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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 ...
, an ordinal number, or ordinal, is a generalization of
ordinal numeral In linguistics, ordinal numerals or ordinal number words are words representing position or rank in a sequential order; the order may be of size, importance, chronology, and so on (e.g., "third", "tertiary"). They differ from cardinal numerals ...
s (first, second, th, etc.) aimed to extend
enumeration An enumeration is a complete, ordered listing of all the items in a collection. The term is commonly used in mathematics and computer science to refer to a listing of all of the elements of a set. The precise requirements for an enumeration (fo ...
to
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. A finite set can be enumerated by successively labeling each element with the least
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 ...
that has not been previously used. To extend this process to various infinite sets, ordinal numbers are defined more generally as
linearly 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) ...
labels that include the natural numbers and have the property that every set of ordinals has a least element (this is needed for giving a meaning to "the least unused element"). This more general definition allows us to define an ordinal number \omega that is greater than every natural number, along with ordinal numbers \omega + 1, \omega + 2, etc., which are even greater than \omega. A linear order such that every subset has a least element is called 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 ...
. 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 ...
implies that every set can be well-ordered, and given two well-ordered sets, one is
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
to an
initial segment In mathematics, an upper set (also called an upward closed set, an upset, or an isotone set in ''X'') of a partially ordered set (X, \leq) is a subset S \subseteq X with the following property: if ''s'' is in ''S'' and if ''x'' in ''X'' is larger ...
of the other. So ordinal numbers exist and are essentially unique. Ordinal numbers are distinct from
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. Th ...
s, which measure the size of sets. Although the distinction between ordinals and cardinals is not always apparent on finite sets (one can go from one to the other just by counting labels), they are very different in the infinite case, where different infinite ordinals can correspond to sets having the same cardinal. Like other kinds of numbers, ordinals can be added, multiplied, and exponentiated, although none of these operations are commutative. Ordinals were introduced by
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 ...
in 1883 in order to accommodate infinite sequences and classify derived sets, which he had previously introduced in 1872 while studying the uniqueness of
trigonometric series In mathematics, a trigonometric series is a infinite series of the form : \frac+\displaystyle\sum_^(A_ \cos + B_ \sin), an infinite version of a trigonometric polynomial. It is called the Fourier series of the integrable function f if the term ...
.


Ordinals extend the natural numbers

A
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 ...
(which, in this context, includes the number 0) can be used for two purposes: to describe the ''size'' of a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
, or to describe the ''position'' of an element in a sequence. When restricted to finite sets, these two concepts coincide, since all
linear order 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 ...
s of a finite set are
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
. When dealing with infinite sets, however, one has to distinguish between the notion of size, which leads to
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. Th ...
s, and the notion of position, which leads to the ordinal numbers described here. This is because while any set has only one size (its cardinality), there are many nonisomorphic
well-ordering 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-o ...
s of any infinite set, as explained below. Whereas the notion of cardinal number is associated with a set with no particular structure on it, the ordinals are intimately linked with the special kind of sets that are called
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. A well-ordered set is a
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 ...
set in which every non-empty subset has a least element (a totally ordered set is an ordered set such that, given two distinct elements, one is less than the other). Equivalently, assuming the axiom of dependent choice, it is a totally ordered set without any infinite decreasing sequence — though there may be infinite increasing sequences. Ordinals may be used to label the elements of any given well-ordered set (the smallest element being labelled 0, the one after that 1, the next one 2, "and so on"), and to measure the "length" of the whole set by the least ordinal that is not a label for an element of the set. This "length" is called the ''order type'' of the set. Any ordinal is defined by the set of ordinals that precede it. In fact, the most common definition of ordinals ''identifies'' each ordinal ''as'' the set of ordinals that precede it. For example, the ordinal 42 is generally identified as the set . Conversely, any set ''S'' of ordinals that is downward closed — meaning that for any ordinal α in ''S'' and any ordinal β < α, β is also in ''S'' — is (or can be identified with) an ordinal. This definition of ordinals in terms of sets allows for infinite ordinals. The smallest infinite ordinal is \omega, which can be identified with the set of natural numbers (so that the ordinal associated with every natural number precedes \omega). Indeed, the set of natural numbers is well-ordered—as is any set of ordinals—and since it is downward closed, it can be identified with the ordinal associated with it. Perhaps a clearer intuition of ordinals can be formed by examining a first few of them: as mentioned above, they start with the natural numbers, 0, 1, 2, 3, 4, 5, … After ''all'' natural numbers comes the first infinite ordinal, ω, and after that come ω+1, ω+2, ω+3, and so on. (Exactly what addition means will be defined later on: just consider them as names.) After all of these come ω·2 (which is ω+ω), ω·2+1, ω·2+2, and so on, then ω·3, and then later on ω·4. Now the set of ordinals formed in this way (the ω·''m''+''n'', where ''m'' and ''n'' are natural numbers) must itself have an ordinal associated with it: and that is ω2. Further on, there will be ω3, then ω4, and so on, and ωω, then ωωω, then later ωωωω, and even later ε0 ( epsilon nought) (to give a few examples of relatively small—countable—ordinals). This can be continued indefinitely (as every time one says "and so on" when enumerating ordinals, it defines a larger ordinal). The smallest
uncountable 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 num ...
ordinal is the set of all countable ordinals, expressed as ω1 or \Omega.


