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Ordinal number

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Generalization of "n-th" to infinite cases
This article is about the mathematical concept. For number words denoting a position in a sequence ("first", "second", "third", etc.), seeOrdinal numeral.
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Representation of the ordinal numbers up toωω{\displaystyle \omega ^{\omega }}. One turn of the spiral corresponds to the mapping⁠f(α)=ω(1+α){\displaystyle f(\alpha )=\omega (1+\alpha )}⁠. Sincef{\displaystyle f} hasωω{\displaystyle \omega ^{\omega }} as the least fixed point, larger ordinal numbers cannot be represented on this diagram.

Inset theory, anordinal number, orordinal, is a generalization ofordinal numerals (first, second,nth, etc.) aimed to extendenumeration toinfinite sets.[1] UsuallyGreek letters are used for ordinal numbervariables to help distinguish them fromnatural number variables.

A finite set can be enumerated by successively labeling each element with the leastnatural number that has not been previously used. To extend this process to variousinfinite sets, ordinal numbers are defined more generally as alinearly ordered class of numbers that include the natural numbers and have the property that every non-empty collection (set orproper class) of ordinals has aleast or "smallest" element (this is needed for giving a meaning to "the least unused element"). This more general definition allows us to define an ordinal numberω{\displaystyle \omega } (omega) to be the least element that is greater than every natural number, along with ordinal numbers⁠ω+1{\displaystyle \omega +1}⁠,⁠ω+2{\displaystyle \omega +2}⁠, etc., which are even greater than⁠ω{\displaystyle \omega }⁠.

TheZermelo–Fraenkel set theory asserts that, forany set of ordinals, there exists another ordinal greater than all of them. The answer to the question "What if that set is the set of all ordinals?" (theBurali-Forti paradox) is that the collection of all ordinals is not a set, but a proper class.

A linear order such that every non-empty subset has a least element is called awell-order. Theaxiom of choice implies that every set can be well-ordered. Given two well-ordered sets, one isisomorphic to aninitial segment of the other, and the isomorphism is unique. This allows a unique ordinal to be associated with each well-ordered set, known as itsorder type.

Ordinal numbers are distinct fromcardinal numbers, which measure the size of sets. Although the distinction between ordinals and cardinals is not this 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 beadded, multiplied, and exponentiated, although none of these operations arecommutative.

Ordinals were introduced byGeorg Cantor in 1883[2] to accommodate infinite sequences and classifyderived sets, which he had previously introduced in 1872 while studying the uniqueness oftrigonometric series.[3]

Motivation

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Anatural number (which, in this context, includes the number0) can be used for two purposes: to describe thesize of aset, or to describe theposition of an element in asequence. When generalized to infinite sets, the notion of size leads tocardinal numbers, and the notion of position leads to the ordinal numbers described here.

In a broader mathematical sense, counting can be viewed as the instantiation ofmathematical induction. To enumerate a well-ordered set is effectively to verify a property for its elements sequentially. For the natural numbers, this is standard induction: if a property holds for 0, and its truth for⁠n{\displaystyle n}⁠ implies its truth for⁠n+1{\displaystyle n+1}⁠, then it holds for all natural numbers. This process corresponds to the first infinite ordinal,⁠ω{\displaystyle \omega }⁠.

A graphical "matchstick" representation of the ordinalω2. Each stick corresponds to an ordinal of the formω ·j +i wherej andi are natural numbers. This structure corresponds tonested induction: an inner induction oni and an outer induction onj.

Mathematical contexts often require iterating beyond a single infinite limit. The ordinal⁠ω2{\displaystyle \omega ^{2}}⁠ (represented in the figure) exemplifies the concept ofnested induction. It consists of a sequence of distinct copies of the natural numbers ordered one after another. To verify a property for all ordinals less than⁠ω2{\displaystyle \omega ^{2}}⁠, one performs an "inner" induction (counting through⁠0,1,2,…{\displaystyle 0,1,2,\dots }⁠), establishes the limit at⁠ω{\displaystyle \omega }⁠, and then proceeds to the next sequence (⁠ω+1,ω+2,…{\displaystyle \omega +1,\omega +2,\dots }⁠). This structure parallels a nested loop in computer programming (e.g., iterating through pairs of natural numbers⁠(j,i){\displaystyle (j,i)}⁠ orderedlexicographically). Ordinals allow the definition of processes of arbitrary complexity, such as⁠ω3{\displaystyle \omega ^{3}}⁠ (triple nesting) or⁠ωω{\displaystyle \omega ^{\omega }}⁠ (induction over the depth of nested induction).

