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Beth number

Beth numbers are defined by transfinite recursion:

• ${\displaystyle \beth _{0}=\aleph _{0},}$
• ${\displaystyle \beth _{\alpha +1}=2^{\beth _{\alpha }},}$
• ${\displaystyle \beth _{\lambda }=\sup {\Bigl \{}\beth _{\alpha }:\alpha <\lambda {\Bigr \}},}$

where ${\displaystyle \alpha }$ is an ordinal and ${\displaystyle \lambda }$ is a limit ordinal.^[1]

The cardinal ${\displaystyle \beth _{0}=\aleph _{0}}$ is the cardinality of any countably infinite set such as the set ${\displaystyle \mathbb {N} }$ofnatural numbers, so that ${\displaystyle \beth _{0}=|\mathbb {N} |}$.

Let ${\displaystyle \alpha }$ be an ordinal, and ${\displaystyle A_{\alpha }}$ be a set with cardinality ${\displaystyle \beth _{\alpha }=|A_{\alpha }|}$. Then,

• ${\displaystyle {\mathcal {P}}(A_{\alpha })}$ denotes the power setof${\displaystyle A_{\alpha }}$ (i.e., the set of all subsets of ${\displaystyle A_{\alpha }}$),

• the set ${\displaystyle 2^{A_{\alpha }}\subset {\mathcal {P}}(A_{\alpha }\times 2)}$ denotes the set of all functions from ${\displaystyle A_{\alpha }}$to${\displaystyle \{0,1\}}$,

• the cardinal ${\displaystyle 2^{\beth _{\alpha }}}$ is the result of cardinal exponentiation, and

• ${\displaystyle \beth _{\alpha +1}=2^{\beth _{\alpha }}=\left|2^{A_{\alpha }}\right|=|{\mathcal {P}}(A_{\alpha })|}$ is the cardinality of the power set of ${\displaystyle A_{\alpha }}$.

Given this definition,

${\displaystyle \beth _{0},\beth _{1},\beth _{2},\beth _{3},\dots }$

are respectively the cardinalities of

${\displaystyle \mathbb {N} ,{\mathcal {P}}(\mathbb {N} ),{\mathcal {P}}({\mathcal {P}}(\mathbb {N} )),{\mathcal {P}}({\mathcal {P}}({\mathcal {P}}(\mathbb {N} ))),\dots }$

so that the second beth number ${\displaystyle \beth _{1}}$ is equal to ${\displaystyle {\mathfrak {c}}}$, the cardinality of the continuum (the cardinality of the set of the real numbers), and the third beth number ${\displaystyle \beth _{2}}$ 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 ordinals ${\displaystyle \lambda }$, the corresponding beth number is defined to be the supremum of the beth numbers for all ordinals strictly smaller than ${\displaystyle \lambda }$:

${\displaystyle \beth _{\lambda }=\sup {\Bigl \{}\beth _{\alpha }:\alpha <\lambda {\Bigr \}}.}$

One can show that this definition is equivalent to

${\displaystyle \beth _{\lambda }=|\bigcup {\Bigl \{}A_{\alpha }:\alpha <\lambda {\Bigr \}}|.}$

For instance:

• ${\displaystyle \beth _{\omega }}$ is the cardinality of ${\displaystyle \bigcup {\Bigl \{}\mathbb {N} ,{\mathcal {P}}(\mathbb {N} ),{\mathcal {P}}({\mathcal {P}}(\mathbb {N} )),{\mathcal {P}}({\mathcal {P}}({\mathcal {P}}(\mathbb {N} ))),\dots {\Bigr \}}}$.
• ${\displaystyle \beth _{2\omega }}$ is the cardinality of ${\displaystyle \bigcup {\Bigl \{}\mathbb {N} ,{\mathcal {P}}(\mathbb {N} ),{\mathcal {P}}({\mathcal {P}}(\mathbb {N} )),{\mathcal {P}}({\mathcal {P}}({\mathcal {P}}(\mathbb {N} ))),\dots ,{A_{\omega }},{\mathcal {P}}({A_{\omega }}),{\mathcal {P}}({\mathcal {P}}({A_{\omega }})),{\mathcal {P}}({\mathcal {P}}({\mathcal {P}}({A_{\omega }}))),\dots {\Bigr \}}}$.
• ${\displaystyle \beth _{\omega ^{2}}}$ is the cardinality of ${\displaystyle \bigcup {\Bigl \{}\mathbb {N} ,{\mathcal {P}}(\mathbb {N} ),{\mathcal {P}}({\mathcal {P}}(\mathbb {N} )),{\mathcal {P}}({\mathcal {P}}({\mathcal {P}}(\mathbb {N} ))),\dots ,{A_{\omega }},{\mathcal {P}}({A_{\omega }}),{\mathcal {P}}({\mathcal {P}}({A_{\omega }})),\dots ,{A_{2\omega }},{\mathcal {P}}({A_{2\omega }}),{\mathcal {P}}({\mathcal {P}}({A_{2\omega }})),\dots ,}$ ${\displaystyle {A_{3\omega }},{\mathcal {P}}({A_{3\omega }}),{\mathcal {P}}({\mathcal {P}}({A_{3\omega }})),\dots ,\dots {\Bigr \}}}$.

