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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, particularly in
set theory 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, set theory, as a branch of mathematics, is mostly concern ...
, the aleph numbers are a
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is call ...
of numbers used to represent the
cardinality In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
(or size) of infinite sets that can be well-ordered. They were introduced by the mathematician
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance o ...
and are named after the symbol he used to denote them, the
Hebrew Hebrew (; ; ) is a Northwest Semitic language of the Afroasiatic language family. Historically, it is one of the spoken languages of the Israelites and their longest-surviving descendants, the Jews and Samaritans. It was largely preserve ...
letter
aleph Aleph (or alef or alif, transliterated ʾ) is the first letter of the Semitic abjads, including Phoenician , Hebrew , Aramaic , Syriac , Arabic ʾ and North Arabian 𐪑. It also appears as South Arabian 𐩱 and Ge'ez . These lette ...
($\,\aleph\,$). 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 ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal n ...
s is $\,\aleph_0\,$ (read ''aleph-nought'' or ''aleph-zero''; the term ''aleph-null'' is also sometimes used), the next larger cardinality of a well-orderable set is aleph-one $\,\aleph_1\;,$ then $\,\aleph_2\,$ and so on. Continuing in this manner, it is possible to define a
cardinal number In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. The ...
$\,\aleph_\alpha\,$ for every
ordinal number In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the least n ...
$\,\alpha\;,$ 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 played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance o ...
, 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 natural number. It is often denoted by the infinity symbol . Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions amo ...
($\,\infty\,$) 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 of the
real number line In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a poin ...
(applied to a function or
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is call ...
that " diverges to infinity" or "increases without bound"), or as an extreme point of the extended real number line.

Aleph-nought

$\,\aleph_0\,$ (aleph-nought, also aleph-zero or aleph-null) is the cardinality of the set of all natural numbers, and is an
infinite cardinal In mathematics, transfinite numbers are numbers that are " infinite" in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite. These include the transfinite cardinals, which are cardinal numbers used to q ...
. The set of all finite ordinals, called $\,\omega\,$ or $\,\omega_\,$ (where $\,\omega\,$ is the lowercase Greek letter
omega Omega (; capital: Ω, lowercase: ω; Ancient Greek ὦ, later ὦ μέγα, Modern Greek ωμέγα) is the twenty-fourth and final letter in the Greek alphabet. In the Greek numeric system/isopsephy ( gematria), it has a value of 800. The w ...
), has cardinality $\,\aleph_0\,$. A set has cardinality $\,\aleph_0\,$ if and only if it is
countably infinite In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...
, that is, there is a
bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
(one-to-one correspondence) between it and the natural numbers. Examples of such sets are * the set of all
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language o ...
s, * any infinite subset of the integers, such as the set of all square numbers or the set of all prime numbers, * the set of all
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rati ...
s, * the set of all constructible numbers (in the geometric sense), * the set of all
algebraic number An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of the p ...
s, * the set of all computable numbers, * the set of all binary strings of finite length, and * the set of all finite
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
s of any given countably infinite set. These infinite ordinals: $\,\omega\;,$ $\,\omega+1\;,$ $\,\omega\,\cdot2\,,\,$ $\,\omega^\,,$ $\,\omega^\,$ and $\,\varepsilon_\,$ are among the countably infinite sets. For example, the sequence (with ordinality $\,\omega\,\cdot2\,$) of all positive odd integers followed by all positive even integers :$\,\\,$ is an ordering of the set (with cardinality $\aleph_0$) of positive integers. If the axiom of countable choice (a weaker version of the axiom of choice) holds, then $\,\aleph_0\,$ is smaller than any other infinite cardinal.