The validity of inductive counting rests on the property ofwell-foundedness, specifically the requirement that every process can be traced back to a "foundational" element. A linear order that exhibits this well-foundedness is termed awell-order. The existence of a "least" or minimal element in every non-empty subset of a well-ordered set grounds the principle oftransfinite induction, generalizing standard induction by ensuring that if a property fails to hold, there exists a specific least counterexample.

Ordinals serve as the canonical abstractions of these well-ordered structures.[4] A fundamental theorem in set theory establishes that any two well-ordered sets arecomparable: given two well-orders, either they areisomorphic, or one is isomorphic to a properinitial segment of the other. This uniqueness implies that well-orders can be classified by their structure alone, independent of specific representations. Consequently,ordinal numbers are defined as the representative forms of these isomorphism classes.

Definitions

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Well-ordering

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The construction procedure of transfinite sequences implies that the ordinals used to label their elements arewell-ordered. This means that:

Every non-empty collection of ordinals⁠T{\displaystyle T}⁠ has a unique least element.

Intuitively, the least ordinal in⁠T{\displaystyle T}⁠ is the "next" ordinal introduced after all ordinals strictly less than every ordinal in⁠T{\displaystyle T}⁠ have been used. In contrast,⁠T{\displaystyle T}⁠ need not have a greatest element, for example when⁠T{\displaystyle T}⁠ is the set of all natural numbers.

Being well-ordered is stronger than merely beinglinearly ordered. For example, thereal numbers are naturally linearly ordered, but anyopen interval has no least element.

The importance of well-ordering is that it enablestransfinite induction. If a statement⁠P(α){\displaystyle P(\alpha )}⁠ about an ordinal⁠α{\displaystyle \alpha }⁠ is not universally true for all⁠α{\displaystyle \alpha }⁠, i.e., if it has any counterexamples, then it must have a least counterexample. Conversely, if it can be proven that⁠P(α){\displaystyle P(\alpha )}⁠ is true whenever⁠P(β){\displaystyle P(\beta )}⁠ is true for all⁠β<α{\displaystyle \beta <\alpha }⁠, then it cannot have any least counterexample, and thus it cannot have any counterexample at all, i.e., it is indeed universally true.[5]

Well-ordered sets

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In the usualZermelo–Fraenkel (ZF) formalization of set theory, a well-ordered set is atotally ordered set⁠(S,≤){\displaystyle (S,\leq )}⁠ such that every non-empty subset⁠T⊆S{\displaystyle T\subseteq S}⁠ has a least element. Here “set” means a collection that is itself an object of ZF, as opposed to aproper class.

A well-ordered set can be written as a transfinite sequence by assigning ordinal labels to its elements in a way that respects ordering:⁠aα≤aβ{\displaystyle a_{\alpha }\leq a_{\beta }}⁠ if and only if⁠α≤β{\displaystyle \alpha \leq \beta }⁠. (Such a one-to-one correspondence between two ordered sets is called anorder isomorphism.) The labels used in such an enumeration form aninitial segment of the ordinals, in the sense that if a label is used then every smaller ordinal label is also used. By well-orderedness, there is a unique least ordinal that is not used as a label; this ordinal determines the "length" of the sequence, and is called theorder type of the well-ordering. Thanks to0-based indexing, the order type coincides with thecardinality for finite sets, including the empty set.[1] However, for infinite sets, different well-order relations⁠≤{\displaystyle \leq }⁠ can have different order types.

Definition of an ordinal as an equivalence class

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Order types can be defined without a preexisting notion of ordinals, by working directly with order isomorphisms between general well-ordered sets, just ascardinality can be defined throughbijections between general (unordered) sets. As with bijections, being order-isomorphic is anequivalence relation on well-ordered sets; itsequivalence classes correspond to order types (ordinals).

