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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, the beth numbers are a certain sequence of
infinite Infinite may refer to: Mathematics *Infinite set, a set that is not a finite set *Infinity, an abstract concept describing something without any limit Music *Infinite (band), a South Korean boy band *''Infinite'' (EP), debut EP of American mus ...
cardinal number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s, conventionally written \beth_0,\ \beth_1,\ \beth_2,\ \beth_3,\ \dots, where \beth is the second
Hebrew letter The Hebrew alphabet ( he, wikt:אלפבית, אָלֶף־בֵּית עִבְרִי, ), known variously by scholars as the Ktav Ashuri, Jewish script, square script and block script, is an abjad script used in the writing of the Hebrew language ...

Hebrew letter
(
beth Beth may refer to: Letter and number *Bet (letter), or beth, the second letter of the Semitic abjads (writing systems) *Hebrew word for "house", often used in the name of synagogues and schools (e.g. Beth Israel (disambiguation), Beth Israel) Na ...
). The beth numbers are related to the
aleph number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s (\aleph_0,\ \aleph_1,\ \dots), but there may be numbers indexed by \aleph that are not indexed by \beth.


Definition

To define the beth numbers, start by letting :\beth_0=\aleph_0 be the cardinality of any
countably infinite In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
set; for concreteness, take the set \mathbb of
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
s to be a typical case. Denote by ''P''(''A'') the
power set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
of ''A'' (i.e., the set of all subsets of ''A''), then, define :\beth_=2^, which is the cardinality of the power set of ''A'' (if \beth_ is the cardinality of ''A''). Given this definition, :\beth_0,\ \beth_1,\ \beth_2,\ \beth_3,\ \dots are respectively the cardinalities of :\mathbb,\ P(\mathbb),\ P(P(\mathbb)),\ P(P(P(\mathbb))),\ \dots. so that the second beth number \beth_1 is equal to \mathfrak c, the
cardinality of the continuum In set theory Set theory is the branch of that studies , which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of , is mostly concerned with those that ar ...
(the cardinality of the set of the real numbers), and the third beth number \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, λ, the corresponding beth number, is defined to be the supremum of the beth numbers for all ordinals strictly smaller than λ: :\beth_=\sup\. One can also show that the von Neumann universes V_ have cardinality \beth_ .


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 \aleph_0 and \aleph_1, it follows that :\beth_1 \ge \aleph_1. Repeating this argument (see transfinite induction) yields \beth_\alpha \ge \aleph_\alpha for all ordinals \alpha. The continuum hypothesis is equivalent to :\beth_1=\aleph_1. 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s, i.e., \beth_\alpha = \aleph_\alpha for all ordinals \alpha.


Specific cardinals


Beth null

Since this is defined to be \aleph_0, or aleph null, sets with cardinality \beth_0 include: *the
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
s N *the rational numbers Q *the algebraic numbers *the computable numbers and computable sets *the set of finite sets of integers *the set of Multiset, finite multisets of integers *the set of finite sequences of integers


Beth one

Sets with cardinality \beth_1 include: *the transcendental numbers *the irrational numbers *the real numbers R *the complex numbers C *the uncomputable real numbers *Euclidean space R''n'' *the
power set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
of the
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
s (the set of all subsets of the natural numbers) *the set of sequences of integers (i.e. all functions N → Z, often denoted ZN) *the set of sequences of real numbers, RN *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

\beth_2 (pronounced ''beth two'') is also referred to as 2''c'' (pronounced ''two to the power of c''). Sets with cardinality \beth_2 include: * The
power set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
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 function (mathematics), functions from R to R (RR) * The set of all functions from R''m'' to R''n'' * 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

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


Generalization

The more general symbol \beth_\alpha(\kappa), for ordinals ''α'' and cardinals ''κ'', is occasionally used. It is defined by: :\beth_0(\kappa)=\kappa, :\beth_(\kappa)=2^, :\beth_\lambda(\kappa)=\sup\ if λ is a limit ordinal. So :\beth_\alpha=\beth_\alpha(\aleph_0). In ZF, for any cardinals ''κ'' and ''μ'', there is an ordinal ''α'' such that: :\kappa \le \beth_\alpha(\mu). And in ZF, for any cardinal κ and ordinals ''α'' and ''β'': :\beth_\beta(\beth_\alpha(\kappa)) = \beth_(\kappa). Consequently, in Zermelo–Fraenkel set theory absent ur-elements with or without the axiom of choice, for any cardinals κ and μ, the equality :\beth_\beta(\kappa) = \beth_\beta(\mu) 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 that 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.


See also

* Transfinite number * Uncountable set


References


Bibliography

* T. E. Forster, ''Set Theory with a Universal Set: Exploring an Untyped Universe'', Oxford University Press, 1995 — ''Beth number'' is defined on page 5. * 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. Cardinal numbers Infinity