Definitions


Well-ordered sets

In 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 ...
ed set, every non-empty subset contains a distinct smallest element. Given the axiom of dependent choice, this is equivalent to saying that the set 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 there is no infinite decreasing sequence (the latter being easier to visualize). In practice, the importance of well-ordering is justified by the possibility of applying
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 ...
, which says, essentially, that any property that passes on from the predecessors of an element to that element itself must be true of all elements (of the given well-ordered set). If the states of a computation (computer program or game) can be well-ordered—in such a way that each step is followed by a "lower" step—then the computation will terminate. It is inappropriate to distinguish between two well-ordered sets if they only differ in the "labeling of their elements", or more formally: if the elements of the first set can be paired off with the elements of the second set such that if one element is smaller than another in the first set, then the partner of the first element is smaller than the partner of the second element in the second set, and vice versa. Such a one-to-one correspondence is called an
order isomorphism In the mathematical field of order theory, an order isomorphism is a special kind of monotone function that constitutes a suitable notion of isomorphism for partially ordered sets (posets). Whenever two posets are order isomorphic, they can be cons ...
, and the two well-ordered sets are said to be order-isomorphic or ''similar'' (with the understanding that this is an equivalence relation). Formally, if a
partial order In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...
≤ is defined on the set ''S'', and a partial order ≤' is defined on the set ''S' '', then the
poset In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary r ...
s (''S'',≤) and (''S' '',≤') are order isomorphic if there is a bijection ''f'' that preserves the ordering. That is, ''f''(''a'') ≤' ''f''(''b'') if and only if ''a'' ≤ ''b''. Provided there exists an order isomorphism between two well-ordered sets, the order isomorphism is unique: this makes it quite justifiable to consider the two sets as essentially identical, and to seek a "canonical" representative of the isomorphism type (class). This is exactly what the ordinals provide, and it also provides a canonical labeling of the elements of any well-ordered set. Every ''well-ordered'' set (''S'',<) is order-isomorphic to the set of ordinals less than one specific ordinal number under their natural ordering. This canonical set is the ''order type'' of (''S'',<). Essentially, an ordinal is intended to be defined as an
isomorphism class In mathematics, an isomorphism class is a collection of mathematical objects isomorphic to each other. Isomorphism classes are often defined as the exact identity of the elements of the set is considered irrelevant, and the properties of the stru ...
of well-ordered sets: that is, as an equivalence class for the equivalence relation of "being order-isomorphic". There is a technical difficulty involved, however, in the fact that the equivalence class is too large to be a set in the usual Zermelo–Fraenkel (ZF) formalization of set theory. But this is not a serious difficulty. The ordinal can be said to be the ''
order type 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 ...
'' of any set in the class.


Definition of an ordinal as an equivalence class

The original definition of ordinal numbers, found for example in the ''
Principia Mathematica The ''Principia Mathematica'' (often abbreviated ''PM'') is a three-volume work on the foundations of mathematics written by mathematician–philosophers Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. ...
'', defines the order type of a well-ordering as the set of all well-orderings similar (order-isomorphic) to that well-ordering: in other words, an ordinal number is genuinely an equivalence class of well-ordered sets. This definition must be abandoned in ZF and related systems of axiomatic set theory because these equivalence classes are too large to form a set. However, this definition still can be used in
type theory In mathematics, logic, and computer science, a type theory is the formal presentation of a specific type system, and in general type theory is the academic study of type systems. Some type theories serve as alternatives to set theory as a fou ...
and in Quine's axiomatic set theory New Foundations and related systems (where it affords a rather surprising alternative solution to the
Burali-Forti paradox In set theory, a field of mathematics, the Burali-Forti paradox demonstrates that constructing "the set of all ordinal numbers" leads to a contradiction and therefore shows an antinomy in a system that allows its construction. It is named after Ces ...
of the largest ordinal).


