{"history":[],"article":{"id":"6fe02da2-2389-422c-ab63-99d33bb3ef73","title":"Ordinal number","slug":"ordinal-number","content":":::summary\n:::\n\n# \nIn [set theory](https://bliptext.com/articles/set-theory), an ****ordinal** number**, or **ordinal**, is a generalization of **ordinal** numerals (first, second, nth, etc.) aimed to extend [enumeration](https://bliptext.com/articles/enumeration) to [infinite sets](https://bliptext.com/articles/infinite-set).\nA finite set can be enumerated by successively labeling each element with the least [natural number](https://bliptext.com/articles/natural-number) that has not been previously used.\nTo extend this process to various [infinite sets](https://bliptext.com/articles/infinite-set), ordinal numbers are defined more generally using [linearly ordered](https://bliptext.com/articles/linearly-ordered) [greek letter](https://bliptext.com/articles/greek-letters) [variables](https://bliptext.com/articles/variable-mathematics) that include the natural numbers and have the property that every set of ordinals has a [least or \"smallest\" element](https://bliptext.com/articles/greatest-element-and-least-element) (this is needed for giving a meaning to \"the least unused element\").\nThis more general definition allows us to define an ordinal number \\omega (omega) to be the least element that is greater than every natural number, along with ordinal numbers \\omega + 1, \\omega + 2, etc., which are even greater than \\omega.\nA linear order such that every non-empty subset has a least element is called a [well-order](https://bliptext.com/articles/well-order).\nThe [axiom of choice](https://bliptext.com/articles/axiom-of-choice) implies that every set can be well-ordered, and given two well-ordered sets, one is [isomorphic](https://bliptext.com/articles/isomorphic) to an [initial segment](https://bliptext.com/articles/initial-segment) of the other.\nSo ordinal numbers exist and are essentially unique.\nOrdinal numbers are distinct from [cardinal numbers](https://bliptext.com/articles/cardinal-number), which measure the size of sets.\nAlthough 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.\nLike other kinds of numbers, ordinals can be [added, multiplied, and exponentiated](https://bliptext.com/articles/ordinal-arithmetic), although none of these operations are [commutative](https://bliptext.com/articles/commutative-property).\nOrdinals were introduced by [Georg Cantor](https://bliptext.com/articles/georg-cantor) in 1883 in order to accommodate infinite sequences and classify [derived sets](https://bliptext.com/articles/derived-set-mathematics), which he had previously introduced in 1872 while studying the uniqueness of [trigonometric series](https://bliptext.com/articles/trigonometric-series).\n# [Representation of the ordinal numbers up to ωω. One turn of the spiral corresponds to the mapping f(\\alpha )=\\omega (1+\\alpha ). Since f has as least fixed point, larger ordinal numbers cannot be represented on this diagram. | upload.wikimedia.org/wikipedia/commons/a/af/Omega-exp-omega-normal-dark///svg.svg]\n\n# Ordinals extend the natural numbers\nA [natural number](https://bliptext.com/articles/natural-number) (which, in this context, includes the number [0](https://bliptext.com/articles/0-number)) can be used for two purposes: to describe the size of a [set](https://bliptext.com/articles/set-mathematics), or to describe the position of an element in a sequence.\nWhen restricted to finite sets, these two concepts coincide, since all [linear orders](https://bliptext.com/articles/linear-order) of a finite set are [isomorphic](https://bliptext.com/articles/order-isomorphism).\nWhen dealing with infinite sets, however, one has to distinguish between the notion of size, which leads to [cardinal numbers](https://bliptext.com/articles/cardinal-number), and the notion of position, which leads to the ordinal numbers described here.\nThis is because while any set has only one size (its [cardinality](https://bliptext.com/articles/cardinality)), there are many nonisomorphic [well-orderings](https://bliptext.com/articles/well-ordering) of any infinite set, as explained below.\nWhereas 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-ordered](https://bliptext.com/articles/well-order).\nA well-[ordered](https://bliptext.com/articles/partial-order) set is a [totally ordered](https://bliptext.com/articles/totally-[ordered](https://bliptext.com/articles/partial-order)) set (an [ordered](https://bliptext.com/articles/partial-order) set such that, given two distinct elements, one is less than the other) in which every non-empty subset has a least element.\nEquivalently, assuming the [axiom of dependent choice](https://bliptext.com/articles/axiom-of-dependent-choice), it is a totally ordered set without any infinite decreasing sequence — though there may be infinite increasing sequences.\nOrdinals 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.\nThis \"length\" is called the order type of the set.\nAny ordinal is defined by the set of ordinals that precede it.\nIn fact, the most common definition of ordinals identifies each ordinal as the set of ordinals that precede it.\nFor example, the ordinal 42 is generally identified as the set.\nConversely, any set S of ordinals that is [downward closed](https://bliptext.com/articles/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.\nThis definition of ordinals in terms of sets allows for infinite ordinals.\nThe 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).\nIndeed, 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.\nPerhaps 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.\n(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.