alpha} \right| = |\mathcal{P}(A_\alpha)|</math> is the cardinality of the power set of <math>A_\alpha</math>.

Given this definition,

:<math>\beth_0, \beth_1, \beth_2, \beth_3, \dots</math>

are respectively the cardinalities of

:<math>\mathbb{N}, \mathcal{P}(\mathbb{N}), \mathcal{P}(\mathcal{P}(\mathbb{N})), \mathcal{P}(\mathcal{P}(\mathcal{P}(\mathbb{N}))), \dots</math>

so that the second beth number <math>\beth_1</math> is equal to <math>\mathfrak{c}</math>, the [[cardinality of the continuum]] (the cardinality of the set of the [[real number]]s), and the third beth number <math>\beth_2</math> is the cardinality of the power set of the continuum.

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 <math>\lambda</math>, the corresponding beth number is defined to be the [[supremum]] of the beth numbers for all ordinals strictly smaller than <math>\lambda</math>:

:<math>\beth_\lambda = \sup \Bigl\{ \beth_{\alpha} : \alpha < \lambda \Bigr\}.</math>

One can show that this definition is equivalent to

:<math>\beth_\lambda = |\bigcup \Bigl\{ A_{\alpha} : \alpha < \lambda \Bigr\}|.</math>

For instance:

*<math>\beth_\omega</math> is the cardinality of <math>\bigcup \Bigl\{\mathbb{N}, \mathcal{P}(\mathbb{N}), \mathcal{P}(\mathcal{P}(\mathbb{N})), \mathcal{P}(\mathcal{P}(\mathcal{P}(\mathbb{N}))), \dots \Bigr\}</math>.
*<math>\beth_{\omega2}</math> is the cardinality of <math>\bigcup \Bigl\{\mathbb{N}, \mathcal{P}(\mathbb{N}), \mathcal{P}(\mathcal{P}(\mathbb{N})), \mathcal{P}(\mathcal{P}(\mathcal{P}(\mathbb{N}))), \dots, {A_\omega}, \mathcal{P}({A_\omega}), \mathcal{P}(\mathcal{P}({A_\omega})), \mathcal{P}(\mathcal{P}(\mathcal{P}({A_\omega}))), \dots\Bigr\}</math>.
*<math>\beth_{\omega^2}</math> is the cardinality of <math>\bigcup \Bigl\{\mathbb{N}, \mathcal{P}(\mathbb{N}), \mathcal{P}(\mathcal{P}(\mathbb{N})), \mathcal{P}(\mathcal{P}(\mathcal{P}(\mathbb{N}))), \dots, {A_\omega}, \mathcal{P}({A_\omega}), \mathcal{P}(\mathcal{P}({A_\omega})), \dots, {A_{\omega2}}, \mathcal{P}({A_{\omega2}}), \mathcal{P}(\mathcal{P}({A_{\omega2}})), \dots, </math> <math> {A_{\omega3}}, \mathcal{P}({A_{\omega3}}), \mathcal{P}(\mathcal{P}({A_{\omega3}})), \dots, \dots \Bigr\}</math>.

This equivalence can be shown by seeing that:

*for any set <math>\mathbb{S}</math>, the union set of all its members can be no larger than the supremum of its member cardinalities times its own cardinality, <math>|\bigcup\mathbb{S}|\le \Bigl(|\mathbb{S}| \times \sup\Bigl\{|s|:s\in\mathbb{S}\Bigr\}\Bigr)</math>
*for any two non-zero cardinalities <math>\kappa_a, \kappa_b</math>, if at least one of them is an infinite cardinality, then the product will be the larger of the two, <math>\kappa_a \times \kappa_b = \max\{\kappa_a, \kappa_b\}</math>
*the set <math>\Bigl\{ A_{\alpha} : \alpha < \lambda \Bigr\}</math> will be smaller than most or all of its subsets for any limit ordinal <math>\lambda</math>
*therefore, <math>|\bigcup\Bigl\{ A_{\alpha} : \alpha < \lambda \Bigr\}|=\sup \Bigl\{ \beth_{\alpha} : \alpha < \lambda \Bigr\}</math> for any limit ordinal <math>\lambda</math>

Note that this behavior is different from that of successor ordinals. Cardinalities less than <math>\beth_\beta</math> but greater than any <math>\beth_\alpha: \alpha<\beta</math> can exist when <math>\beta</math> is a successor ordinal (in that case, the existence is undecidable in ZFC and controlled by the [[Generalized Continuum Hypothesis]]); but cannot exist when <math>\beta</math> is a limit ordinal, even under the second definition presented.