This equivalence can be shown by seeing that:

• for any set ${\displaystyle \mathbb {S} }$, the union set of all its members can be no larger than the supremum of its member cardinalities times its own cardinality, ${\displaystyle |\bigcup \mathbb {S} |\leq {\Bigl (}|\mathbb {S} |\times \sup {\Bigl \{}|s|:s\in \mathbb {S} {\Bigr \}}{\Bigr )}}$
• for any two non-zero cardinalities ${\displaystyle \kappa _{a},\kappa _{b}}$, if at least one of them is an infinite cardinality, then the product will be the larger of the two, ${\displaystyle \kappa _{a}\times \kappa _{b}=\max\{\kappa _{a},\kappa _{b}\}}$
• the set ${\displaystyle {\Bigl \{}A_{\alpha }:\alpha <\lambda {\Bigr \}}}$ will be smaller than most or all of its subsets for any limit ordinal ${\displaystyle \lambda }$
• therefore, ${\displaystyle |\bigcup {\Bigl \{}A_{\alpha }:\alpha <\lambda {\Bigr \}}|=\sup {\Bigl \{}\beth _{\alpha }:\alpha <\lambda {\Bigr \}}}$ for any limit ordinal ${\displaystyle \lambda }$

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

One can also show that the von Neumann universes ${\displaystyle V_{\omega +\alpha }}$ have cardinality ${\displaystyle \beth _{\alpha }}$.

Assuming the axiom of choice, infinite cardinalities are linearly ordered; no two cardinalities can fail to be comparable. Thus, since by definition no infinite cardinalities are between ${\displaystyle \aleph _{0}}$ and ${\displaystyle \aleph _{1}}$, it follows that

${\displaystyle \beth _{1}\geq \aleph _{1}.}$

Repeating this argument (see transfinite induction) yields ${\displaystyle \beth _{\alpha }\geq \aleph _{\alpha }}$ for all ordinals ${\displaystyle \alpha }$.

The continuum hypothesis is equivalent to

${\displaystyle \beth _{1}=\aleph _{1}.}$

The generalized continuum hypothesis says the sequence of beth numbers thus defined is the same as the sequence of aleph numbers, i.e., ${\displaystyle \beth _{\alpha }=\aleph _{\alpha }}$ for all ordinals ${\displaystyle \alpha }$.

Since this is defined to be ${\displaystyle \aleph _{0}}$, or aleph null, sets with cardinality ${\displaystyle \beth _{0}}$ include:

• the natural numbers ${\displaystyle \mathbb {N} }$
• the rational numbers ${\displaystyle \mathbb {Q} }$
• the algebraic numbers ${\displaystyle {\mathcal {A}}}$
• the computable numbers and computable sets
• the set of finite setsofintegers or of rationals or of algebraic numbers
• the set of finite multisetsofintegers
• the set of finite sequencesofintegers.