\nNow 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.\nFurther on, there will be ω3, then ω4, and so on, and ωω, then ωω ω, then later ωω ω ω , and even later ε0 ([epsilon nought](https://bliptext.com/articles/epsilon-number)) (to give a few examples of relatively small—countable—ordinals).\nThis can be continued indefinitely (as every time one says \"and so on\" when enumerating ordinals, it defines a larger ordinal).\nThe smallest [uncountable](https://bliptext.com/articles/uncountable-set) ordinal is the set of all countable ordinals, expressed as [ω1](https://bliptext.com/articles/first-uncountable-ordinal) or \\Omega.\n# [A graphical \"matchstick\" representation of the ordinal ω2. Each stick corresponds to an ordinal of the form ω·m+n where m and n are natural numbers. | upload.wikimedia.org/wikipedia/commons/1/18/Ordinal///ww.svg]\n\n# Definitions\n\n## Well-ordered sets\nIn a [well-ordered](https://bliptext.com/articles/well-order) set, every non-empty subset contains a distinct smallest element.\nGiven the [axiom of dependent choice](https://bliptext.com/articles/axiom-of-dependent-choice), this is equivalent to saying that the set is [totally ordered](https://bliptext.com/articles/totally-ordered) and there is no infinite decreasing sequence (the latter being easier to visualize).\nIn practice, the importance of well-ordering is justified by the possibility of applying [transfinite induction](https://bliptext.com/articles/transfinite-induction), 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).\nIf 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.\nIt 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.\nSuch a one-to-one correspondence is called an [order isomorphism](https://bliptext.com/articles/order-isomorphism), and the two well-ordered sets are said to be order-isomorphic or similar (with the understanding that this is an [equivalence relation](https://bliptext.com/articles/equivalence-relation)).\nFormally, if a [partial order](https://bliptext.com/articles/partial-order) ≤ is defined on the set S, and a [partial order](https://bliptext.com/articles/partial-order) ≤' is defined on the set S' , then the [posets](https://bliptext.com/articles/partially-ordered-set) (S,≤) and (S' ,≤') are [order isomorphic](https://bliptext.com/articles/order-isomorphic) if there is a [bijection](https://bliptext.com/articles/bijection) f that preserves the ordering.\nThat is, f(a) ≤' f(b) if and only if a ≤ b.\nProvided 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](https://bliptext.com/articles/representative-mathematics) of the isomorphism type (class).\nThis is exactly what the ordinals provide, and it also provides a canonical labeling of the elements of any well-ordered set.\nEvery well-ordered set (S,<) is order-isomorphic to the set of ordinals less than one specific ordinal number under their natural ordering.\nThis canonical set is the order type of (S,<).\nEssentially, an ordinal is intended to be defined as an [isomorphism class](https://bliptext.com/articles/isomorphism-class) of well-ordered sets: that is, as an [equivalence class](https://bliptext.com/articles/equivalence-class) for the [equivalence relation](https://bliptext.com/articles/equivalence-relation) of \"being order-isomorphic\".\nThere 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](https://bliptext.com/articles/zermelo-fraenkel-set-theory) (ZF) formalization of set theory.\nBut this is not a serious difficulty.\nThe ordinal can be said to be the [order type](https://bliptext.com/articles/order-type) of any set in the class.\n\n## Definition of an ordinal as an equivalence class\nThe original definition of ordinal numbers, found for example in the [Principia Mathematica](https://bliptext.com/articles/principia-mathematica), 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.\nThis definition must be abandoned in [ZF](https://bliptext.com/articles/zermelo-fraenkel-set-theory) and related systems of [axiomatic set theory](https://bliptext.com/articles/axiomatic-set-theory) because these equivalence classes are too large to form a set.\nHowever, this definition still can be used in [type theory](https://bliptext.com/articles/type-theory) and in Quine's axiomatic set theory [New Foundations](https://bliptext.com/articles/new-foundations) and related systems (where it affords a rather surprising alternative solution to the [Burali-Forti paradox](https://bliptext.com/articles/burali-forti-paradox) of the largest ordinal).\n\n## Von Neumann definition of ordinals\nRather 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.\nThus, an ordinal number will be a well-ordered set; and every well-ordered set will be order-isomorphic to exactly one ordinal number.\nFor each well-ordered set T, defines an [order isomorphism](https://bliptext.com/articles/order-isomorphism) between T and the set of all subsets of T having the form ordered by inclusion.\nThis motivates the standard definition, suggested by [John von Neumann](https://bliptext.com/articles/john-von-neumann) at the age of 19, now called definition of **von Neumann ordinals**: \"each ordinal is the well-ordered set of all smaller ordinals\".\nIn symbols, \\lambda = [0,\\lambda). Formally:\nThe natural numbers are thus ordinals by this definition.\nFor instance, 2 is an element of\n4 = ,\nand 2 is equal to\nand so it is a subset of\n.\nIt can be shown by [transfinite induction](https://bliptext.com/articles/transfinite-induction) that every well-ordered set is order-isomorphic to exactly one of these ordinals, that is, there is an order preserving [bijective function](https://bliptext.com/articles/bijective-function) between them.