Nevanlinna theory: Difference between revisions

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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 ==
To define the '''beth numbers''', start by letting
 
:<math>\beth_0=\aleph_0</math>
 
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
 
:<math>\beth_{\alpha+1}=2^{\beth_{\alpha}},</math>
 
which is the cardinality of the power set of ''A'' if <math>\beth_{\alpha}</math> is the cardinality of ''A''.
 
Given this definition,
 
:<math>\beth_0,\ \beth_1,\ \beth_2,\ \beth_3,\ \dots</math>
 
are respectively the cardinalities of
 
:<math>\mathbb{N},\ P(\mathbb{N}),\ P(P(\mathbb{N})),\ P(P(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]], 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 λ the corresponding beth number is defined as the [[supremum]] of the beth numbers for all ordinals strictly smaller than λ:
 
:<math>\beth_{\lambda}=\sup\{ \beth_{\alpha}:\alpha<\lambda \}.</math>
 
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#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.,
<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]] then sets with cardinality <math>\beth_0</math> include:
 
*the [[natural number]]s '''N'''
*the [[rational number]]s '''Q'''
*the [[algebraic number]]s
*the [[computable number]]s and [[computable set]]s
*the set of [[finite set]]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 '''R'''
*the [[complex number]]s '''C'''
*[[Euclidean space]] '''R'''<sup>''n''</sup>
*the [[power set]] of the [[natural number]]s (the set of all subsets of the natural numbers)
*the set of [[sequence]]s of integers (i.e. all functions '''N''' → '''Z''', often denoted '''Z'''<sup>'''N'''</sup>)
*the set of sequences of real numbers, '''R'''<sup>'''N'''</sup>
*the set of all [[continuous function]]s from '''R''' to '''R'''
*the set of finite subsets of real numbers
 
=== Beth two ===
<math>\beth_2</math> (pronounced ''beth two'') is also referred to as '''2<sup>''c''</sup>''' (pronounced ''two to the power of c'').
 
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 '''R''' to '''R''' ('''R'''<sup>'''R'''</sup>)
* The set of all functions from '''R'''<sup>''m''</sup> to '''R'''<sup>''n''</sup>
* 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
* The [[Stone–Čech compactification]]s of '''R''', '''Q''', and '''N'''
 
=== 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 α and cardinals κ, 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 ZF, for any cardinals κ and μ, there is an ordinal α such that:
 
:<math>\kappa \le \beth_{\alpha}(\mu).</math>
 
And in ZF, for any cardinal κ and ordinals α and β:
 
:<math>\beth_{\beta}(\beth_{\alpha}(\kappa)) = \beth_{\alpha+\beta}(\kappa).</math>
 
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
 
:<math>\beth_{\beta}(\kappa) = \beth_{\beta}(\mu)</math>
 
holds for all sufficiently large ordinals β (that is, there is an ordinal α such that the equality holds for every ordinal β ≥ α).
 
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.
 
==References==
 
* 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 | 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.
* {{cite book
  | last = Roitman
  | first = Judith
  | title = Introduction to Modern Set Theory
  | date = 2011
  | publisher = [[Virginia Commonwealth University]]
  | isbn = 978-0-9824062-4-3 }} See page 109 for beth numbers.
 
[[Category:Cardinal numbers]]
[[Category:Infinity]]

Revision as of 13:02, 24 February 2014

I'm Salina and I live with my husband and our three children in Tredington, in the south part. My hobbies are Herpetoculture, Basket Weaving and Locksport.

my blog; Fifa 15 coin generator