In

axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ...

holds, this is the next larger cardinal).
We can then define the aleph numbers as follows:
and for λ, an infinite

axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ...

. ZFC set theory, which includes the axiom of choice, implies that every infinite set has an aleph number as its cardinality (i.e. is equinumerous with its initial ordinal), and thus the initial ordinals of the aleph numbers serve as a class of representatives for all possible infinite cardinal numbers.
When cardinality is studied in ZF without the axiom of choice, it is no longer possible to prove that each infinite set has some aleph number as its cardinality; the sets whose cardinality is an aleph number are exactly the infinite sets that can be well-ordered. The method of Scott's trick is sometimes used as an alternative way to construct representatives for cardinal numbers in the setting of ZF. For example, one can define card(''S'') to be the set of sets with the same cardinality as ''S'' of minimum possible rank. This has the property that card(''S'') = card(''T'') if and only if ''S'' and ''T'' have the same cardinality. (The set card(''S'') does not have the same cardinality of ''S'' in general, but all its elements do.)

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 ...

, particularly in set theory
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, 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 mathematics, i ...

, the aleph numbers are a sequence
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 t ...

of numbers used to represent the cardinality
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

(or size) of infinite set
In set theory
illustrating the intersection (set theory), intersection of two set (mathematics), sets.
Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections of objects. Although any ...

s that can be well-ordered
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

. They were introduced by the mathematician Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor ( , ; – January 6, 1918) was a German mathematician. He created set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence be ...

and are named after the symbol he used to denote them, the Hebrew
Hebrew (, , or ) is a Northwest Semitic languages, Northwest Semitic language of the Afroasiatic languages, Afroasiatic language family. Historically, it is regarded as the language of the Israelites, Judeans and their ancestors. It is the o ...

letter aleph
Aleph (or alef or alif, transliterated ʾ) is the first letter
Letter, letters, or literature may refer to:
Characters typeface
* Letter (alphabet)
A letter is a segmental symbol
A symbol is a mark, sign, or word that indicates, sig ...

($\backslash aleph$).
(Though in older mathematics books, the letter aleph is often printed upside down by accident,For example, in the letter aleph appears both the right way up and upside down partly because a monotype
Monotyping is a type of printmaking
300px, Rembrandt, ''Self-portrait'', etching">Self-portrait.html" ;"title="Rembrandt, ''Self-portrait">Rembrandt, ''Self-portrait'', etching, c.1630
Printmaking is the process of creating artworks by ...

matrix for aleph was mistakenly constructed the wrong way up).
The cardinality 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 is $\backslash aleph\_0$ (read ''aleph-nought'' or ''aleph-zero''; the term ''aleph-null'' is also sometimes used), the next larger cardinality of a well-order
In mathematics, a well-order (or well-ordering or well-order relation) on a Set (mathematics), set ''S'' is a total order on ''S'' with the property that every non-empty subset of ''S'' has a least element in this ordering. The set ''S'' together ...

able set is aleph-one $\backslash aleph\_1$, then $\backslash aleph\_2$ and so on. Continuing in this manner, it is possible to define a cardinal number
150px, Aleph null, the smallest infinite cardinal
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ca ...

$\backslash aleph\_\backslash alpha$ for every ordinal number
In set theory
illustrating the intersection of two sets
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a s ...

$\backslash alpha$, as described below.
The concept and notation are due to Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor ( , ; – January 6, 1918) was a German mathematician. He created set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence be ...

, who defined the notion of cardinality and realized that infinite sets can have different cardinalities.
The aleph numbers differ from the infinity
Infinity is that which is boundless, endless, or larger than any number
A number is a mathematical object
A mathematical object is an abstract concept arising in mathematics.
In the usual language of mathematics, an ''object'' is anything ...

($\backslash infty$) commonly found in algebra and calculus, in that the alephs measure the sizes of sets, while infinity is commonly defined either as an extreme limit
Limit or Limits may refer to:
Arts and media
* Limit (music), a way to characterize harmony
* Limit (song), "Limit" (song), a 2016 single by Luna Sea
* Limits (Paenda song), "Limits" (Paenda song), 2019 song that represented Austria in the Eurov ...

of the real number line
Real may refer to:
Currencies
* Brazilian real
The Brazilian real ( pt, real, plural, pl. '; currency symbol, sign: R$; ISO 4217, code: BRL) is the official currency of Brazil. It is subdivided into 100 centavos. The Central Bank of Brazil i ...

