Beth number
Encyclopedia
In mathematics
, the infinite cardinal number
s are represented by the Hebrew letter
(aleph
) indexed with a subscript that runs over the ordinal number
s (see aleph number
). The second Hebrew letter
(beth
) is used in a related way, but does not necessarily index all of the numbers indexed by
.
be the cardinality of any countably infinite set; for concreteness, take the set
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
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 universe
s
have cardinality 
.
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 number
s, i.e.,
for all ordinals
.
or aleph null then sets with cardinality 
include:
include:
is also referred to as 2c .
Sets with cardinality
include:
is the smallest uncountable strong limit cardinal.
, for ordinals α and cardinals κ, is occasionally used. It is defined by:
if λ is a limit ordinal.
So
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-element
s 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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the infinite cardinal number
Cardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite...
s are represented by the Hebrew letter
(alephAleph (letter)
' is the reconstructed name of the first letter of the Proto-Canaanite alphabet, continued in descended Semitic alphabets as Phoenician ' , Syriac ' , Hebrew Aleph , and Arabic ' ....
) indexed with a subscript that runs over the ordinal number
Ordinal number
In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...
s (see aleph number
Aleph number
In set theory, a discipline within mathematics, the aleph numbers are a sequence of numbers used to represent the cardinality of infinite sets. They are named after the symbol used to denote them, the Hebrew letter aleph...
). The second Hebrew letter
Hebrew alphabet
The Hebrew alphabet , known variously by scholars as the Jewish script, square script, block script, or more historically, the Assyrian script, is used in the writing of the Hebrew language, as well as other Jewish languages, most notably Yiddish, Ladino, and Judeo-Arabic. There have been two...
(bethBet (letter)
Bet, Beth, Beh, or Vet is the second letter of many Semitic abjads, including Arabic alphabet , Aramaic, Hebrew , Phoenician and Syriac...
) is used in a related way, but does not necessarily index all of the numbers indexed by
.Definition
To define the beth numbers, start by lettingbe the cardinality of any countably infinite set; for concreteness, take the set
of natural numberNatural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
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
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 continuumCardinality of the continuum
In set theory, the cardinality of the continuum is the cardinality or “size” of the set of real numbers \mathbb R, sometimes called the continuum. It is an infinite cardinal number and is denoted by |\mathbb R| or \mathfrak c ....
, and the third beth number
is the cardinality of the power set of the continuum.Because of Cantor's theorem
Cantor's theorem
In elementary set theory, Cantor's theorem states that, for any set A, the set of all subsets of A has a strictly greater cardinality than A itself...
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
Supremum
In mathematics, given a subset S of a totally or partially ordered set T, the supremum of S, if it exists, is the least element of T that is greater than or equal to every element of S. Consequently, the supremum is also referred to as the least upper bound . If the supremum exists, it is unique...
of the beth numbers for all ordinals strictly smaller than λ:
One can also show that the von Neumann universe
Von Neumann universe
In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted V, is the class of hereditary well-founded sets...
s
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 thatRepeating this argument (see transfinite induction
Transfinite induction
Transfinite induction is an extension of mathematical induction to well-ordered sets, for instance to sets of ordinal numbers or cardinal numbers.- Transfinite induction :Let P be a property defined for all ordinals α...
) yields
for all ordinals
.The continuum hypothesis
Continuum hypothesis
In mathematics, the continuum hypothesis is a hypothesis, advanced by Georg Cantor in 1874, about the possible sizes of infinite sets. It states:Establishing the truth or falsehood of the continuum hypothesis is the first of Hilbert's 23 problems presented in the year 1900...
is equivalent to
The generalized continuum hypothesis says the sequence of beth numbers thus defined is the same as the sequence of aleph number
Aleph number
In set theory, a discipline within mathematics, the aleph numbers are a sequence of numbers used to represent the cardinality of infinite sets. They are named after the symbol used to denote them, the Hebrew letter aleph...
s, i.e.,
for all ordinals
.Beth null
Since this is defined to be or aleph null then sets with cardinality include:- the natural numberNatural numberIn mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
s N - the rational numberRational numberIn mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
s Q - the algebraic numberAlgebraic numberIn mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with rational coefficients. Numbers such as π that are not algebraic are said to be transcendental; almost all real numbers are transcendental...
s - the computable numberComputable numberIn mathematics, particularly theoretical computer science and mathematical logic, the computable numbers, also known as the recursive numbers or the computable reals, are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm...
s and computable sets - the set of finite sets of integerIntegerThe integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s
Beth one
Sets with cardinality include:- the transcendental numbers
- the irrational numberIrrational numberIn mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b non-zero, and is therefore not a rational number....
s - the real numberReal numberIn mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s R - the complex numberComplex numberA complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s C - Euclidean spaceEuclidean spaceIn mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
Rn - the power set of the natural numberNatural numberIn mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
s (the set of all subsets of the natural numbers) - the set of sequenceSequenceIn mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
s of integers (i.e. all functions N → Z, often denoted ZN) - the set of sequences of real numbers, RN
- the set of all continuous functionContinuous functionIn mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
s from R to R - the set of finite subsets of real numbers
Beth two
is also referred to as 2c .Sets with cardinality
include:- The power set of the set of real numberReal numberIn mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s, so it is the number of subsetSubsetIn mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
s of the real lineReal lineIn mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...
, 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 functionsFunction (mathematics)In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
from R to R (RR) - The set of all functions from Rm 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 compactificationStone–Cech compactificationIn the mathematical discipline of general topology, Stone–Čech compactification is a technique for constructing a universal map from a topological space X to a compact Hausdorff space βX...
s of R, Q, and N
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.So
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
Zermelo–Fraenkel set theory
In mathematics, Zermelo–Fraenkel set theory with the axiom of choice, named after mathematicians Ernst Zermelo and Abraham Fraenkel and commonly abbreviated ZFC, is one of several axiomatic systems that were proposed in the early twentieth century to formulate a theory of sets without the paradoxes...
absent ur-element
Ur-element
In set theory, a branch of mathematics, an urelement or ur-element is an object which is not a set, but that may be an element of a set...
s 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
Transitive closure
In mathematics, the transitive closure of a binary relation R on a set X is the transitive relation R+ on set X such that R+ contains R and R+ is minimal . If the binary relation itself is transitive, then the transitive closure will be that same binary relation; otherwise, the transitive closure...
contains no ur-elements). If the axiom of choice holds, then any set of ur-elements is equinumerous with a pure set.