One can also show that the [[von Neumann universe]]s <math>V_{\omega+\alpha}</math> have cardinality <math>\beth_{\alpha}</math>.

== Relation to the aleph numbers ==
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 
:<math>\beth_1 \ge \aleph_1.</math>
Repeating this argument (see [[transfinite induction]]) yields 
<math>\beth_\alpha \ge \aleph_\alpha</math> 
for all ordinals <math>\alpha</math>.

The [[continuum hypothesis]] is equivalent to
:<math>\beth_1=\aleph_1.</math>

The [[Continuum hypothesis#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., 
<math>\beth_\alpha = \aleph_\alpha</math> 
for all ordinals <math>\alpha</math>.

== Specific cardinals ==

=== Beth null ===
Since this is defined to be <math>\aleph_0</math>, or [[aleph null]], sets with cardinality <math>\beth_0</math> include:

* the [[natural number]]s <math>\mathbb{N}</math>
* the [[rational number]]s <math>\mathbb{Q}</math>
* the [[algebraic number]]s <math>\mathbb{A}</math>
* the [[computable number]]s and [[computable set]]s
* the set of [[finite set]]s of [[integer]]s or of [[rational number|rationals]] or of [[algebraic number]]s
* the set of [[Multiset|finite multiset]]s of [[integer]]s
* the set of [[finite sequence]]s of [[integer]]s.

=== Beth one ===

{{main|cardinality of the continuum}}

Sets with cardinality <math>\beth_1</math> include:

* the [[transcendental numbers]]
* the [[irrational number]]s
* the [[real number]]s <math>\mathbb{R}</math>
* the [[complex number]]s <math>\mathbb{C}</math>
* the [[uncomputable real number]]s
* [[Euclidean space]] <math>\mathbb{R}^n</math>
* the [[power set]] of the [[natural number]]s <math>2^\mathbb{N}</math> (the set of all subsets of the natural numbers)
* the set of [[sequence]]s of integers (i.e., <math>\mathbb{Z}^\mathbb{N}</math>, which includes all functions from <math>\mathbb{N}</math> to <math>\mathbb{Z}</math>)
* the set of sequences of real numbers, <math>\mathbb{R}^\mathbb{N}</math> 
* the set of all [[real analytic function]]s from <math>\mathbb{R}</math> to <math>\mathbb{R}</math>
* the set of all [[continuous function]]s from <math>\mathbb{R}</math> to <math>\mathbb{R}</math>
* the set of all functions from <math>\mathbb{R}</math> to <math>\mathbb{R}</math> with at most countable discontinuities <ref name=":3">{{cite journal |last=Soltanifar |first=Mohsen |year=2023 |title= A classification of elements of function space F(R,R) |journal=Mathematics |volume=11 |issue=17 |page=3715 |doi=10.3390/math11173715 |doi-access=free |arxiv=2308.06297 }}</ref>
*the set of finite subsets of real numbers
*the set of all [[analytic function]]s from <math>\mathbb{C}</math> to <math>\mathbb{C}</math> (the [[holomorphic]] functions)
*the set of all functions from the natural numbers to the natural numbers (<math>\mathbb{N}^\mathbb{N}</math>).

=== Beth two ===
<math>\beth_2</math> (pronounced ''beth two'') is also referred to as <math>2^\mathfrak{c}</math> (pronounced ''two to the power of <math>\mathfrak{c}</math>'').

Sets with cardinality <math>\beth_2</math> include:

* 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
* the power set of the power set of the set of natural numbers
* the set of all [[function (mathematics)|functions]] from <math>\mathbb{R}</math> to <math>\mathbb{R}</math> (<math>\mathbb{R}^\mathbb{R}</math>)
* the set of all functions from <math>\mathbb{R}^m</math> to <math>\mathbb{R}^n</math>
* the set of all functions from <math>\mathbb{R}</math> to <math>\mathbb{R}</math> with uncountably many discontinuities <ref name=":3"/>
* the power set of the set of all functions from the set of natural numbers to itself, or the number of sets of sequences of natural numbers
* the [[Stone–Čech compactification]]s of <math>\mathbb{R}</math>, <math>\mathbb{Q}</math>, and <math>\mathbb{N}</math>
* the set of deterministic [[fractal]]s in <math>\mathbb{R}^n</math> <ref name=":4">{{cite journal  |last=Soltanifar |first=Mohsen |year=2021 |title=A generalization of the Hausdorff dimension theorem for deterministic fractals |journal=Mathematics |volume=9 |issue=13 |page=1546 |arxiv=2007.07991 |doi=10.3390/math9131546 |doi-access=free }}</ref>
* the set of random [[fractal]]s in <math>\mathbb{R}^n</math>.<ref name=":5">{{cite journal |last=Soltanifar |first=Mohsen |year=2022 |title=The second generalization of the Hausdorff dimension theorem for random fractals |journal=Mathematics |volume=10 |issue=5 |page=706 |hdl=1807/110291 |hdl-access=free |doi=10.3390/math10050706 |doi-access=free }}</ref>

=== Beth omega ===
<math>\beth_\omega</math> (pronounced ''beth omega'') is the smallest [[uncountable]] [[strong limit cardinal]].

==Generalization==
The more general symbol <math>\beth_\alpha(\kappa)</math>, for ordinals <math>\alpha</math> and cardinals <math>\kappa</math>, is occasionally used. It is defined by:
:<math>\beth_0(\kappa)=\kappa,</math>
:<math>\beth_{\alpha+1}(\kappa)=2^{\beth_\alpha(\kappa)},</math>
:<math>\beth_\lambda(\kappa)=\sup\{ \beth_\alpha(\kappa):\alpha<\lambda \}</math> if ''λ'' is a limit ordinal.

So
:<math>\beth_\alpha=\beth_\alpha(\aleph_0).</math>

In [[Zermelo–Fraenkel set theory]] (ZF), for any cardinals <math>\kappa</math> and <math>\mu</math>, there is an ordinal <math>\alpha</math> such that:

:<math>\kappa \le \beth_\alpha(\mu).</math>

And in ZF, for any cardinal <math>\kappa</math> and ordinals <math>\alpha</math> and <math>\beta</math>:

:<math>\beth_\beta(\beth_\alpha(\kappa)) = \beth_{\alpha+\beta}(\kappa).</math>

Consequently, in ZF absent [[ur-element]]s, with or without the [[axiom of choice]], for any cardinals <math>\kappa</math> and <math>\mu</math>, the equality

:<math>\beth_\beta(\kappa) = \beth_\beta(\mu)</math>

holds for all sufficiently large ordinals <math>\beta</math>. That is, there is an ordinal <math>\alpha</math> such that the equality holds for every ordinal <math>\beta \geq \alpha</math>.

This also holds in Zermelo–Fraenkel set theory with ur-elements (with or without the axiom of choice), provided that 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.

== Borel determinacy ==

[[Borel determinacy]] is implied by the existence of all beths of countable index.<ref>{{cite web
 |url=https://golem.ph.utexas.edu/category/2021/07/borel_determinacy_does_not_require_replacement.html
 |title=Borel Determinacy Does Not Require Replacement
 |last=Leinster
 |first=Tom
 |date=23 July 2021
 |website=The n-Category Café
 |publisher=The University of Texas at Austin
 |access-date=25 August 2021
 |quote=}}</ref>

== See also ==

* [[Transfinite number]]
* [[Uncountable set]]

== References ==
<references />

==Bibliography==
{{refbegin}}
* [[Thomas Forster|T. E. Forster]], ''Set Theory with a Universal Set: Exploring an Untyped Universe'', [[Oxford University Press]], 1995 &mdash; ''Beth number'' is defined on page 5.
* {{ cite book | last=Bell | first=John Lane |author1link = John Lane Bell|author2=Slomson, Alan B. | year=2006 | title=Models and Ultraproducts: An Introduction | edition=reprint of 1974 | orig-year=1969 | publisher=[[Dover Publications]] | isbn=0-486-44979-3 }} See pages 6 and 204–205 for beth numbers.
* {{cite book
  | last = Roitman
  | first = Judith
|authorlink = Judith Roitman
  | title = Introduction to Modern Set Theory
  | date = 2011
  | publisher = [[Virginia Commonwealth University]]
  | isbn = 978-0-9824062-4-3 }} See page 109 for beth numbers.
{{refend}}

[[Category:Cardinal numbers]]
[[Category:Infinity]]