In thePrincipia Mathematica approach, the order type of a well-ordered set⁠(S,≤){\displaystyle (S,\leq )}⁠ isidentified with itsisomorphism class, i.e., the set of all well-ordered sets⁠(S′,≤′){\displaystyle (S',\leq ')}⁠ order-isomorphic to⁠(S,≤){\displaystyle (S,\leq )}⁠. Since elements of⁠S′{\displaystyle S'}⁠ are allowed to beanything, this definition has the “set of all sets” flavor, and in ZF such collections are generally too large to be sets. This definition can still be used intype theory and in Quine's axiomatic set theoryNew Foundations and related systems.[a]

InZF and related systems ofaxiomatic set theory, these equivalence classes are generally too large to form sets. Consequently, it is necessary to select a unique, canonical representative from each class—a single set that embodies the structure of the well-ordering. Since the fundamental relation in set theory is set membership (⁠∈{\displaystyle \in }⁠), the ideal representation is one where the abstract order relation⁠<{\displaystyle <}⁠ is translated directly into the membership relation⁠∈{\displaystyle \in }⁠.

Von Neumann definition of ordinals

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Thevon Neumann representation[6] provides this canonical form. It relies on the observation that anywell-founded relation satisfying certain properties can be mapped to a specific set where the relation becomes set membership. This mapping is known as theMostowski collapse lemma.

When applied to a well-ordering, the Mostowski collapse yields a specific set⁠S{\displaystyle S}⁠ where the order relation⁠x<y{\displaystyle x<y}⁠ is literally true if and only if⁠x∈y{\displaystyle x\in y}⁠. The resulting sets have the property that they aretransitive: every element of⁠S{\displaystyle S}⁠ is also a subset of⁠S{\displaystyle S}⁠ (i.e., the union of the set is contained within the set). In this representation, each ordinal is identified with the set of all preceding ordinals.

Thus the finitevon Neumann ordinals are definedrecursively as⁠0=∅{\displaystyle 0=\emptyset }⁠,⁠1={0}{\displaystyle 1=\{0\}}⁠,⁠2={0,1}{\displaystyle 2=\{0,1\}}⁠, etc. The first infinite ordinal⁠ω{\displaystyle \omega }⁠ is represented by the set of all finite ordinals, i.e., the set ofvon Neumann natural numbers⁠N={0,1,2,…}{\displaystyle \mathbb {N} =\{0,1,2,\ldots \}}⁠. Then⁠ω+1={0,1,2,…,ω}=N∪{ω}{\displaystyle \omega +1=\{0,1,2,\ldots ,\omega \}=\mathbb {N} \cup \{\omega \}}⁠, and so on.

Informally, one may define an ordinal recursively as adownward closed set of ordinals. Such a recursive definition is usually justified with thetransitive closure. However, the von Neumann ordinals are alreadytransitive sets, allowing them to be formally defined by a concise statement:

A setS{\displaystyle S} is an ordinal if and only ifS{\displaystyle S} istransitive (every element ofS{\displaystyle S} is a subset ofS{\displaystyle S}) and strictly well-ordered[b] by set membership (∈{\displaystyle \in }).

The set⁠ω≡N{\displaystyle \omega \equiv \mathbb {N} }⁠ is usually defined as the smallest inductive set (containing⁠∅{\displaystyle \emptyset }⁠ and closed under successor). The "smallest" constraint guarantees that each element of⁠ω{\displaystyle \omega }⁠ is either zero or the successor of another element of⁠ω{\displaystyle \omega }⁠, which allows induction to demonstrate that⁠ω{\displaystyle \omega }⁠ indeed satisfies the formal definition of ordinals stated above.

Basic properties

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Defining the strict order relation⁠<{\displaystyle <}⁠ as the membership relation⁠∈{\displaystyle \in }⁠ restricted to the class of all ordinals, the recursive characterization⁠γ={α∣α<γ}{\displaystyle \gamma =\{\alpha \mid \alpha <\gamma \}}⁠ is reduced to the following statement:

The non-strict order relation⁠≤{\displaystyle \leq }⁠ has an alternative characterization:⁠α≤β{\displaystyle \alpha \leq \beta }⁠ if and only if⁠α⊆β{\displaystyle \alpha \subseteq \beta }⁠ for ordinals⁠α,β{\displaystyle \alpha ,\beta }⁠. The "if" direction follows from transitivity of⁠β{\displaystyle \beta }⁠, and the "only if" direction follows from:

This implies that⁠≤{\displaystyle \leq }⁠ is apartial order. It is in fact atotal order, and awell-order:

So all ordinal numbers form a well-ordered class⁠ON{\displaystyle \mathrm {ON} }⁠, and thus any nonempty set of ordinals equipped with⁠≤{\displaystyle \leq }⁠ is a well-ordered set.