Von Neumann definition of ordinals

Rather than defining an ordinal as an ''equivalence class'' of well-ordered sets, it will be defined as a particular well-ordered set that (canonically) represents the class. Thus, an ordinal number will be a well-ordered set; and every well-ordered set will be order-isomorphic to exactly one ordinal number. For each well-ordered set T, a\mapsto T_ defines an
order isomorphism In the mathematical field of order theory, an order isomorphism is a special kind of monotone function that constitutes a suitable notion of isomorphism for partially ordered sets (posets). Whenever two posets are order isomorphic, they can be cons ...
between T and the set of all subsets of T having the form T_:=\ ordered by inclusion. This motivates the standard definition, suggested by
John von Neumann John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest cove ...
at the age of 19, now called definition of von Neumann ordinals: "each ordinal is the well-ordered set of all smaller ordinals." In symbols, \lambda = _Formally: :A_set_''S''_is_an_ordinal_
_Formally: :A_set_''S''_is_an_ordinal_if_and_only_if">,\lambda)._Formally: :A_set_''S''_is_an_ordinal_if_and_only_if_''S''_is_strict_order.html" "title="if_and_only_if.html" ;"title=",\lambda). Formally: :A set ''S'' is an ordinal if and only if">,\lambda). Formally: :A set ''S'' is an ordinal if and only if ''S'' is strict order">strictly In mathematical writing, the term strict refers to the property of excluding equality and equivalence and often occurs in the context of inequality and monotonic functions. It is often attached to a technical term to indicate that the exclusive ...
well-ordered with respect to set membership and every element of ''S'' is also a subset of ''S''. The natural numbers are thus ordinals by this definition. For instance, 2 is an element of 4 = , and 2 is equal to and so it is a subset of . It can be shown by
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 ...
that every well-ordered set is order-isomorphic to exactly one of these ordinals, that is, there is an order preserving bijective function between them. Furthermore, the elements of every ordinal are ordinals themselves. Given two ordinals ''S'' and ''T'', ''S'' is an element of ''T'' if and only if ''S'' is a proper subset of ''T''. Moreover, either ''S'' is an element of ''T'', or ''T'' is an element of ''S'', or they are equal. So every set of ordinals 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 ...
. Further, every set of ordinals is well-ordered. This generalizes the fact that every set of natural numbers is well-ordered. Consequently, every ordinal ''S'' is a set having as elements precisely the ordinals smaller than ''S''. For example, every set of ordinals has a supremum, the ordinal obtained by taking the union of all the ordinals in the set. This union exists regardless of the set's size, by the
axiom of union In axiomatic set theory, the axiom of union is one of the axioms of Zermelo–Fraenkel set theory. This axiom was introduced by Ernst Zermelo. The axiom states that for each set ''x'' there is a set ''y'' whose elements are precisely the elemen ...
. The class of all ordinals is not a set. If it were a set, one could show that it was an ordinal and thus a member of itself, which would contradict its ''strict'' ordering by membership. This is the
Burali-Forti paradox In set theory, a field of mathematics, the Burali-Forti paradox demonstrates that constructing "the set of all ordinal numbers" leads to a contradiction and therefore shows an antinomy in a system that allows its construction. It is named after Ces ...
. The class of all ordinals is variously called "Ord", "ON", or "∞". An ordinal is
finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marke ...
if and only if the opposite order is also well-ordered, which is the case if and only if each of its non-empty subsets has a
maximum In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given r ...
.


Other definitions

There are other modern formulations of the definition of ordinal. For example, assuming the
axiom of regularity In mathematics, the axiom of regularity (also known as the axiom of foundation) is an axiom of Zermelo–Fraenkel set theory that states that every non-empty set ''A'' contains an element that is disjoint from ''A''. In first-order logic, the ...
, the following are equivalent for a set ''x'': *''x'' is a (von Neumann) ordinal, *''x'' is a
transitive set In set theory, a branch of mathematics, a set A is called transitive if either of the following equivalent conditions hold: * whenever x \in A, and y \in x, then y \in A. * whenever x \in A, and x is not an urelement, then x is a subset of A. Simil ...
, and set membership is trichotomous on ''x'', *''x'' is a transitive set
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 ...
by set inclusion, *''x'' is a transitive set of transitive sets. These definitions cannot be used in
non-well-founded set theories Non-well-founded set theories are variants of axiomatic set theory that allow sets to be elements of themselves and otherwise violate the rule of well-foundedness. In non-well-founded set theories, the foundation axiom of ZFC is replaced by axio ...
. In set theories with
urelement In set theory, a branch of mathematics, an urelement or ur-element (from the German prefix ''ur-'', 'primordial') is an object that is not a set, but that may be an element of a set. It is also referred to as an atom or individual. Theory There ...
s, one has to further make sure that the definition excludes urelements from appearing in ordinals.