(applied to a function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...

or sequence
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 t ...

that " diverges to infinity" or "increases without bound"), or as an extreme point of the extended real number line
In mathematics, the affinely extended real number system is obtained from the real number system \R by adding two infinity elements: +\infty and where the infinities are treated as actual numbers. It is useful in describing the algebra on infiniti ...

.
Aleph-nought

$\backslash aleph\_0$ (aleph-nought, also aleph-zero or aleph-null) is the cardinality of the set of all natural numbers, and is an infinite cardinal. The set of all finite ordinals, called $\backslash omega$ or $\backslash omega\_$ (where $\backslash omega$ is the lowercase Greek letteromega
Omega (; capital
Capital most commonly refers to:
* Capital letter
Letter case (or just case) is the distinction between the letters that are in larger uppercase or capitals (or more formally ''majuscule'') and smaller lowercase (o ...

), has cardinality $\backslash aleph\_0$. A set has cardinality $\backslash aleph\_0$ if and only if it is countably infinite
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

, that is, there is a bijection
In , a bijection, bijective function, one-to-one correspondence, or invertible function, is a between the elements of two , where each element of one set is paired with exactly one element of the other set, and each element of the other set is p ...

(one-to-one correspondence) between it and the natural numbers. Examples of such sets are
* the set of all integer
An integer (from the Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Re ...

s,
* any infinite subset of the integers, such as the set of all square numbers
In Euclidean geometry
Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method c ...

or the set of all prime numbers
A prime number (or a prime) is a natural number
In mathematics, the natural numbers are those 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 coun ...

,
* the set of all rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...

s,
* the set of all constructible number
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space ...

s (in the geometric sense),
* the set of all algebraic number
An algebraic number is any complex number
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus ...

s,
* the set of all computable number
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s,
* the set of all binary string
String or strings may refer to:
*String (structure), a long flexible structure made from threads twisted together, which is used to tie, bind, or hang other objects
Arts, entertainment, and media Films
* Strings (1991 film), ''Strings'' (1991 fil ...

s of finite length, and
* the set of all finite subset
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s of any given countably infinite set.
These infinite ordinals: $\backslash omega$, $\backslash omega+1$, $\backslash omega\backslash cdot2$, $\backslash omega^$, $\backslash omega^$ and $\backslash varepsilon\_$ are among the countably infinite sets. For example, the sequence (with ordinality ω·2) of all positive odd integers followed by all positive even integers
:$\backslash $
is an ordering of the set (with cardinality $\backslash aleph\_0$) of positive integers.
If the axiom of countable choice
The axiom of countable choice or axiom of denumerable choice, denoted ACω, is an axiom
An axiom, postulate or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The ...

(a weaker version of the axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ...

) holds, then $\backslash aleph\_0$ is smaller than any other infinite cardinal.
Aleph-one

$\backslash aleph\_1$ is the cardinality of the set of all countableordinal number
In set theory
illustrating the intersection of two sets
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a s ...

s, called $\backslash omega\_$ or sometimes $\backslash Omega$. This $\backslash omega\_$ is itself an ordinal number larger than all countable ones, so it is an uncountable set
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

. Therefore, $\backslash aleph\_1$ is distinct from $\backslash aleph\_0$. The definition of $\backslash aleph\_1$ implies (in ZF, Zermelo–Fraenkel set theory
In set theory
illustrating the intersection (set theory), intersection of two set (mathematics), sets.
Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections of objects. Although any t ...

''without'' the axiom of choice) that no cardinal number is between $\backslash aleph\_0$ and $\backslash aleph\_1$. If the axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ...

is used, it can be further proved that the class of cardinal numbers is totally ordered
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

, and thus $\backslash aleph\_1$ is the second-smallest infinite cardinal number. Using the axiom of choice, one can show one of the most useful properties of the set $\backslash omega\_$: any countable subset of $\backslash omega\_$ has an upper bound in $\backslash omega\_.$ (This follows from the fact that the union of a countable number of countable sets is itself countable—one of the most common applications of the axiom of choice.) This fact is analogous to the situation in $\backslash aleph\_0$: every finite set of natural numbers has a maximum which is also a natural number, and finite unions of finite sets are finite.
$\backslash omega\_$ is actually a useful concept, if somewhat exotic-sounding. An example application is "closing" with respect to countable operations; e.g., trying to explicitly describe the $\backslash sigma$-algebra generated by an arbitrary collection of subsets (see e.g. Borel hierarchyIn mathematical logic, the Borel hierarchy is a stratification of the Borel algebra generated by the open subsets of a Polish space; elements of this algebra are called Borel sets. Each Borel set is assigned a unique countable ordinal number called ...