\nFurthermore, the elements of every ordinal are ordinals themselves.\nGiven two ordinals S and T, S is an element of T if and only if S is a [proper subset](https://bliptext.com/articles/proper-subset) of T.\nMoreover, either S is an element of T, or T is an element of S, or they are equal.\nSo every set of ordinals is [totally ordered](https://bliptext.com/articles/total-order).\nFurther, every set of ordinals is well-ordered.\nThis generalizes the fact that every set of natural numbers is well-ordered.\nConsequently, every ordinal S is a set having as elements precisely the ordinals smaller than S.\nFor example, every set of ordinals has a [supremum](https://bliptext.com/articles/supremum), the ordinal obtained by taking the union of all the ordinals in the set.\nThis union exists regardless of the set's size, by the [axiom of union](https://bliptext.com/articles/axiom-of-union).\nThe class of all ordinals is not a set.\nIf 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.\nThis is the [Burali-Forti paradox](https://bliptext.com/articles/burali-forti-paradox).\nThe class of all ordinals is variously called \"Ord\", \"ON\", or \"∞\".\nAn ordinal is [finite](https://bliptext.com/articles/finite-set) 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 [greatest element](https://bliptext.com/articles/greatest-element-and-least-element).\n\n## Other definitions\nThere are other modern formulations of the definition of ordinal.\nFor example, assuming the [axiom of regularity](https://bliptext.com/articles/axiom-of-regularity), the following are equivalent for a set x:\nThese definitions cannot be used in [non-well-founded set theories](https://bliptext.com/articles/non-well-founded-set-theory).\nIn set theories with [urelements](https://bliptext.com/articles/urelement), one has to further make sure that the definition excludes [urelements](https://bliptext.com/articles/urelement) from appearing in ordinals.\n\n# Transfinite sequence\nIf α is any ordinal and X is a set, an α-indexed sequence of elements of X is a function from α to X.\nThis concept, a **transfinite [sequence](https://bliptext.com/articles/sequence)** (if α is infinite) or ordinal-indexed [sequence](https://bliptext.com/articles/sequence), is a generalization of the concept of a [sequence](https://bliptext.com/articles/sequence).\nAn ordinary sequence corresponds to the case α = ω, while a finite α corresponds to a [tuple](https://bliptext.com/articles/tuple-mathematics), a.k.a. [string](https://bliptext.com/articles/string-computer-science).\n\n# Transfinite induction\nTransfinite induction holds in any [well-ordered](https://bliptext.com/articles/well-order) set, but it is so important in relation to ordinals that it is worth restating here.\nThat is, if P(α) is true whenever P(β) is true for all β < α, then P(α) is true for all α.\nOr, more practically: in order to prove a property P for all ordinals α, one can assume that it is already known for all smaller β < α.\n\n## Transfinite recursion\nTransfinite induction can be used not only to prove things, but also to define them.\nSuch a definition is normally said to be by [transfinite recursion](https://bliptext.com/articles/transfinite-recursion) – the proof that the result is well-defined uses transfinite induction.\nLet F denote a (class) function F to be defined on the ordinals.\nThe idea now is that, in defining F(α) for an unspecified ordinal α, one may assume that F(β) is already defined for all β < α and thus give a formula for F(α) in terms of these F(β).\nIt then follows by transfinite induction that there is one and only one function satisfying the recursion formula up to and including α.\nHere 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 β < α.\nThis 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.\nIn fact, F(0) makes sense since there is no ordinal β < 0, and the set is empty.\nSo F(0) is equal to 0 (the smallest ordinal of all).\nNow 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).\nIt turns out that this example is not very exciting, since provably F(α) = α for all ordinals α, which can be shown, precisely, by transfinite induction.\n\n## Successor and limit ordinals\nAny nonzero ordinal has the minimum element, zero.\nIt may or may not have a maximum element.\nFor example, 42 has maximum 41 and ω+6 has maximum ω+5.\nOn the other hand, ω does not have a maximum since there is no largest natural number.\nIf an ordinal has a maximum α, then it is the next ordinal after α, and it is called a [successor ordinal](https://bliptext.com/articles/successor-ordinal), namely the successor of α, written α+1.\nIn the von Neumann definition of ordinals, the successor of α is since its elements are those of α and α itself.\nA nonzero ordinal that is not a successor is called a [limit ordinal](https://bliptext.com/articles/limit-ordinal).\nOne justification for this term is that a [limit](https://bliptext.com/articles/limit-point) ordinal is the [limit](https://bliptext.com/articles/limit-point) in a topological sense of all smaller ordinals (under the [order topology](https://bliptext.com/articles/order-topology)).\nWhen is an ordinal-indexed sequence, indexed by a limit \\gamma and the sequence is increasing, i.e. whenever 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.\nIn this sense, a limit ordinal is the limit of all smaller ordinals (indexed by itself).\nPut more directly, it is the supremum of the set of smaller ordinals.\nAnother way of defining a limit ordinal is to say that α is a limit ordinal if and only if:\nSo in the following sequence:\nω 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.\nThus, every ordinal is either zero, or a successor (of a well-defined predecessor), or a limit.