Transfinite sequence

If α is any ordinal and ''X'' is a set, an α-indexed sequence of elements of ''X'' is a function from α to ''X''. This concept, a transfinite sequence (if α is infinite) or ordinal-indexed sequence, is a generalization of the concept of 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 ...
. An ordinary sequence corresponds to the case α = ω, while a finite α corresponds to a
tuple In mathematics, a tuple is a finite ordered list (sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is defi ...
, a.k.a. string.


Transfinite induction

Transfinite induction holds in any
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 set, but it is so important in relation to ordinals that it is worth restating here. : Any property that passes from the set of ordinals smaller than a given ordinal α to α itself, is true of all ordinals. That is, if ''P''(α) is true whenever ''P''(β) is true for all , then ''P''(α) is true for ''all'' α. Or, more practically: in order to prove a property ''P'' for all ordinals α, one can assume that it is already known for all smaller .


Transfinite recursion

Transfinite induction can be used not only to prove things, but also to define them. Such a definition is normally said to be by
transfinite recursion 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 ...
– the proof that the result is well-defined uses transfinite induction. Let ''F'' denote a (class) function ''F'' to be defined on the ordinals. The idea now is that, in defining ''F''(α) for an unspecified ordinal α, one may assume that ''F''(β) is already defined for all and thus give a formula for ''F''(α) in terms of these ''F''(β). It then follows by transfinite induction that there is one and only one function satisfying the recursion formula up to and including α. Here is an example of definition by transfinite recursion on the ordinals (more will be given later): define function ''F'' by letting ''F''(α) be the smallest ordinal not in the set , that is, the set consisting of all ''F''(β) for . This definition assumes the ''F''(β) known in the very process of defining ''F''; this apparent vicious circle is exactly what definition by transfinite recursion permits. In fact, ''F''(0) makes sense since there is no ordinal , and the set is empty. So ''F''(0) is equal to 0 (the smallest ordinal of all). Now that ''F''(0) is known, the definition applied to ''F''(1) makes sense (it is the smallest ordinal not in the singleton set ), and so on (the ''and so on'' is exactly transfinite induction). It turns out that this example is not very exciting, since provably for all ordinals α, which can be shown, precisely, by transfinite induction.


Successor and limit ordinals

Any nonzero ordinal has the minimum element, zero. It may or may not have a maximum element. For example, 42 has maximum 41 and ω+6 has maximum ω+5. On the other hand, ω does not have a maximum since there is no largest natural number. If an ordinal has a maximum α, then it is the next ordinal after α, and it is called 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 ...
'', namely the successor of α, written α+1. In the von Neumann definition of ordinals, the successor of α is \alpha\cup\ since its elements are those of α and α itself. A nonzero ordinal that is ''not'' a successor is called 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 ...
''. One justification for this term is that a limit ordinal is the
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
in a topological sense of all smaller ordinals (under the
order topology In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets. If ''X'' is a totally ordered set, t ...
). When \langle \alpha_ , \iota < \gamma \rangle is an ordinal-indexed sequence, indexed by a limit \gamma and the sequence is ''increasing'', i.e. \alpha_ < \alpha_ whenever \iota < \rho, its ''limit'' is defined as the least upper bound of the set \, that is, the smallest ordinal (it always exists) greater than any term of the sequence. In this sense, a limit ordinal is the limit of all smaller ordinals (indexed by itself). Put more directly, it is the supremum of the set of smaller ordinals. Another way of defining a limit ordinal is to say that α is a limit ordinal if and only if: : There is an ordinal less than α and whenever ζ is an ordinal less than α, then there exists an ordinal ξ such that ζ < ξ < α. So in the following sequence: : 0, 1, 2, …, ω, ω+1 ω is a limit ordinal because for any smaller ordinal (in this example, a natural number) there is another ordinal (natural number) larger than it, but still less than ω. Thus, every ordinal is either zero, or a successor (of a well-defined predecessor), or a limit. This distinction is important, because many definitions by transfinite recursion rely upon it. Very often, when defining a function ''F'' by transfinite recursion on all ordinals, one defines ''F''(0), and ''F''(α+1) assuming ''F''(α) is defined, and then, for limit ordinals δ one defines ''F''(δ) as the limit of the ''F''(β) for all β<δ (either in the sense of ordinal limits, as previously explained, or for some other notion of limit if ''F'' does not take ordinal values). Thus, the interesting step in the definition is the successor step, not the limit ordinals. Such functions (especially for ''F'' nondecreasing and taking ordinal values) are called continuous. Ordinal addition, multiplication and exponentiation are continuous as functions of their second argument (but can be defined non-recursively).