Specific ordinals can be constructed explicitly with the following principles:

The explicit forms of⁠succ{\displaystyle \mathrm {succ} }⁠ and⁠sup{\displaystyle \sup }⁠ imply that there exists an ordinal greater than any set of ordinals. In other words,

Order types

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Every well-ordered setS{\displaystyle S} is order-isomorphic to exactly one ordinal, known as itsorder type. Uniqueness is guaranteed because a well-ordered set cannot be isomorphic to a proper initial segment of itself, preventing isomorphism to two distinct ordinals. The existence of this ordinal is proven by defining a map pairing eachx∈S{\displaystyle x\in S} with the ordinal representing the order type of the initial segment⁠Sx={y∈S∣y<x}{\displaystyle S_{x}=\{y\in S\mid y<x\}}⁠. By theAxiom schema of replacement, the range of this map is a set of ordinals. Because this range is downward closed (the order type of a segment of a segment is a smaller ordinal), the range is itself an ordinalγ{\displaystyle \gamma }. The domain of the isomorphism must be all ofS{\displaystyle S}; otherwise, the least elementz∈S{\displaystyle z\in S} outside the domain would implySz≅γ{\displaystyle S_{z}\cong \gamma }, which would effectively includez{\displaystyle z} in the domain, a contradiction. Thus,S≅γ{\displaystyle S\cong \gamma }.[10]

Successor and limit ordinals

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Every ordinal number is one of three types: the ordinal zero, a successor ordinal, or a limit ordinal.

There is variation in the definition of limit ordinals regarding the inclusion of zero. Some texts, such asIntroduction to Cardinal Arithmetic by Holz et al., define a limit ordinal as a non-zero ordinal that is not a successor.[11] In contrast, other standard set theory texts, including Jech'sSet Theory and Just and Weese'sDiscovering Modern Set Theory, define a limit ordinal simply as any ordinal that is not a successor, which implies that 0 is a limit ordinal.[12][13] When the topological definition is used (based on theorder topology), 0 is not a limit ordinal because it is not alimit point of the set of smaller ordinals (which is empty); Rosenstein'sLinear Orderings uses this definition.[14] When 0 is included as a limit, ordinals that are strictly greater than 0 and not successors are usually referred to as "nonzero limit ordinals".

The following properties characterize nonzero limit ordinals:

λ{\displaystyle \lambda } is a nonzero limit ordinal if and only ifλ≠0{\displaystyle \lambda \neq 0} and for every ordinalα<λ{\displaystyle \alpha <\lambda }, the successorS(α){\displaystyle S(\alpha )} is also less thanλ{\displaystyle \lambda }.

This implies that a nonzero limit ordinal is equal to the supremum of all ordinals strictly less than it:

λ{\displaystyle \lambda } is a nonzero limit ordinal if and only ifλ=sup{α∣α<λ}=⋃λ{\displaystyle \lambda =\sup\{\alpha \mid \alpha <\lambda \}=\bigcup \lambda } andλ≠0{\displaystyle \lambda \neq 0}.

For example,ω{\displaystyle \omega } is a limit ordinal because any natural number is less than⁠ω{\displaystyle \omega }⁠, and the successor of any natural number is also a natural number (hence less than⁠ω{\displaystyle \omega }⁠). It is the least limit ordinal because each natural number⁠n∈ω{\displaystyle n\in \omega }⁠ is either zero or a successor.

Termination of decreasing sequences

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Any strictly decreasing sequence of ordinals⁠α0>α1>α2>⋯{\displaystyle \alpha _{0}>\alpha _{1}>\alpha _{2}>\cdots }⁠ must be finite. This follows directly from the natural ordering of ordinals being a well-order: if there existed such an infinite decreasing sequence, then the set⁠{αi∣i∈N}{\displaystyle \{\alpha _{i}\mid i\in \mathbb {N} \}}⁠ would be a set of ordinals without a least element. By the same argument, a well-ordered set has no infinite strictly descending chains; in fact, assuming theaxiom of dependent choice, anytotal order that satisfies this condition is a well-order, giving an alternative characterization of well-ordered sets.[15] The fact this is true for natural numbers is the basis of Fermat's method ofproof by infinite descent, which can be generalized to ordinals and other well-ordered classes too, as a special case oftransfinite induction where the proof of any⁠P(α){\displaystyle P(\alpha )}⁠ only requires⁠P(β){\displaystyle P(\beta )}⁠ for at most one specific⁠β<α{\displaystyle \beta <\alpha }⁠.

