![Aleph0](https://upload.wikimedia.org/wikipedia/commons/e/e0/Aleph0.svg)
In
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 conce ...
, 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 calle ...
of numbers used to represent the
cardinality (or size) of
infinite set
In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable.
Properties
The set of natural numbers (whose existence is postulated by the axiom of infinity) is infinite. It is the only s ...
s that can be
well-ordered
In mathematics, a well-order (or well-ordering or well-order relation) on a 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 with the well-or ...
. 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 of ...
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 preserved ...
letter
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 ...
s is
(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 ''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 with the well-or ...
able set is aleph-one
then
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. T ...
for every
ordinal number 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 of ...
,
who defined the notion of cardinality and realized that
infinite sets can have different cardinalities.
The aleph numbers differ from the
infinity (
) 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
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
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 calle ...
that "
diverges to infinity" or "increases without bound"), or as an extreme point of the
extended real number line.
Aleph-nought
(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
or
(where
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 ...
), has cardinality
. A set has cardinality
if and only if it is
countably infinite, that is, there is a
bijection (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 languag ...
s,
* any infinite subset of the integers, such as the set of all
square numbers
In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For example, 9 is a square number, since it equals and can be written as .
The us ...
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 rat ...
s,
* the set of all
constructible number
In geometry and algebra, a real number r is constructible if and only if, given a line segment of unit length, a line segment of length , r, can be constructed with compass and straightedge in a finite number of steps. Equivalently, r is cons ...
s (in the geometric sense),
* the set of all
algebraic numbers,
* the set of all
computable number
In mathematics, computable numbers are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm. They are also known as the recursive numbers, effective numbers or the computable reals or recursive ...
s,
* the set of all binary
strings of finite length, and
* the set of all finite
subsets of any given countably infinite set.
These infinite ordinals:
and
are among the countably infinite sets. For example, the sequence (with
ordinality
In mathematics, especially in set theory, two ordered sets and are said to have the same order type if they are order isomorphic, that is, if there exists a bijection (each element pairs with exactly one in the other set) f\colon X \to Y such th ...
) of all positive odd integers followed by all positive even integers
:
is an ordering of the set (with cardinality
) of positive integers.
If the
axiom of countable choice
The axiom of countable choice or axiom of denumerable choice, denoted ACω, is an axiom of set theory that states that every countable collection of non-empty sets must have a choice function. That is, given a function ''A'' with domain N (where ...
(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 of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
) holds, then
is smaller than any other infinite cardinal.
Aleph-one
is the cardinality of the set of all countable
ordinal numbers, called
or sometimes
. This
is itself an ordinal number larger than all countable ones, so it is an
uncountable set
In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal nu ...
. Therefore,
is distinct from
. The definition of
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
and
. 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 of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
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
is the second-smallest infinite cardinal number. Using the axiom of choice, one can show one of the most useful properties of the set
: any countable subset of
has an upper bound in
. (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
: every finite set of natural numbers has a maximum which is also a natural number, and
finite unions of finite sets are finite.
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 In 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 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 can ...
s,
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
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 to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers. Its correctness is a theorem of ZFC.
Induction by cases
Let P(\alpha) be a property defined for ...
, a set by "throwing in" all possible countable unions and complements, and taking the union of all that over all of
.
Continuum hypothesis
The
cardinality 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) is
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
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
) 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, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory 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
Paul Joseph Cohen (April 2, 1934 – March 23, 2007) was an American mathematician. He is best known for his proofs that the continuum hypothesis and the axiom of choice are independent from Zermelo–Fraenkel set theory, for which he was award ...
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
:
where the smallest infinite ordinal is denoted . That is, the cardinal number is the least upper bound
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest lo ...
of
:
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 and moreover it is possible to assume is as large as we like. We are only forced to avoid setting it to certain special cardinals with cofinality
In mathematics, especially in order theory, the cofinality cf(''A'') of a partially ordered set ''A'' is the least of the cardinalities of the cofinal subsets of ''A''.
This definition of cofinality relies on the axiom of choice, as it uses the ...
meaning there is an unbounded function from to it (see Easton's theorem).
Aleph-α for general α
To define for arbitrary ordinal number we must define the successor cardinal operation, which assigns to any cardinal number the next larger well-order
In mathematics, a well-order (or well-ordering or well-order relation) on a 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 with the well-or ...
ed cardinal (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 of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
holds, this is the next larger cardinal).
We can then define the aleph numbers as follows:
:
:
and for , an infinite 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 a ...
,
:
The α-th infinite initial ordinal
In a written or published work, an initial capital, also referred to as a drop capital or simply an initial cap, initial, initcapital, initcap or init or a drop cap or drop, is a letter at the beginning of a word, a chapter, or a paragraph that ...
is written . Its cardinality is written
In ZFC, the aleph function is a bijection from the ordinals to the infinite cardinals.
Fixed points of omega
For any ordinal α we have
:
In many cases 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 fixed-point lemma for normal functions is a basic result in axiomatic set theory stating that any normal function has arbitrarily large fixed points (Levy 1979: p. 117). It was first proved by Oswald Veblen in 1908.
Background and form ...
. 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 is a weakly inaccessible cardinal. If were a successor ordinal In set theory, the successor of an ordinal number ''α'' is the smallest ordinal number greater than ''α''. An ordinal number that is a successor is called a successor ordinal. Properties
Every ordinal other than 0 is either a successor ordin ...
, then would be a successor cardinal and hence not weakly inaccessible. If 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 a ...
less than then its cofinality
In mathematics, especially in order theory, the cofinality cf(''A'') of a partially ordered set ''A'' is the least of the cardinalities of the cofinal subsets of ''A''.
This definition of cofinality relies on the axiom of choice, as it uses the ...
(and thus the cofinality of ) would be less than and so would not be regular and thus not weakly inaccessible. Thus and consequently which makes it a fixed point.
Role of axiom of choice
The cardinality of any infinite ordinal number is an aleph number. Every aleph is the cardinality of some ordinal. The least of these is its initial ordinal
In a written or published work, an initial capital, also referred to as a drop capital or simply an initial cap, initial, initcapital, initcap or init or a drop cap or drop, is a letter at the beginning of a word, a chapter, or a paragraph that ...
. Any set whose cardinality is an aleph is equinumerous
In mathematics, two sets or classes ''A'' and ''B'' are equinumerous if there exists a one-to-one correspondence (or bijection) between them, that is, if there exists a function from ''A'' to ''B'' such that for every element ''y'' of ''B'', the ...
with an ordinal and is thus well-order
In mathematics, a well-order (or well-ordering or well-order relation) on a 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 with the well-or ...
able.
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
In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable.
Properties
The set of natural numbers (whose existence is postulated by the axiom of infinity) is infinite. It is the only s ...
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
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
. 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
In set theory, Scott's trick is a method for giving a definition of equivalence classes for equivalence relations on a proper class (Jech 2003:65) by referring to levels of the cumulative hierarchy.
The method relies on the axiom of regularity but ...
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.)
See also
* Beth number
In mathematics, particularly in set theory, the beth numbers are a certain sequence of infinite cardinal numbers (also known as transfinite numbers), conventionally written \beth_0,\ \beth_1,\ \beth_2,\ \beth_3,\ \dots, where \beth is the second H ...
* Gimel function
* Regular cardinal
In 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 ...
* Transfinite number
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 ...
* Ordinal number
Notes
Citations
External links
*
*
{{DEFAULTSORT:Aleph Number
Cardinal numbers
Hebrew alphabet
Infinity