Indexing classes of ordinals

Any well-ordered set is similar (order-isomorphic) to a unique ordinal number \alpha; in other words, its elements can be indexed in increasing fashion by the ordinals less than \alpha. This applies, in particular, to any set of ordinals: any set of ordinals is naturally indexed by the ordinals less than some \alpha. The same holds, with a slight modification, for ''classes'' of ordinals (a collection of ordinals, possibly too large to form a set, defined by some property): any class of ordinals can be indexed by ordinals (and, when the class is unbounded in the class of all ordinals, this puts it in class-bijection with the class of all ordinals). So the \gamma-th element in the class (with the convention that the "0-th" is the smallest, the "1-st" is the next smallest, and so on) can be freely spoken of. Formally, the definition is by transfinite induction: the \gamma-th element of the class is defined (provided it has already been defined for all \beta<\gamma), as the smallest element greater than the \beta-th element for all \beta<\gamma. This could be applied, for example, to the class of limit ordinals: the \gamma-th ordinal, which is either a limit or zero is \omega\cdot\gamma (see
ordinal arithmetic In the mathematical field of set theory, ordinal arithmetic describes the three usual operations on ordinal numbers: addition, multiplication, and exponentiation. Each can be defined in essentially two different ways: either by constructing an expl ...
for the definition of multiplication of ordinals). Similarly, one can consider ''
additively indecomposable ordinal In set theory, a branch of mathematics, an additively indecomposable ordinal ''α'' is any ordinal number that is not 0 such that for any \beta,\gamma<\alpha, we have \beta+\gamma<\alpha. Additively indecomposable ordi ...
s'' (meaning a nonzero ordinal that is not the sum of two strictly smaller ordinals): the \gamma-th additively indecomposable ordinal is indexed as \omega^\gamma . The technique of indexing classes of ordinals is often useful in the context of fixed points: for example, the \gamma-th ordinal \alpha such that \omega^\alpha = \alpha is written \varepsilon_\gamma. These are called the " epsilon numbers".


Closed unbounded sets and classes

A class C of ordinals is said to be unbounded, or cofinal, when given any ordinal \alpha, there is a \beta in C such that \alpha < \beta (then the class must be a proper class, i.e., it cannot be a set). It is said to be closed when the limit of a sequence of ordinals in the class is again in the class: or, equivalently, when the indexing (class-)function F is continuous in the sense that, for \delta a limit ordinal, F(\delta) (the \delta-th ordinal in the class) is the limit of all F(\gamma) for \gamma < \delta; this is also the same as being closed, in the
topological In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
sense, for the
order topology In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets. If ''X'' is a totally ordered set, t ...
(to avoid talking of topology on proper classes, one can demand that the intersection of the class with any given ordinal is closed for the order topology on that ordinal, this is again equivalent). Of particular importance are those classes of ordinals that are closed and unbounded, sometimes called clubs. For example, the class of all limit ordinals is closed and unbounded: this translates the fact that there is always a limit ordinal greater than a given ordinal, and that a limit of limit ordinals is a limit ordinal (a fortunate fact if the terminology is to make any sense at all!). The class of additively indecomposable ordinals, or the class of \varepsilon_\cdot ordinals, or the class of
cardinals Cardinal or The Cardinal may refer to: Animals * Cardinal (bird) or Cardinalidae, a family of North and South American birds **''Cardinalis'', genus of cardinal in the family Cardinalidae **''Cardinalis cardinalis'', or northern cardinal, the ...
, are all closed unbounded; the set of regular cardinals, however, is unbounded but not closed, and any finite set of ordinals is closed but not unbounded. A class is stationary if it has a nonempty intersection with every closed unbounded class. All superclasses of closed unbounded classes are stationary, and stationary classes are unbounded, but there are stationary classes that are not closed and stationary classes that have no closed unbounded subclass (such as the class of all limit ordinals with countable cofinality). Since the intersection of two closed unbounded classes is closed and unbounded, the intersection of a stationary class and a closed unbounded class is stationary. But the intersection of two stationary classes may be empty, e.g. the class of ordinals with cofinality ω with the class of ordinals with uncountable cofinality. Rather than formulating these definitions for (proper) classes of ordinals, one can formulate them for sets of ordinals below a given ordinal \alpha: A subset of a limit ordinal \alpha is said to be unbounded (or cofinal) under \alpha provided any ordinal less than \alpha is less than some ordinal in the set. More generally, one can call a subset of any ordinal \alpha cofinal in \alpha provided every ordinal less than \alpha is less than ''or equal to'' some ordinal in the set. The subset is said to be closed under \alpha provided it is closed for the order topology ''in'' \alpha, i.e. a limit of ordinals in the set is either in the set or equal to \alpha itself.