This property may be surprising when the initial value is an infinite ordinal. Indeed, for sequences starting from a natural number⁠n{\displaystyle n}⁠, the longest sequence is always one that decreases by 1 every step, leading to a sequence with⁠n{\displaystyle n}⁠ steps (⁠n+1{\displaystyle n+1}⁠ elements). This strategy of descending to the immediate predecessor remains valid for successor ordinals. However, a limit ordinal⁠λ{\displaystyle \lambda }⁠ has no immediate predecessor to descend to, so any next term must jump to some⁠β<λ{\displaystyle \beta <\lambda }⁠, skipping infinitely many ordinals strictly between⁠β{\displaystyle \beta }⁠ and⁠λ{\displaystyle \lambda }⁠. For example, when descending from⁠ω{\displaystyle \omega }⁠, one must choose a finite natural number, and thus "commit" to the number of maximum remaining steps. Descending from⁠ω⋅k{\displaystyle \omega \cdot k}⁠ allows one to make such a commitment⁠k{\displaystyle k}⁠ times, and descending from⁠ω2{\displaystyle \omega ^{2}}⁠ allows one to commit to a finite value of⁠k{\displaystyle k}⁠. Larger ordinals may allow more complicated decision structures, but the number of descending steps remains unbounded but finite.[16]

This property is useful for proving termination for any procedure. 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.

Transfinite sequence

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Ifα{\displaystyle \alpha } is any ordinal andX{\displaystyle X} is a set, anα{\displaystyle \alpha }-indexed sequence of elements ofX{\displaystyle X} is a function fromα{\displaystyle \alpha } toX{\displaystyle X}. This concept, atransfinite sequence (ifα{\displaystyle \alpha } is infinite) orordinal-indexed sequence, is a generalization of the concept of asequence. An ordinary sequence corresponds to the caseα=ω{\displaystyle \alpha =\omega }, while a finiteα{\displaystyle \alpha } corresponds to atuple, a.k.a.string.

While a sequence indexed by a specific ordinalα{\displaystyle \alpha } is a set, a sequence indexed by the class of all ordinals is aproper class. TheAxiom schema of replacement guarantees that any initial segment of such a class-sequence (the restriction of the function to some specific ordinalδ{\displaystyle \delta }) is a set.

When⟨xι∣ι<λ⟩{\displaystyle \langle x_{\iota }\mid \iota <\lambda \rangle } is a transfinite sequence of ordinals indexed by a limit ordinalλ{\displaystyle \lambda } and the sequence isincreasing (i.e.ι<ρ⟹xι<xρ{\displaystyle \iota <\rho \implies x_{\iota }<x_{\rho }}), itslimit is defined as the least upper bound of the set{xι∣ι<λ}{\displaystyle \{x_{\iota }\mid \iota <\lambda \}}.

A transfinite sequencef{\displaystyle f} mapping ordinals to ordinals is said to becontinuous (in the order topology) if for every limit ordinalλ{\displaystyle \lambda } in its domain,

  • iff (λ) is a limit ordinal and for every ε <f (λ) there exists a δ < λ such that for every γ, if δ < γ < λ, then ε <f (γ) ≤f (λ), and
  • iff (λ) isnot a limit ordinal, there exists a δ < λ such that for every γ, if δ < γ < λ, thenf (γ) =f (λ).

A sequence is callednormal if it is both strictly increasing and continuous. If a sequencef is increasing (not necessarily strictly) and continuous and λ is a limit ordinal, thenf(λ)=⋃β<λf(β){\displaystyle f(\lambda )=\bigcup _{\beta <\lambda }f(\beta )}.

Transfinite induction

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Main article:Transfinite induction

Transfinite induction holds in anywell-ordered 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, ifP(α) is true wheneverP(β) is true for allβ < α, thenP(α) is true forall α. Or, more practically: in order to prove a propertyP for all ordinals α, one can assume that it is already known for all smallerβ < α.[5]

Transfinite recursion

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Transfinite induction can be used not only to prove theorems but also to define functions on ordinals. This is known astransfinite recursion.