\nThis distinction is important, because many definitions by transfinite recursion rely upon it.\nVery 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).\nThus, the interesting step in the definition is the successor step, not the limit ordinals.\nSuch functions (especially for F nondecreasing and taking ordinal values) are called continuous.\nOrdinal addition, multiplication and exponentiation are continuous as functions of their second argument (but can be defined non-recursively).\n\n## Indexing classes of ordinals\nAny 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.\nThis applies, in particular, to any set of ordinals: any set of ordinals is naturally indexed by the ordinals less than some \\alpha.\nThe 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).\nSo 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.\nFormally, the definition is by transfinite induction: the \\gamma-th element of the class is defined (provided it has already been defined for all ), as the smallest element greater than the \\beta-th element for all \\beta<\\gamma.\nThis could be applied, for example, to the class of limit ordinals: the \\gamma-th ordinal, which is either a limit or zero is (see [ordinal arithmetic](https://bliptext.com/articles/ordinal-arithmetic) for the definition of multiplication of ordinals).\nSimilarly, one can consider [additively indecomposable ordinals](https://bliptext.com/articles/additively-indecomposable-ordinal) (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.\nThe technique of indexing classes of ordinals is often useful in the context of fixed points: for example, the \\gamma-th ordinal \\alpha such that is written \\varepsilon_\\gamma.\nThese are called the \"[epsilon numbers](https://bliptext.com/articles/epsilon-numbers-mathematics)\".\n\n## Closed unbounded sets and classes\nA class C of ordinals is said to be unbounded, or cofinal, when given any ordinal \\alpha, there is a \\beta in C such that (then the class must be a proper class, i.e., it cannot be a set).\nIt 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 ; this is also the same as being closed, in the [topological](https://bliptext.com/articles/topological-space) sense, for the [order topology](https://bliptext.com/articles/order-topology) (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](https://bliptext.com/articles/order-topology) on that ordinal, this is again equivalent).\nOf particular importance are those classes of ordinals that are [closed and unbounded](https://bliptext.com/articles/club-set), sometimes called clubs.\nFor 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!).\nThe class of additively indecomposable ordinals, or the class of ordinals, or the class of [cardinals](https://bliptext.com/articles/), are all closed unbounded; the set of [regular](https://bliptext.com/articles/) [cardinals](https://bliptext.com/articles/), however, is unbounded but not closed, and any finite set of ordinals is closed but not unbounded.\nA class is stationary if it has a nonempty intersection with every closed unbounded class.\nAll 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).\nSince the intersection of two closed unbounded classes is closed and unbounded, the intersection of a stationary class and a closed unbounded class is stationary.\nBut 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.\nRather 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.\nMore 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.\nThe 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.\n\n# Arithmetic of ordinals\nThere are three usual operations on ordinals: addition, multiplication, and exponentiation.\nEach 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.\nThe [Cantor normal form](https://bliptext.com/articles/ordinal-arithmetic) provides a standardized way of writing ordinals.\nIt uniquely represents each ordinal as a finite sum of ordinal powers of ω.\nHowever, this cannot form the basis of a universal ordinal notation due to such self-referential representations as ε0 = ωε0.\nOrdinals are a subclass of the class of [surreal numbers](https://bliptext.com/articles/surreal-number), and the so-called \"natural\" arithmetical operations for [surreal numbers](https://bliptext.com/articles/surreal-number) are an alternative way to combine ordinals arithmetically.\nThey retain commutativity at the expense of continuity.\nInterpreted as [nimbers](https://bliptext.com/articles/nimber), a game-theoretic variant of numbers, ordinals can also be combined via nimber arithmetic operations.\nThese operations are commutative but the restriction to natural numbers is generally not the same as ordinary addition of natural numbers.\n\n# Ordinals and cardinals\n\n## Initial ordinal of a cardinal\nEach ordinal associates with one [cardinal](https://bliptext.com/articles/cardinal-number), its [cardinal](https://bliptext.com/articles/cardinal-number)ity.\nIf there is a bijection between two ordinals (e.g. and ω + 1 > ω), then they associate with the same cardinal.\nAny well-ordered set having an ordinal as its order-type has the same cardinality as that ordinal.\nThe least ordinal associated with a given cardinal is called the initial ordinal of that cardinal.\nEvery finite ordinal (natural number) is initial, and no other ordinal associates with its cardinal.\nBut most infinite ordinals are not initial, as many infinite ordinals associate with the same cardinal.\nThe [axiom of choice](https://bliptext.com/articles/axiom-of-choice) is equivalent to the statement that every set can be well-ordered, i.e. that every cardinal has an initial ordinal.