Arithmetic of ordinals

There are three usual operations on ordinals: addition, multiplication, and (ordinal) exponentiation. Each can be defined in essentially two different ways: either by constructing an explicit well-ordered set that represents the operation or by using transfinite recursion. The
Cantor normal form In the mathematical field of set theory, ordinal arithmetic describes the three usual operations on ordinal numbers: addition, multiplication, and exponentiation. Each can be defined in essentially two different ways: either by constructing an e ...
provides a standardized way of writing ordinals. It uniquely represents each ordinal as a finite sum of ordinal powers of ω. However, this cannot form the basis of a universal ordinal notation due to such self-referential representations as ε0 = ωε0. The so-called "natural" arithmetical operations retain commutativity at the expense of continuity. Interpreted as ''
nimber In mathematics, the nimbers, also called ''Grundy numbers'', are introduced in combinatorial game theory, where they are defined as the values of heaps in the game Nim. The nimbers are the ordinal numbers endowed with ''nimber addition'' and ' ...
s'' (a game-theoretic variant of numbers), ordinals are also subject to nimber arithmetic operations.


Ordinals and cardinals


Initial ordinal of a cardinal

Each ordinal associates with one cardinal, its cardinality. If there is a bijection between two ordinals (e.g. and ), then they associate with the same cardinal. Any well-ordered set having an ordinal as its order-type has the same cardinality as that ordinal. The least ordinal associated with a given cardinal is called the ''initial ordinal'' of that cardinal. Every finite ordinal (natural number) is initial, and no other ordinal associates with its cardinal. But most infinite ordinals are not initial, as many infinite ordinals associate with the same cardinal. 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 equivalent to the statement that every set can be well-ordered, i.e. that every cardinal has an initial ordinal. In theories with the axiom of choice, the cardinal number of any set has an initial ordinal, and one may employ the
Von Neumann cardinal assignment The von Neumann cardinal assignment is a cardinal assignment that uses ordinal numbers. For a well-orderable set ''U'', we define its cardinal number to be the smallest ordinal number equinumerous to ''U'', using the von Neumann definition of an or ...
as the cardinal's representation. (However, we must then be careful to distinguish between cardinal arithmetic and ordinal arithmetic.) In set theories without the axiom of choice, a cardinal may be represented by the set of sets with that cardinality having minimal rank (see
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 ...
). One issue with Scott's trick is that it identifies the cardinal number 0 with \, which in some formulations is the ordinal number 1. It may be clearer to apply Von Neumann cardinal assignment to finite cases and to use Scott's trick for sets which are infinite or do not admit well orderings. Note that cardinal and ordinal arithmetic agree for finite numbers. The α-th infinite initial ordinal is written \omega_\alpha, it is always a limit ordinal. Its cardinality is written \aleph_\alpha. For example, the cardinality of ω0 = ω is \aleph_0, which is also the cardinality of ω2 or ε0 (all are countable ordinals). So ω can be identified with \aleph_0, except that the notation \aleph_0 is used when writing cardinals, and ω when writing ordinals (this is important since, for example, \aleph_0^2 = \aleph_0 whereas \omega^2 > \omega). Also, \omega_1 is the smallest uncountable ordinal (to see that it exists, consider the set of equivalence classes of well-orderings of the natural numbers: each such well-ordering defines a countable ordinal, and \omega_1 is the order type of that set), \omega_2 is the smallest ordinal whose cardinality is greater than \aleph_1, and so on, and \omega_\omega is the limit of the \omega_n for natural numbers ''n'' (any limit of cardinals is a cardinal, so this limit is indeed the first cardinal after all the \omega_n).


Cofinality

The
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 ...
of an ordinal \alpha is the smallest ordinal \delta that is the order type of a cofinal subset of \alpha. Notice that a number of authors define cofinality or use it only for limit ordinals. The cofinality of a set of ordinals or any other well-ordered set is the cofinality of the order type of that set. Thus for a limit ordinal, there exists a \delta-indexed strictly increasing sequence with limit \alpha. For example, the cofinality of ω2 is ω, because the sequence ω·''m'' (where ''m'' ranges over the natural numbers) tends to ω2; but, more generally, any countable limit ordinal has cofinality ω. An uncountable limit ordinal may have either cofinality ω as does \omega_\omega or an uncountable cofinality. The cofinality of 0 is 0. And the cofinality of any successor ordinal is 1. The cofinality of any limit ordinal is at least \omega. An ordinal that is equal to its cofinality is called regular and it is always an initial ordinal. Any limit of regular ordinals is a limit of initial ordinals and thus is also initial even if it is not regular, which it usually is not. If the Axiom of Choice, then \omega_ is regular for each α. In this case, the ordinals 0, 1, \omega, \omega_1, and \omega_2 are regular, whereas 2, 3, \omega_\omega, and ωω·2 are initial ordinals that are not regular. The cofinality of any ordinal ''α'' is a regular ordinal, i.e. the cofinality of the cofinality of ''α'' is the same as the cofinality of ''α''. So the cofinality operation is
idempotent Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...
.