Sets with cardinality ${\displaystyle \beth _{1}}$ include:

• the transcendental numbers
• the irrational numbers
• the real numbers ${\displaystyle \mathbb {R} }$
• the complex numbers ${\displaystyle \mathbb {C} }$
• the uncomputable real numbers
• Euclidean space ${\displaystyle \mathbb {R} ^{n}}$
• the power set of the natural numbers ${\displaystyle 2^{\mathbb {N} }}$ (the set of all subsets of the natural numbers)
• the set of sequences of integers (i.e., ${\displaystyle \mathbb {Z} ^{\mathbb {N} }}$, which includes all functions from ${\displaystyle \mathbb {N} }$to${\displaystyle \mathbb {Z} }$)
• the set of sequences of real numbers, ${\displaystyle \mathbb {R} ^{\mathbb {N} }}$
• the set of all real analytic functions from ${\displaystyle \mathbb {R} }$to${\displaystyle \mathbb {R} }$
• the set of all continuous functions from ${\displaystyle \mathbb {R} }$to${\displaystyle \mathbb {R} }$
• the set of all functions from ${\displaystyle \mathbb {R} }$to${\displaystyle \mathbb {R} }$ with at most countable discontinuities ^[2]
• the set of finite subsets of real numbers
• the set of all analytic functions from ${\displaystyle \mathbb {C} }$to${\displaystyle \mathbb {C} }$ (the holomorphic functions)
• the set of all functions from the natural numbers to the natural numbers (${\displaystyle \mathbb {N} ^{\mathbb {N} }}$).

${\displaystyle \beth _{2}}$ (pronounced beth two) is also referred to as ${\displaystyle 2^{\mathfrak {c}}}$ (pronounced two to the power of ${\displaystyle {\mathfrak {c}}}$).

Sets with cardinality ${\displaystyle \beth _{2}}$ include:

• the power set of the set of real numbers, so it is the number of subsets 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 functions from ${\displaystyle \mathbb {R} }$to${\displaystyle \mathbb {R} }$ (${\displaystyle \mathbb {R} ^{\mathbb {R} }}$)
• the set of all functions from ${\displaystyle \mathbb {R} ^{m}}$to${\displaystyle \mathbb {R} ^{n}}$
• the set of all functions from ${\displaystyle \mathbb {R} }$to${\displaystyle \mathbb {R} }$ with uncountably many discontinuities ^[2]
• 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 compactificationsof${\displaystyle \mathbb {R} }$, ${\displaystyle \mathbb {Q} }$, and ${\displaystyle \mathbb {N} }$
• the set of deterministic fractalsin${\displaystyle \mathbb {R} ^{n}}$ ^[3]
• the set of random fractalsin${\displaystyle \mathbb {R} ^{n}}$.^[4]

${\displaystyle \beth _{\omega }}$ (pronounced beth omega) is the smallest uncountable strong limit cardinal.

The more general symbol ${\displaystyle \beth _{\alpha }(\kappa )}$, for ordinals ${\displaystyle \alpha }$ and cardinals ${\displaystyle \kappa }$, is occasionally used. It is defined by:

${\displaystyle \beth _{0}(\kappa )=\kappa ,}$

${\displaystyle \beth _{\alpha +1}(\kappa )=2^{\beth _{\alpha }(\kappa )},}$

${\displaystyle \beth _{\lambda }(\kappa )=\sup\{\beth _{\alpha }(\kappa ):\alpha <\lambda \}}$ifλ is a limit ordinal.

So

${\displaystyle \beth _{\alpha }=\beth _{\alpha }(\aleph _{0}).}$

InZermelo–Fraenkel set theory (ZF), for any cardinals ${\displaystyle \kappa }$ and ${\displaystyle \mu }$, there is an ordinal ${\displaystyle \alpha }$ such that:

${\displaystyle \kappa \leq \beth _{\alpha }(\mu ).}$

And in ZF, for any cardinal ${\displaystyle \kappa }$ and ordinals ${\displaystyle \alpha }$ and ${\displaystyle \beta }$:

${\displaystyle \beth _{\beta }(\beth _{\alpha }(\kappa ))=\beth _{\alpha +\beta }(\kappa ).}$

Consequently, in ZF absent ur-elements, with or without the axiom of choice, for any cardinals ${\displaystyle \kappa }$ and ${\displaystyle \mu }$, the equality

${\displaystyle \beth _{\beta }(\kappa )=\beth _{\beta }(\mu )}$

holds for all sufficiently large ordinals ${\displaystyle \beta }$. That is, there is an ordinal ${\displaystyle \alpha }$ such that the equality holds for every ordinal ${\displaystyle \beta \geq \alpha }$.

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 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 is implied by the existence of all beths of countable index.^[5]

• Transfinite number
• Uncountable set