). This is harder than most explicit descriptions of "generation" in algebra (vector space
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s, group
A group is a number
A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...

s, etc.) because in those cases we only have to close with respect to finite operations—sums, products, and the like. The process involves defining, for each countable ordinal, via transfinite induction
Transfinite induction is an extension of mathematical induction
Image:Dominoeffect.png, Mathematical induction can be informally illustrated by reference to the sequential effect of falling Domino effect, dominoes.
Mathematical induction is a m ...

, a set by "throwing in" all possible countable unions and complements, and taking the union of all that over all of $\backslash omega\_$.
Every uncountable coanalytic subset of a Polish space
In the mathematical discipline of general topology, a Polish space is a separable space, separable Completely metrizable space, completely metrizable topological space; that is, a space homeomorphic to a Complete space, complete metric space that ha ...

$X$ has cardinality $\backslash aleph\_1$ or $2^$.
Continuum hypothesis

Thecardinality
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

of the set of real number
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

s (cardinality of the continuum
In set theory
illustrating the intersection of two sets
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, ...

) is $2^$. It cannot be determined from ZFC (Zermelo–Fraenkel set theory
In set theory
illustrating the intersection (set theory), intersection of two set (mathematics), sets.
Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections of objects. Although any t ...

with the axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ...

) where this number fits exactly in the aleph number hierarchy, but it follows from ZFC that the continuum hypothesis, CH, is equivalent to the identity
The CH states that there is no set whose cardinality is strictly between that of the integers and the real numbers. CH is independent of ZFC: it can be neither proven nor disproven within the context of that axiom system (provided that ZFC is consistent
In classical deductive logic
Deductive reasoning, also deductive logic, is the process of reasoning from one or more statements (premises) to reach a logical conclusion.
Deductive reasoning goes in the same direction as that of the conditiona ...

). That CH is consistent with ZFC was demonstrated by Kurt Gödel
Kurt Friedrich Gödel ( , ; April 28, 1906 – January 14, 1978) was a logician
Logic is an interdisciplinary field which studies truth
Truth is the property of being in accord with fact or reality.Merriam-Webster's Online Dict ...

in 1940, when he showed that its negation is not a theorem of ZFC. That it is independent of ZFC was demonstrated by Paul Cohen
:''For other people named Paul Cohen, see Paul Cohen (disambiguation). Not to be confused with Paul Cohn.''
Paul Joseph Cohen (April 2, 1934 – March 23, 2007) was an United States, American mathematician. He is best known for his proofs that t ...

in 1963, when he showed conversely that the CH itself is not a theorem of ZFC—by the (then novel) method of forcing.
Aleph-omega

Aleph-omega is :$\backslash aleph\_\backslash omega\; =\; \backslash sup\; \backslash \; =\; \backslash sup\; \backslash $ where the smallest infinite ordinal is denoted ω. That is, the cardinal number $\backslash aleph\_\backslash omega$ is the least upper bound of $\backslash aleph\_\backslash omega$ is the first uncountable cardinal number that can be demonstrated within Zermelo–Fraenkel set theory ''not'' to be equal to the cardinality of the set of allreal number
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

s; for any positive integer ''n'' we can consistently assume that $2^\; =\; \backslash aleph\_n$, and moreover it is possible to assume $2^$ is as large as we like. We are only forced to avoid setting it to certain special cardinals with cofinality
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

$\backslash aleph\_0$, meaning there is an unbounded function from $\backslash aleph\_0$ to it (see Easton's theoremIn set theory
illustrating the intersection (set theory), intersection of two set (mathematics), sets.
Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections of objects. Although any typ ...

).
Aleph-$\backslash alpha$ for general $\backslash alpha$

To define $\backslash aleph\_\backslash alpha$ for arbitrary ordinal number $\backslash alpha$, we must define the successor cardinal operation, which assigns to any cardinal number $\backslash rho$ the next largerwell-order
In mathematics, a well-order (or well-ordering or well-order relation) on a Set (mathematics), set ''S'' is a total order on ''S'' with the property that every non-empty subset of ''S'' has a least element in this ordering. The set ''S'' together ...

ed cardinal $\backslash rho^$ (if the limit ordinal
In set theory, a limit ordinal is an ordinal number that is neither zero nor a successor ordinal. Alternatively, an ordinal λ is a limit ordinal if there is an ordinal less than λ, and whenever β is an ordinal less than λ, then there exists an ...

