Editing Beth number (section)

== 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]].