\nIn 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](https://bliptext.com/articles/von-neumann-cardinal-assignment) as the cardinal's representation.\n(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](https://bliptext.com/articles/scott-s-trick)).\nOne issue with Scott's trick is that it identifies the cardinal number 0 with \\{\\emptyset\\}, which in some formulations is the ordinal number 1.\nIt 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.\nNote that cardinal and ordinal arithmetic agree for finite numbers.\nThe α-th infinite initial ordinal is written \\omega_\\alpha, it is always a limit ordinal.\nIts cardinality is written \\aleph_\\alpha.\nFor example, the cardinality of ω0 = ω is \\aleph_0, which is also the cardinality of ω2 or ε0 (all are countable ordinals).\nSo 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 ).\nAlso, \\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 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).\n\n## Cofinality\nThe [cofinality](https://bliptext.com/articles/[cofinal](https://bliptext.com/articles/cofinal-mathematics)ity) of an ordinal \\alpha is the smallest ordinal \\delta that is the order type of a [cofinal](https://bliptext.com/articles/cofinal-mathematics) subset of \\alpha.\nNotice that a number of authors define cofinality or use it only for limit ordinals.\nThe cofinality of a set of ordinals or any other well-ordered set is the cofinality of the order type of that set.\nThus for a limit ordinal, there exists a \\delta-indexed strictly increasing sequence with limit \\alpha.\nFor 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.\nAn uncountable limit ordinal may have either cofinality ω as does or an uncountable cofinality.\nThe cofinality of 0 is 0.\nAnd the cofinality of any successor ordinal is 1.\nThe cofinality of any limit ordinal is at least \\omega.\nAn ordinal that is equal to its cofinality is called regular and it is always an initial ordinal.\nAny 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.\nIf the Axiom of Choice, then is regular for each α.\nIn 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.\nThe 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.\nSo the cofinality operation is [idempotent](https://bliptext.com/articles/idempotent).\n\n# Some \"large\" countable ordinals\nAs mentioned above (see [Cantor normal form](https://bliptext.com/articles/ordinal-arithmetic)), 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^{\\omega^\\omega}, etc.\nMany ordinals can be defined in such a manner as fixed points of certain ordinal functions (the \\iota-th ordinal such that 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\").\nOne 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.\nPerhaps the most important ordinal that limits a system of construction in this manner is the [Church–Kleene ordinal](https://bliptext.com/articles/church-kleene-ordinal), (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](https://bliptext.com/articles/computable-function) (this can be made rigorous, of course).\nConsiderably large ordinals can be defined below \\omega_1^{\\mathrm{CK} }, however, which measure the \"proof-theoretic strength\" of certain [formal systems](https://bliptext.com/articles/formal-system) (for example, measures the strength of [Peano arithmetic](https://bliptext.com/articles/peano-axioms)).\nLarge countable ordinals such as countable [admissible ordinals](https://bliptext.com/articles/admissible-ordinal) can also be defined above the Church-Kleene ordinal, which are of interest in various parts of logic.\n\n# Topology and ordinals\nAny ordinal number can be made into a [topological space](https://bliptext.com/articles/topological-space) by endowing it with the [order topology](https://bliptext.com/articles/order-topology); this topology is [discrete](https://bliptext.com/articles/discrete-topology) if and only if it is less than or equal to ω.\nA subset of ω + 1 is open in the order topology if and only if either it is [cofinite](https://bliptext.com/articles/cofinite) or it does not contain ω as an element.\nSee the [Topology and ordinals](https://bliptext.com/articles/order-topology) section of the \"Order topology\" article.\n\n# History\nThe transfinite ordinal numbers, which first appeared in 1883, originated in Cantor's work with [derived sets](https://bliptext.com/articles/derived-set-mathematics).\nIf P is a set of real numbers, the derived set is the set of [limit points](https://bliptext.com/articles/limit-point) of P.\nIn 1872, Cantor generated the sets P(n) by applying the derived set operation n times to P.\nIn 1880, he pointed out that these sets form the sequence\nP' ⊇ ··· ⊇ P(n) ⊇ P(n + 1) ⊇ ···,\nand he continued the derivation process by defining P(∞) as the intersection of these sets.\nThen he iterated the derived set operation and intersections to extend his sequence of sets into the infinite:\nP(∞) ⊇ P(∞ + 1) ⊇ P(∞ + 2) ⊇ ··· ⊇ P(2∞) ⊇ ··· ⊇ P(∞ 2) ⊇ ···.\nThe superscripts containing ∞ are just indices defined by the derivation process.\nCantor used these sets in the theorems:\nThese theorems are proved by partitioning into [pairwise disjoint](https://bliptext.com/articles/pairwise-disjoint) sets:\n' = ('\\ P(2)) ∪ (P(2) \\ P(3)) ∪ ··· ∪ (P(∞) \\ P(∞ + 1)) ∪ ··· ∪ P(α)\n. For\nβ < α:\nsince\nP(β + 1)\ncontains the limit points of P(β), the sets\nP(β) \\ P(β + 1)\nhave no limit points.\nHence, they are [discrete sets](https://bliptext.com/articles/discrete-set), so they are countable.\nProof of first theorem: If\nP(α) = ∅\nfor some index α, then is the countable union of countable sets.\nTherefore, is countable.\nThe second theorem requires proving the existence of an α such that\nP(α) = ∅\n. To prove this, Cantor considered the set of all α having countably many predecessors.