Formally, a functionF is defined by transfinite recursion on the ordinals if, for every ordinalα, the value⁠F(α){\displaystyle F(\alpha )}⁠ is specified using the set of values{F(β)∣β<α}{\displaystyle \{F(\beta )\mid \beta <\alpha \}}.

Very often, when defining a functionF by transfinite recursion on all ordinals, the definition is separated into cases based on the type of the ordinal:

  1. Base case: DefineF(0){\displaystyle F(0)}.
  2. Successor step: DefineF(α+1){\displaystyle F(\alpha +1)} assumingF(α){\displaystyle F(\alpha )} is defined.
  3. Limit step: For a limit ordinalλ{\displaystyle \lambda }, defineF(λ){\displaystyle F(\lambda )} as the limit ofF(β){\displaystyle F(\beta )} for allβ<λ{\displaystyle \beta <\lambda } (either in the sense of ordinal limits or some other notion of limit if the codomain allows it).

The interesting step in the definition is usually the successor step. IfF(α){\displaystyle F(\alpha )} for limit ordinalsα{\displaystyle \alpha } is defined as thelimsup ofF(β){\displaystyle F(\beta )} forβ<α{\displaystyle \beta <\alpha } andF{\displaystyle F} takes ordinal values and is non-decreasing, the functionF{\displaystyle F} will be continuous as defined above. Ordinal addition, multiplication and exponentiation are continuous as functions of their second argument.

The existence and uniqueness of such a function are proven by constructing it as the union of partial approximations. The proof proceeds in three steps:

  1. Local Existence: For any specific ordinalδ, one proves the existence of a unique "recursion segment"—a function defined onδ that satisfies the recursive rule for all⁠β<δ{\displaystyle \beta <\delta }⁠.
  2. Uniqueness and Compatibility: One proves that any two recursion segments agree on their common domain. If⁠g1{\displaystyle g_{1}}⁠ is a segment on⁠δ1{\displaystyle \delta _{1}}⁠ and⁠g2{\displaystyle g_{2}}⁠ is a segment on⁠δ2{\displaystyle \delta _{2}}⁠ with⁠δ1<δ2{\displaystyle \delta _{1}<\delta _{2}}⁠, then⁠g2{\displaystyle g_{2}}⁠ restricted to⁠δ1{\displaystyle \delta _{1}}⁠ is identical to⁠g1{\displaystyle g_{1}}⁠.
  3. Global Definition: The global class functionF is defined as the union of all such unique recursion segments. For any ordinalα, the value⁠F(α){\displaystyle F(\alpha )}⁠ is the value assigned toα by any recursion segment defined on a domain larger thanα.

The rigorous justification for local existence relies on theaxiom schema of replacement for the step of limit ordinals in order to collect the recursion segments into a set.

This construction allows definitions such as ordinal addition, multiplication, and exponentiation to be rigorous. For example, exponentiation⁠αβ{\displaystyle \alpha ^{\beta }}⁠ is defined recursively onβ:

The principle of transfinite recursion also implies that recursion can be performed up to a specific ordinalδ{\displaystyle \delta } (defining a set rather than a proper class). This is often used to define sequences of lengthω{\displaystyle \omega }. For example, to prove that anormal functionf{\displaystyle f} has arbitrarily largefixed points, one constructs a sequence starting with any ordinalγ0{\displaystyle \gamma _{0}} and definesγn+1=f(γn){\displaystyle \gamma _{n+1}=f(\gamma _{n})} by recursion onn<ω{\displaystyle n<\omega }. Then the limitδ=supn<ωγn{\displaystyle \delta =\sup _{n<\omega }\gamma _{n}} is a fixed point off{\displaystyle f} because continuity ensuresf(δ)=f(supγn)=supf(γn)=supγn+1=δ{\displaystyle f(\delta )=f(\sup \gamma _{n})=\sup f(\gamma _{n})=\sup \gamma _{n+1}=\delta }.

Indexing classes of ordinals

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Any well-ordered set is similar (order-isomorphic) to a unique ordinal numberα{\displaystyle \alpha }; in other words, its elements can be indexed in increasing fashion by the ordinals less than⁠α{\displaystyle \alpha }⁠. This applies, in particular, to any set of ordinals: any set of ordinals is naturally indexed by the ordinals less than some⁠α{\displaystyle \alpha }⁠. The same holds, with a slight modification, forclasses 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 theγ{\displaystyle \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: theγ{\displaystyle \gamma }-th element of the class is defined (provided it has already been defined for allβ<γ{\displaystyle \beta <\gamma }), as the smallest element greater than theβ{\displaystyle \beta }-th element for all⁠β<γ{\displaystyle \beta <\gamma }⁠.

