Editing Beth number (section)

== 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>.