\nTo 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.\nCantor called the set of finite ordinals the first **number class**.\nThe second number class is the set of ordinals whose predecessors form a countably infinite set.\nThe set of all α having countably many predecessors—that is, the set of countable ordinals—is the union of these two number classes.\nCantor proved that the cardinality of the second number class is the first uncountable cardinality.\nCantor's second theorem becomes: If is countable, then there is a countable ordinal α such that\nP(α) = ∅\n. Its proof uses [proof by contradiction](https://bliptext.com/articles/proof-by-contradiction).\nLet be countable, and assume there is no such α.\nThis assumption produces two cases.\nindent=1\nP(β) \\ P(β + 1)\nis non-empty for all countable β.\nSince there are uncountably many of these pairwise disjoint sets, their union is uncountable.\nThis union is a subset of , so P' is uncountable.\nP(β) \\ P(β + 1)\nis empty for some countable β.\nSince\nP(β + 1) ⊆ P(β)\n, this implies\nP(β + 1) = P(β)\n. Thus, P(β) is a [perfect set](https://bliptext.com/articles/derived-set-mathematics), so it is uncountable.\nSince\nP(β) ⊆\n, the set is uncountable.\nIn both cases, **' is uncountable, which contradicts '** being countable.\nTherefore, there is a countable ordinal α such that\nP(α) = ∅\n. Cantor's work with derived sets and ordinal numbers led to the [Cantor-Bendixson theorem](https://bliptext.com/articles/cantor-bendixson-theorem).\nUsing successors, limits, and cardinality, Cantor generated an unbounded sequence of ordinal numbers and number classes.\nThe\n(α + 1)\n-th number class is the set of ordinals whose predecessors form a set of the same cardinality as the α-th number class.\nThe cardinality of the\n(α + 1)\n-th number class is the cardinality immediately following that of the α-th number class.\nFor a limit ordinal α, the α-th number class is the union of the β-th number classes for\nβ < α\n. Its cardinality is the limit of the cardinalities of these number classes.\nIf n is finite, the n-th number class has cardinality \\aleph_{n-1}.\nIf\nα ≥ ω\n, the α-th number class has cardinality \\aleph_\\alpha.\nTherefore, the cardinalities of the number classes correspond one-to-one with the [aleph numbers](https://bliptext.com/articles/aleph-number).\nAlso, 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.\nTherefore, the non-limit number classes partition the ordinals into pairwise disjoint sets.","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-26T12:54:42.861Z","updated_at":"2025-01-26T12:54:42.861Z","search_vector":"'number':2 'ordin':1","search_content":" \n \n\n# \nIn [set theory](https://bliptext.com/articles/set-theory), an ****ordinal** number**, or **ordinal**, is a generalization of **ordinal** numerals (first, second, nth, etc.) aimed to extend [enumeration](https://bliptext.com/articles/enumeration) to [infinite sets](https://bliptext.com/articles/infinite-set).\nA finite set can be enumerated by successively labeling each element with the least [natural number](https://bliptext.com/articles/natural-number) that has not been previously used.\nTo extend this process to various [infinite sets](https://bliptext.com/articles/infinite-set), ordinal numbers are defined more generally using [linearly ordered](https://bliptext.com/articles/linearly-ordered) [greek letter](https://bliptext.com/articles/greek-letters) [variables](https://bliptext.com/articles/variable-mathematics) that include the natural numbers and have the property that every set of ordinals has a [least or \"smallest\" element](https://bliptext.com/articles/greatest-element-and-least-element) (this is needed for giving a meaning to \"the least unused element\").\nThis more general definition allows us to define an ordinal number \\omega (omega) to be the least element that is greater than every natural number, along with ordinal numbers \\omega + 1, \\omega + 2, etc., which are even greater than \\omega.\nA linear order such that every non-empty subset has a least element is called a [well-order](https://bliptext.com/articles/well-order).\nThe [axiom of choice](https://bliptext.com/articles/axiom-of-choice) implies that every set can be well-ordered, and given two well-ordered sets, one is [isomorphic](https://bliptext.com/articles/isomorphic) to an [initial segment](https://bliptext.com/articles/initial-segment) of the other.\nSo ordinal numbers exist and are essentially unique.\nOrdinal numbers are distinct from [cardinal numbers](https://bliptext.com/articles/cardinal-number), which measure the size of sets.\nAlthough 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.\nLike other kinds of numbers, ordinals can be [added, multiplied, and exponentiated](https://bliptext.com/articles/ordinal-arithmetic), although none of these operations are [commutative](https://bliptext.com/articles/commutative-property).\nOrdinals were introduced by [Georg Cantor](https://bliptext.com/articles/georg-cantor) in 1883 in order to accommodate infinite sequences and classify [derived sets](https://bliptext.com/articles/derived-set-mathematics), which he had previously introduced in 1872 while studying the uniqueness of [trigonometric series](https://bliptext.com/articles/trigonometric-series).