Aleph-one

$\,\aleph_1\,$ is the cardinality of the set of all countable
ordinal number In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the least n ...
s, called $\,\omega_\,$ or sometimes $\,\Omega\,$. This $\,\omega_\,$ is itself an ordinal number larger than all countable ones, so it is an uncountable set. Therefore, $\,\aleph_1\,$ is distinct from $\,\aleph_0\,$. The definition of $\,\aleph_1\,$ implies (in ZF,
Zermelo–Fraenkel set theory In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such ...
''without'' the axiom of choice) that no cardinal number is between $\,\aleph_0\,$ and $\,\aleph_1\,$. If the axiom of choice is used, it can be further proved that the class of cardinal numbers is
totally ordered In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive ...
, and thus $\,\aleph_1\,$ is the second-smallest infinite cardinal number. Using the axiom of choice, one can show one of the most useful properties of the set $\,\omega_\,$: any countable subset of $\,\omega_\,$ has an upper bound in $\,\omega_\,$. (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 $\,\aleph_0\;$ : every finite set of natural numbers has a maximum which is also a natural number, and finite unions of finite sets are finite. $\,\omega_~$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 σ-algebra generated by an arbitrary collection of subsets (see e.g. Borel hierarchy). This is harder than most explicit descriptions of "generation" in algebra (
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
s, groups, 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, a set by "throwing in" all possible countable unions and complements, and taking the union of all that over all of $\, \omega_$.

Continuum hypothesis

The
cardinality In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
of the set of
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s (
cardinality 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 \mathfrak c (lowercase fraktur "c") or , \math ...
) is $\, 2^ ~.$ It cannot be determined from ZFC (
Zermelo–Fraenkel set theory In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such ...
augmented with the axiom of choice) 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 :$2^ = \aleph_1.$ 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). That CH is consistent with ZFC was demonstrated by Kurt Gödel 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 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 :$\aleph_\omega = \sup \, \ = \sup \, \~$ where the smallest infinite ordinal is denoted . That is, the cardinal number $\,\aleph_\omega\,$ is the least upper bound of :$\left\ ~.$ $\,\aleph_\omega$ 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 all
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s; for any positive integer ''n'' we can consistently assume that $\,2^ = \aleph_n~,$ 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 $\, \aleph_0 ~ ,$ meaning there is an unbounded function from $\, \aleph_0 \,$ to it (see Easton's theorem).

Aleph-α for general α

To define $\,\aleph_\alpha\,$ for arbitrary ordinal number $\,\alpha~,$ we must define the successor cardinal operation, which assigns to any cardinal number $\,\rho\,$ the next larger well-ordered cardinal $\,\rho^\,$ (if the axiom of choice holds, this is the next larger cardinal). We can then define the aleph numbers as follows: :$\aleph_ = \omega$ :$\aleph_ = \aleph_^+ ~$ and for , an infinite limit ordinal, :$\aleph_ = \bigcup_ \aleph_\beta ~.$ The α-th infinite initial ordinal is written $\omega_\alpha$. Its cardinality is written $\,\aleph_\alpha~.$ In ZFC, the aleph function $\,\aleph\,$ is a bijection from the ordinals to the infinite cardinals.

Fixed points of omega

For any ordinal α we have :$\alpha \leq \omega_\alpha ~.$ In many cases $\omega_$ is strictly greater than . 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 the fixed-point lemma for normal functions. The first such is the limit of the sequence :$\omega, \, \omega_\omega, \, \omega_, \, \ldots ~.$ Any weakly inaccessible cardinal is also a fixed point of the aleph function. This can be shown in ZFC as follows. Suppose $\,\kappa = \aleph_\lambda\,$ is a weakly inaccessible cardinal. If $\lambda$ were a successor ordinal, then $\,\aleph_\lambda\,$ would be a
successor cardinal In 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 th ...
and hence not weakly inaccessible. If $\,\lambda\,$ were a limit ordinal less than $\,\kappa~,$ then its cofinality (and thus the cofinality of $\aleph_\lambda$) would be less than $\,\kappa\,$ and so $\,\kappa\,$ would not be regular and thus not weakly inaccessible. Thus $\,\lambda \geq \kappa\,$ and consequently $\,\lambda = \kappa\,$ which makes it a fixed point.

Role of axiom of choice

The cardinality of any infinite
ordinal number In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the least n ...
is an aleph number. Every aleph is the cardinality of some ordinal. The least of these is its initial ordinal. Any set whose cardinality is an aleph is equinumerous with an ordinal and is thus well-orderable. Each
finite set In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, :\ is a finite set with five elements. T ...
is well-orderable, but does not have an aleph as its cardinality. The assumption that the cardinality of each infinite set 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 axiom of choice. 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 to be the set of sets with the same cardinality as of minimum possible rank. This has the property that if and only if and have the same cardinality. (The set does not have the same cardinality of in general, but all its elements do.)