Some "large" countable ordinals

As mentioned above (see
Cantor normal form In the mathematical field of set theory, ordinal arithmetic describes the three usual operations on ordinal numbers: addition, multiplication, and exponentiation. Each can be defined in essentially two different ways: either by constructing an e ...
), the ordinal ε0 is the smallest satisfying the equation \omega^\alpha = \alpha, so it is the limit of the sequence 0, 1, \omega, \omega^\omega, \omega^, etc. Many ordinals can be defined in such a manner as fixed points of certain ordinal functions (the \iota-th ordinal such that \omega^\alpha = \alpha is called \varepsilon_\iota, then one could go on trying to find the \iota-th ordinal such that \varepsilon_\alpha = \alpha, "and so on", but all the subtlety lies in the "and so on"). One could try to do this systematically, but no matter what system is used to define and construct ordinals, there is always an ordinal that lies just above all the ordinals constructed by the system. Perhaps the most important ordinal that limits a system of construction in this manner is the
Church–Kleene ordinal In mathematics, particularly set theory, non-recursive ordinals are large countable ordinals greater than all the recursive ordinals, and therefore can not be expressed using ordinal collapsing functions. The Church–Kleene ordinal and varian ...
, \omega_1^ (despite the \omega_1 in the name, this ordinal is countable), which is the smallest ordinal that cannot in any way be represented by a
computable function Computable functions are the basic objects of study in computability theory. Computable functions are the formalized analogue of the intuitive notion of algorithms, in the sense that a function is computable if there exists an algorithm that can do ...
(this can be made rigorous, of course). Considerably large ordinals can be defined below \omega_1^, however, which measure the "proof-theoretic strength" of certain
formal system A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A form ...
s (for example, \varepsilon_0 measures the strength of Peano arithmetic). Large countable ordinals such as countable
admissible ordinal In set theory, an ordinal number ''α'' is an admissible ordinal if L''α'' is an admissible set (that is, a transitive model of Kripke–Platek set theory); in other words, ''α'' is admissible when ''α'' is a limit ordinal and L''α'' ⊧ Σ0- ...
s can also be defined above the Church-Kleene ordinal, which are of interest in various parts of logic.


Topology and ordinals

Any ordinal number can be made into a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
by endowing it with the
order topology In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets. If ''X'' is a totally ordered set, t ...
; this topology is
discrete Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory *Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit *Discrete group, a g ...
if and only if the ordinal is a countable cardinal, i.e. at most ω. A subset of ω + 1 is open in the order topology if and only if either it is
cofinite In mathematics, a cofinite subset of a set X is a subset A whose complement in X is a finite set. In other words, A contains all but finitely many elements of X. If the complement is not finite, but it is countable, then one says the set is coco ...
or it does not contain ω as an element. See the Topology and ordinals section of the "Order topology" article.