\n# [Representation of the ordinal numbers up to ωω. One turn of the spiral corresponds to the mapping f(\\alpha )=\\omega (1+\\alpha ). Since f has as least fixed point, larger ordinal numbers cannot be represented on this diagram. | upload.wikimedia.org/wikipedia/commons/a/af/Omega-exp-omega-normal-dark///svg.svg]\n\n# Ordinals extend the natural numbers\nA [natural number](https://bliptext.com/articles/natural-number) (which, in this context, includes the number [0](https://bliptext.com/articles/0-number)) can be used for two purposes: to describe the size of a [set](https://bliptext.com/articles/set-mathematics), or to describe the position of an element in a sequence.\nWhen restricted to finite sets, these two concepts coincide, since all [linear orders](https://bliptext.com/articles/linear-order) of a finite set are [isomorphic](https://bliptext.com/articles/order-isomorphism).\nWhen dealing with infinite sets, however, one has to distinguish between the notion of size, which leads to [cardinal numbers](https://bliptext.com/articles/cardinal-number), and the notion of position, which leads to the ordinal numbers described here.\nThis is because while any set has only one size (its [cardinality](https://bliptext.com/articles/cardinality)), there are many nonisomorphic [well-orderings](https://bliptext.com/articles/well-ordering) of any infinite set, as explained below.\nWhereas 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-ordered](https://bliptext.com/articles/well-order).\nA well-[ordered](https://bliptext.com/articles/partial-order) set is a [totally [ordered](https://bliptext.com/articles/partial-order)](https://bliptext.com/articles/totally-[ordered](https://bliptext.com/articles/partial-order)) set (an [ordered](https://bliptext.com/articles/partial-order) set such that, given two distinct elements, one is less than the other) in which every non-empty subset has a least element.\nEquivalently, assuming the [axiom of dependent choice](https://bliptext.com/articles/axiom-of-dependent-choice), it is a totally ordered set without any infinite decreasing sequence — though there may be infinite increasing sequences.\nOrdinals 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.\nThis \"length\" is called the order type of the set.\nAny ordinal is defined by the set of ordinals that precede it.\nIn fact, the most common definition of ordinals identifies each ordinal as the set of ordinals that precede it.\nFor example, the ordinal 42 is generally identified as the set.\nConversely, any set S of ordinals that is [downward closed](https://bliptext.com/articles/downward-closed) — meaning that for any ordinal α in S and any ordinal β ω), then they associate with the same cardinal.\nAny well-ordered set having an ordinal as its order-type has the same cardinality as that ordinal.\nThe least ordinal associated with a given cardinal is called the initial ordinal of that cardinal.\nEvery finite ordinal (natural number) is initial, and no other ordinal associates with its cardinal.\nBut most infinite ordinals are not initial, as many infinite ordinals associate with the same cardinal.\nThe [axiom of choice](https://bliptext.com/articles/axiom-of-choice) is equivalent to the statement that every set can be well-ordered, i.e. that every cardinal has an initial ordinal.\nIn 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](https://bliptext.com/articles/von-neumann-cardinal-assignment) as the cardinal's representation.\n(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](https://bliptext.com/articles/scott-s-trick)).\nOne issue with Scott's trick is that it identifies the cardinal number 0 with \\{\\emptyset\\}, which in some formulations is the ordinal number 1.\nIt 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.\nNote that cardinal and ordinal arithmetic agree for finite numbers.\nThe α-th infinite initial ordinal is written \\omega_\\alpha, it is always a limit ordinal.\nIts cardinality is written \\aleph_\\alpha.\nFor example, the cardinality of ω0 = ω is \\aleph_0, which is also the cardinality of ω2 or ε0 (all are countable ordinals).\nSo 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 ).\nAlso, \\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 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).\n\n## Cofinality\nThe [[cofinal](https://bliptext.com/articles/cofinal-mathematics)ity](https://bliptext.com/articles/[cofinal](https://bliptext.com/articles/cofinal-mathematics)ity) of an ordinal \\alpha is the smallest ordinal \\delta that is the order type of a [cofinal](https://bliptext.com/articles/cofinal-mathematics) subset of \\alpha.\nNotice that a number of authors define cofinality or use it only for limit ordinals.\nThe cofinality of a set of ordinals or any other well-ordered set is the cofinality of the order type of that set.\nThus for a limit ordinal, there exists a \\delta-indexed strictly increasing sequence with limit \\alpha.\nFor 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.\nAn uncountable limit ordinal may have either cofinality ω as does or an uncountable cofinality.\nThe cofinality of 0 is 0.\nAnd the cofinality of any successor ordinal is 1.\nThe cofinality of any limit ordinal is at least \\omega.\nAn ordinal that is equal to its cofinality is called regular and it is always an initial ordinal.\nAny 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.\nIf the Axiom of Choice, then is regular for each α.\nIn 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.\nThe 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.\nSo the cofinality operation is [idempotent](https://bliptext.com/articles/idempotent).\n\n# Some \"large\" countable ordinals\nAs mentioned above (see [Cantor normal form](https://bliptext.com/articles/ordinal-arithmetic)), 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^{\\omega^\\omega}, etc.\nMany ordinals can be defined in such a manner as fixed points of certain ordinal functions (the \\iota-th ordinal such that 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\").