“Beth number”的意思、由来-开放百科全书

词条

Beth number

释义

Definition
Relation to the aleph numbers
Specific cardinals
Beth null Beth one Beth two Beth omega
Generalization
References

In mathematics, the beth numbers are a certain sequence of infinite cardinal numbers, conventionally written , where is the second Hebrew letter (beth). The beth numbers are related to the aleph numbers (), but there may be numbers indexed by that are not indexed by .

Definition

To define the beth numbers, start by letting

be the cardinality of any countably infinite set; for concreteness, take the set of natural numbers 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

which is the cardinality of the power set of A if is the cardinality of A.

Given this definition,

are respectively the cardinalities of

so that the second beth number is equal to , the cardinality of the continuum, and the third beth number 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 λ the corresponding beth number is defined as the supremum of the beth numbers for all ordinals strictly smaller than λ:

One can also show that the von Neumann universes have cardinality .

Relation to the aleph numbers

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 and , it follows that

Repeating this argument (see transfinite induction) yields

for all ordinals .

The continuum hypothesis is equivalent to

The generalized continuum hypothesis says the sequence of beth numbers thus defined is the same as the sequence of aleph numbers, i.e.,

for all ordinals .

Specific cardinals

Beth null

Since this is defined to be or aleph null then sets with cardinality include:

the natural numbers N
the rational numbers Q
the algebraic numbers
the computable numbers and computable sets
the set of finite sets of integers
the set of finite multisets of integers
the set of finite sequences of integers

Beth one

Sets with cardinality include:

the transcendental numbers
the irrational numbers
the real numbers R
the complex numbers C
the uncomputable real numbers
Euclidean space Rⁿ
the power set of the natural numbers (the set of all subsets of the natural numbers)
the set of sequences of integers (i.e. all functions N → Z, often denoted Z^N)
the set of sequences of real numbers, R^N
the set of all real analytic functions from R to R
the set of all continuous functions from R to R
the set of finite subsets of real numbers
the set of all analytic functions from C to C

Beth two

(pronounced beth two) is also referred to as 2^c (pronounced two to the power of c).

Sets with cardinality 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 R to R (R^R)
The set of all functions from R^m to Rⁿ
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 compactifications of R, Q, and N

Beth omega

(pronounced beth omega) is the smallest uncountable strong limit cardinal.

Generalization

The more general symbol , for ordinals α and cardinals κ, is occasionally used. It is defined by:

if λ is a limit ordinal.

In ZF, for any cardinals κ and μ, there is an ordinal α such that:

And in ZF, for any cardinal κ and ordinals α and β:

Consequently, in Zermelo–Fraenkel set theory absent ur-elements with or without the axiom of choice, for any cardinals κ and μ, the equality

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 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 — Beth number is defined on page 5.
{{ cite book | last=Bell | first=John Lane |author2=Slomson, Alan B. | year=2006 | title=Models and Ultraproducts: An Introduction | edition=reprint of 1974 | 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.

2 : Cardinal numbers|Infinity

随便看

开放百科全书收录14589846条英语、德语、日语等多语种百科知识，基本涵盖了大多数领域的百科知识，是一部内容自由、开放的电子版国际百科全书。