History

The transfinite ordinal numbers, which first appeared in 1883, originated in Cantor's work with derived sets. If ''P'' is a set of real numbers, the derived set ''P' '' is the set of
limit point In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contai ...
s of ''P''. In 1872, Cantor generated the sets ''P''(''n'') by applying the derived set operation ''n'' times to ''P''. In 1880, he pointed out that these sets form the sequence ''P' ''⊇ ··· ⊇ ''P''(''n'') ⊇ ''P''(''n'' + 1) ⊇ ···, and he continued the derivation process by defining ''P''(∞) as the intersection of these sets. Then he iterated the derived set operation and intersections to extend his sequence of sets into the infinite: ''P''(∞) ⊇ ''P''(∞ + 1) ⊇ ''P''(∞ + 2) ⊇ ··· ⊇ ''P''(2∞) ⊇ ··· ⊇ ''P''(∞2) ⊇ ···. The superscripts containing ∞ are just indices defined by the derivation process. Cantor used these sets in the theorems: (1) If ''P''(α) = ∅ for some index α, then ''P' '' is countable; (2) Conversely, if ''P' '' is countable, then there is an index α such that ''P''(α) = ∅. These theorems are proved by partitioning ''P' '' into
pairwise disjoint In mathematics, two sets are said to be disjoint sets if they have no element in common. Equivalently, two disjoint sets are sets whose intersection is the empty set.. For example, and are ''disjoint sets,'' while and are not disjoint. A c ...
sets: ''P = (''P' ''∖ ''P''(2)) ∪ (''P''(2) ∖ ''P''(3)) ∪ ··· ∪ (''P''(∞) ∖ ''P''(∞ + 1)) ∪ ··· ∪ ''P''(α). For β < α: since ''P''(β + 1) contains the limit points of ''P''(β), the sets ''P''(β) ∖ ''P''(β + 1) have no limit points. Hence, they are discrete sets, so they are countable. Proof of first theorem: If ''P''(α) = ∅ for some index α, then ''P' '' is the countable union of countable sets. Therefore, ''P' '' is countable. The second theorem requires proving the existence of an α such that ''P''(α) = ∅. To prove this, Cantor considered the set of all α having countably many predecessors. To define this set, he defined the transfinite ordinal numbers and transformed the infinite indices into ordinals by replacing ∞ with ω, the first transfinite ordinal number. Cantor called the set of finite ordinals the first number class. The second number class is the set of ordinals whose predecessors form a countably infinite set. The set of all α having countably many predecessors—that is, the set of countable ordinals—is the union of these two number classes. Cantor proved that the cardinality of the second number class is the first uncountable cardinality. Cantor's second theorem becomes: If ''P' '' is countable, then there is a countable ordinal α such that ''P''(α) = ∅. Its proof uses
proof by contradiction In logic and mathematics, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition, by showing that assuming the proposition to be false leads to a contradiction. Proof by contradiction is also known ...
. Let ''P' '' be countable, and assume there is no such α. This assumption produces two cases. * Case 1: ''P''(β) ∖ ''P''(β + 1) is non-empty for all countable β. Since there are uncountably many of these pairwise disjoint sets, their union is uncountable. This union is a subset of ''P, so ''P' '' is uncountable. * Case 2: ''P''(β) ∖ ''P''(β + 1) is empty for some countable β. Since ''P''(β + 1) ⊆ ''P''(β), this implies ''P''(β + 1) = ''P''(β). Thus, ''P''(β) is a
perfect set In general topology, a subset of a topological space is perfect if it is closed and has no isolated points. Equivalently: the set S is perfect if S=S', where S' denotes the set of all Limit point, limit points of S, also known as the derived set ...
, so it is uncountable. Since ''P''(β) ⊆ ''P, the set ''P' '' is uncountable. In both cases, ''P' '' is uncountable, which contradicts ''P' '' being countable. Therefore, there is a countable ordinal α such that ''P''(α) = ∅. Cantor's work with derived sets and ordinal numbers led to the
Cantor-Bendixson theorem In descriptive set theory, a subset of a Polish space has the perfect set property if it is either countable or has a nonempty perfect subset (Kechris 1995, p. 150). Note that having the perfect set property is not the same as being a p ...
. Using successors, limits, and cardinality, Cantor generated an unbounded sequence of ordinal numbers and number classes. The (α + 1)-th number class is the set of ordinals whose predecessors form a set of the same cardinality as the α-th number class. The cardinality of the (α + 1)-th number class is the cardinality immediately following that of the α-th number class. For a limit ordinal α, the α-th number class is the union of the β-th number classes for β < α. Its cardinality is the limit of the cardinalities of these number classes. If ''n'' is finite, the ''n''-th number class has cardinality \aleph_. If α ≥ ω, the α-th number class has cardinality \aleph_\alpha.The first number class has cardinality \aleph_0.
Mathematical induction Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ...  all hold. Informal metaphors help ...
proves that the ''n''-th number class has cardinality \aleph_. Since the ω-th number class is the union of the ''n''-th number classes, its cardinality is \aleph_\omega, the limit of the \aleph_.
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 ...
proves that if α ≥ ω, the α-th number class has cardinality \aleph_\alpha.
Therefore, the cardinalities of the number classes correspond one-to-one with the
aleph number In mathematics, particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets that can be well-ordered. They were introduced by the mathematician Georg Cantor and are named a ...
s. Also, the α-th number class consists of ordinals different from those in the preceding number classes if and only if α is a non-limit ordinal. Therefore, the non-limit number classes partition the ordinals into pairwise disjoint sets.


See also

*
Counting Counting is the process of determining the number of elements of a finite set of objects, i.e., determining the size of a set. The traditional way of counting consists of continually increasing a (mental or spoken) counter by a unit for every ele ...
* Even and odd ordinals * First uncountable ordinal *
Ordinal space In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets. If ''X'' is a totally ordered set, th ...
*
Surreal number In mathematics, the surreal number system is a totally ordered proper class containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number. The surreals ...
, a generalization of ordinals which includes negatives


Notes


References

* . Published separately as: ''Grundlagen einer allgemeinen Mannigfaltigkeitslehre''. *
English translation: Contributions to the Founding of the Theory of Transfinite Numbers II
* * . * . * . * . * . * . *. *. *. * Also defines ordinal operations in terms of the Cantor Normal Form. *. * . * * - English translation of .


External links

*

GPL'd free software for computing with ordinals and ordinal notations * Chapter 4 o
Don Monk's
on set theory is an introduction to ordinals. {{Authority control Wellfoundedness