,
The α-th infinite initial ordinal
The von Neumann cardinal assignment is a cardinal assignment which uses ordinal number
In set theory, an ordinal number, or ordinal, is one generalization of the concept of a natural number that is used to describe a way to arrange a (possibly ...

is written $\backslash omega\_\backslash alpha$. Its cardinality is written $\backslash aleph\_\backslash alpha$.
In ZFC, the aleph function $\backslash aleph$ is a bijection from the ordinals to the infinite cardinals.
Fixed points of omega

For any ordinal α we have In many cases $\backslash omega\_$ is strictly greater than α. For example, for any successor ordinal α this holds. There are, however, some limit ordinals which are fixed points of the omega function, because of thefixed-point lemma for normal functions
The fixed-point lemma for normal functions is a basic result in axiomatic set theory stating that any normal function has arbitrarily large fixed point (mathematics), fixed points (Levy 1979: p. 117). It was first proved by Oswald Veblen in 19 ...

. The first such is the limit of the sequence
Any weakly inaccessible cardinal is also a fixed point of the aleph function. This can be shown in ZFC as follows. Suppose $\backslash kappa\; =\; \backslash aleph\_\backslash lambda$ is a weakly inaccessible cardinal. If $\backslash lambda$ were a successor ordinalIn set theory
illustrating the intersection (set theory), intersection of two set (mathematics), sets.
Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections of objects. Although any typ ...

, then $\backslash aleph\_\backslash lambda$ would be a successor cardinalIn set theory, one can define a successor operation on cardinal numbers in a similar way to the successor operation on the ordinal numbers. The cardinal successor coincides with the ordinal successor for finite cardinals, but in the infinite case the ...

and hence not weakly inaccessible. If $\backslash lambda$ were a limit ordinal
In set theory, a limit ordinal is an ordinal number that is neither zero nor a successor ordinal. Alternatively, an ordinal λ is a limit ordinal if there is an ordinal less than λ, and whenever β is an ordinal less than λ, then there exists an ...

less than $\backslash kappa$, then its cofinality
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

(and thus the cofinality of $\backslash aleph\_\backslash lambda$) would be less than $\backslash kappa$ and so $\backslash kappa$ would not be regular and thus not weakly inaccessible. Thus $\backslash lambda\; \backslash geq\; \backslash kappa$ and consequently $\backslash lambda\; =\; \backslash kappa$ which makes it a fixed point.
Role of axiom of choice

The cardinality of any infiniteordinal number
In set theory
illustrating the intersection of two sets
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a s ...

is an aleph number. Every aleph is the cardinality of some ordinal. The least of these is its initial ordinal
The von Neumann cardinal assignment is a cardinal assignment which uses ordinal number
In set theory, an ordinal number, or ordinal, is one generalization of the concept of a natural number that is used to describe a way to arrange a (possibly ...

. Any set whose cardinality is an aleph is equinumerous
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

with an ordinal and is thus well-order
In mathematics, a well-order (or well-ordering or well-order relation) on a Set (mathematics), set ''S'' is a total order on ''S'' with the property that every non-empty subset of ''S'' has a least element in this ordering. The set ''S'' together ...

able.
Each finite 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 and t ...

is well-orderable, but does not have an aleph as its cardinality.
The assumption that the cardinality of each infinite set
In set theory
illustrating the intersection (set theory), intersection of two set (mathematics), sets.
Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections of objects. Although any ...

is an aleph number is equivalent over ZF to the existence of a well-ordering of every set, which in turn is equivalent to the See also

*Beth number
In mathematics, the beth numbers are a certain sequence of infinite set, infinite cardinal numbers, conventionally written \beth_0,\ \beth_1,\ \beth_2,\ \beth_3,\ \dots, where \beth is the second Hebrew alphabet, Hebrew letter (bet (letter), beth). ...

* Gimel function
* Regular cardinalIn set theory, a regular cardinal is a cardinal number that is equal to its own cofinality. More explicitly, this means that \kappa is a regular cardinal if and only if every unbounded subset C \subseteq \kappa has cardinality \kappa. Infinite wel ...

* Transfinite number
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

* Ordinal number
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 ...

Notes

Citations

References

*External links

* * {{DEFAULTSORT:Aleph Number Cardinal numbers Infinity