\nOne 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.\nPerhaps the most important ordinal that limits a system of construction in this manner is the [Church–Kleene ordinal](https://bliptext.com/articles/church-kleene-ordinal), (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](https://bliptext.com/articles/computable-function) (this can be made rigorous, of course).\nConsiderably large ordinals can be defined below \\omega_1^{\\mathrm{CK} }, however, which measure the \"proof-theoretic strength\" of certain [formal systems](https://bliptext.com/articles/formal-system) (for example, measures the strength of [Peano arithmetic](https://bliptext.com/articles/peano-axioms)).\nLarge countable ordinals such as countable [admissible ordinals](https://bliptext.com/articles/admissible-ordinal) can also be defined above the Church-Kleene ordinal, which are of interest in various parts of logic.\n\n# Topology and ordinals\nAny ordinal number can be made into a [topological space](https://bliptext.com/articles/topological-space) by endowing it with the [order topology](https://bliptext.com/articles/order-topology); this topology is [discrete](https://bliptext.com/articles/discrete-topology) if and only if it is less than or equal to ω.\nA subset of ω + 1 is open in the order topology if and only if either it is [cofinite](https://bliptext.com/articles/cofinite) or it does not contain ω as an element.\nSee the [Topology and ordinals](https://bliptext.com/articles/order-topology) section of the \"Order topology\" article.\n\n# History\nThe transfinite ordinal numbers, which first appeared in 1883, originated in Cantor's work with [derived sets](https://bliptext.com/articles/derived-set-mathematics).\nIf P is a set of real numbers, the derived set is the set of [limit points](https://bliptext.com/articles/limit-point) of P.\nIn 1872, Cantor generated the sets P(n) by applying the derived set operation n times to P.\nIn 1880, he pointed out that these sets form the sequence\nP' ⊇ ··· ⊇ P(n) ⊇ P(n + 1) ⊇ ···,\nand he continued the derivation process by defining P(∞) as the intersection of these sets.\nThen he iterated the derived set operation and intersections to extend his sequence of sets into the infinite:\nP(∞) ⊇ P(∞ + 1) ⊇ P(∞ + 2) ⊇ ··· ⊇ P(2∞) ⊇ ··· ⊇ P(∞ 2) ⊇ ···.\nThe superscripts containing ∞ are just indices defined by the derivation process.\nCantor used these sets in the theorems:\nThese theorems are proved by partitioning into [pairwise disjoint](https://bliptext.com/articles/pairwise-disjoint) sets:\n' = ('\\ P(2)) ∪ (P(2) \\ P(3)) ∪ ··· ∪ (P(∞) \\ P(∞ + 1)) ∪ ··· ∪ P(α)\n. For\nβ < α:\nsince\nP(β + 1)\ncontains the limit points of P(β), the sets\nP(β) \\ P(β + 1)\nhave no limit points.\nHence, they are [discrete sets](https://bliptext.com/articles/discrete-set), so they are countable.\nProof of first theorem: If\nP(α) = ∅\nfor some index α, then is the countable union of countable sets.\nTherefore, is countable.\nThe second theorem requires proving the existence of an α such that\nP(α) = ∅\n. To prove this, Cantor considered the set of all α having countably many predecessors.\nTo 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.\nCantor called the set of finite ordinals the first **number class**.\nThe second number class is the set of ordinals whose predecessors form a countably infinite set.\nThe set of all α having countably many predecessors—that is, the set of countable ordinals—is the union of these two number classes.\nCantor proved that the cardinality of the second number class is the first uncountable cardinality.\nCantor's second theorem becomes: If is countable, then there is a countable ordinal α such that\nP(α) = ∅\n. Its proof uses [proof by contradiction](https://bliptext.com/articles/proof-by-contradiction).\nLet be countable, and assume there is no such α.\nThis assumption produces two cases.\nindent=1\nP(β) \\ P(β + 1)\nis non-empty for all countable β.\nSince there are uncountably many of these pairwise disjoint sets, their union is uncountable.\nThis union is a subset of , so P' is uncountable.\nP(β) \\ P(β + 1)\nis empty for some countable β.\nSince\nP(β + 1) ⊆ P(β)\n, this implies\nP(β + 1) = P(β)\n. Thus, P(β) is a [perfect set](https://bliptext.com/articles/derived-set-mathematics), so it is uncountable.\nSince\nP(β) ⊆\n, the set is uncountable.\nIn both cases, **' is uncountable, which contradicts '** being countable.\nTherefore, there is a countable ordinal α such that\nP(α) = ∅\n. Cantor's work with derived sets and ordinal numbers led to the [Cantor-Bendixson theorem](https://bliptext.com/articles/cantor-bendixson-theorem).\nUsing successors, limits, and cardinality, Cantor generated an unbounded sequence of ordinal numbers and number classes.\nThe\n(α + 1)\n-th number class is the set of ordinals whose predecessors form a set of the same cardinality as the α-th number class.\nThe cardinality of the\n(α + 1)\n-th number class is the cardinality immediately following that of the α-th number class.\nFor a limit ordinal α, the α-th number class is the union of the β-th number classes for\nβ < α\n. Its cardinality is the limit of the cardinalities of these number classes.\nIf n is finite, the n-th number class has cardinality \\aleph_{n-1}.\nIf\nα ≥ ω\n, the α-th number class has cardinality \\aleph_\\alpha.\nTherefore, the cardinalities of the number classes correspond one-to-one with the [aleph numbers](https://bliptext.com/articles/aleph-number).\nAlso, 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.\nTherefore, the non-limit number classes partition the ordinals into pairwise disjoint sets.","revisionCount":0},"hasMore":false,"nextCursor":null}