This could be applied, for example, to the class of limit ordinals: theγ{\displaystyle \gamma }-th ordinal, which is either a limit or zero isω⋅γ{\displaystyle \omega \cdot \gamma } (seeordinal arithmetic for the definition of multiplication of ordinals). Similarly, one can consideradditively indecomposable ordinals (meaning a nonzero ordinal that is not the sum of two strictly smaller ordinals): theγ{\displaystyle \gamma }-th additively indecomposable ordinal is indexed as⁠ωγ{\displaystyle \omega ^{\gamma }}⁠. The technique of indexing classes of ordinals is often useful in the context of fixed points: for example, theγ{\displaystyle \gamma }-th ordinalα{\displaystyle \alpha } such thatωα=α{\displaystyle \omega ^{\alpha }=\alpha } is written⁠εγ{\displaystyle \varepsilon _{\gamma }}⁠. These are called the "epsilon numbers".

Arithmetic of ordinals

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Main article:Ordinal arithmetic

There are three usual operations on ordinals: addition, multiplication, and 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. TheCantor normal form 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.

Ordinals are a subclass of the class ofsurreal numbers, and the so-called "natural" arithmetical operations for surreal numbers are an alternative way to combine ordinals arithmetically. They retain commutativity at the expense of continuity.

Interpreted asnimbers, a game-theoretic variant of numbers, ordinals can also be combined via nimber arithmetic operations. These operations are commutative but the restriction to natural numbers is generally not the same as ordinary addition of natural numbers.

Ordinals and cardinals

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Initial ordinal of a cardinal

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Each ordinal associates with onecardinal, its cardinality. If there is a bijection between two ordinals (e.g.ω = 1 + ω andω + 1 > ω), 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 theinitial 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. Theaxiom of choice 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 theVon Neumann cardinal assignment 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 (seeScott's trick).

One issue with Scott's trick is that it identifies the cardinal number0{\displaystyle 0} with⁠{∅}{\displaystyle \{\emptyset \}}⁠, which in some formulations is the ordinal number⁠1{\displaystyle 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 written⁠ωα{\displaystyle \omega _{\alpha }}⁠, it is always a limit ordinal. Its cardinality is written⁠ℵα{\displaystyle \aleph _{\alpha }}⁠. For example, the cardinality of ω0 = ω is⁠ℵ0{\displaystyle \aleph _{0}}⁠, which is also the cardinality of ω2 or ε0 (all are countable ordinals). So ω can be identified with⁠ℵ0{\displaystyle \aleph _{0}}⁠, except that the notationℵ0{\displaystyle \aleph _{0}} is used when writing cardinals, and ω when writing ordinals (this is important since, for example,ℵ02{\displaystyle \aleph _{0}^{2}} =ℵ0{\displaystyle \aleph _{0}} whereasω2>ω{\displaystyle \omega ^{2}>\omega }). Also,ω1{\displaystyle \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, andω1{\displaystyle \omega _{1}} is the order type of that set),ω2{\displaystyle \omega _{2}} is the smallest ordinal whose cardinality is greater than⁠ℵ1{\displaystyle \aleph _{1}}⁠, and so on, andωω{\displaystyle \omega _{\omega }} is the limit of theωn{\displaystyle \omega _{n}} for natural numbersn (any limit of cardinals is a cardinal, so this limit is indeed the first cardinal after all theωn{\displaystyle \omega _{n}}).

Cofinality

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Thecofinality of an ordinalα{\displaystyle \alpha } is the smallest ordinalδ{\displaystyle \delta } that is the order type of acofinal subset of⁠α{\displaystyle \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 aδ{\displaystyle \delta }-indexed strictly increasing sequence with limit⁠α{\displaystyle \alpha }⁠. For example, the cofinality of ω2 is ω, because the sequence ω·m (wherem 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 doesωω{\displaystyle \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 least⁠ω{\displaystyle \omega }⁠.

An ordinal that is equal to its cofinality is calledregular 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 holds, thenωα+1{\displaystyle \omega _{\alpha +1}} is regular for eachα. In this case, the ordinals 0, 1,⁠ω{\displaystyle \omega }⁠,⁠ω1{\displaystyle \omega _{1}}⁠, andω2{\displaystyle \omega _{2}} are regular, whereas 2, 3,⁠ωω{\displaystyle \omega _{\omega }}⁠, and ωω·2 are initial ordinals that are not regular.

The cofinality of any ordinalα is a regular ordinal, i.e. the cofinality of the cofinality ofα is the same as the cofinality ofα. So the cofinality operation isidempotent.

Closed unbounded sets and classes

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The concepts of closed and unbounded sets are typically formulated for subsets of aregular cardinalκ{\displaystyle \kappa } that is uncountable. A subsetC⊆κ{\displaystyle C\subseteq \kappa } is said to beunbounded (or cofinal) inκ{\displaystyle \kappa } if for every ordinalα<κ{\displaystyle \alpha <\kappa }, there exists someβ∈C{\displaystyle \beta \in C} such thatα<β{\displaystyle \alpha <\beta }. To define the property of being closed, one first defines a limit point: a non-zero ordinalδ<κ{\displaystyle \delta <\kappa } is a limit point ofC{\displaystyle C} ifsup(C∩δ)=δ{\displaystyle \sup(C\cap \delta )=\delta }. The setC{\displaystyle C} isclosed inκ{\displaystyle \kappa } if it contains all of its limit points belowκ{\displaystyle \kappa }. A set that is both closed and unbounded is commonly referred to as aclub set.

Examples of club sets are fundamental to set theory. The set of alllimit ordinals less thanκ{\displaystyle \kappa } is a club set, as there is always a limit ordinal greater than any given ordinal belowκ{\displaystyle \kappa }, and a limit of limit ordinals is itself a limit ordinal. Ifκ{\displaystyle \kappa } is a limit cardinal, the set of allcardinals belowκ{\displaystyle \kappa } is unbounded, and its set of limit points—thelimit cardinals—forms a closed unbounded set. Furthermore, ifκ{\displaystyle \kappa } is astrong limit cardinal (such as aninaccessible cardinal), the set of strong limit cardinals belowκ{\displaystyle \kappa } is also a club set. Another significant example arises fromnormal functions (functionsf:κ→κ{\displaystyle f:\kappa \to \kappa } that are strictly increasing and continuous); the range of any normal function is a closed unbounded subset ofκ{\displaystyle \kappa }.

Club sets possess structural properties that allow them to generate afilter. Becauseκ{\displaystyle \kappa } is regular and uncountable, the intersection of any two club sets is also a club set. More generally, the intersection of fewer thanκ{\displaystyle \kappa } club sets is a club set. Consequently, the collection of all subsets ofκ{\displaystyle \kappa } that contain a club set forms aκ{\displaystyle \kappa }-complete non-principal filter, known as theclosed unbounded filter (orclub filter).

A subsetS⊆κ{\displaystyle S\subseteq \kappa } is termedstationary if it has a non-empty intersection with every closed unbounded set inκ{\displaystyle \kappa }. Intuitively, stationary sets are "large" enough that they cannot be avoided by any club set. Using the notation of filters, a set is stationary if and only if it does not belong to the dual ideal of the club filter (the ideal of non-stationary sets). While every club set is stationary, not every stationary set is a club; for instance, a stationary setS{\displaystyle S} may fail to be closed. Furthermore, while the intersection of a stationary set and a club set is stationary, the intersection of two stationary sets may be empty.

The distinction between club sets and stationary sets is central to the definitions of certainlarge cardinals. Ifκ{\displaystyle \kappa } is the smallestinaccessible cardinal, the set of singular strong limit cardinals belowκ{\displaystyle \kappa } forms a closed unbounded set. Because this club set contains no regular cardinals, the set of regular cardinals below the first inaccessible is not stationary. This remains true ifκ{\displaystyle \kappa } is then{\displaystyle n}-th inaccessible cardinal for somen<κ{\displaystyle n<\kappa }; the regular cardinals below it will not form a stationary set. A cardinalκ{\displaystyle \kappa } is defined as aMahlo cardinal precisely when the set of regular cardinals below it is stationary. By relaxing the condition on the limit cardinals, one defines a cardinal asweakly Mahlo if it isweakly inaccessible and the set of regular cardinals below it is stationary.

The closed unbounded filter is not anultrafilter under the standard Zermelo–Fraenkel set theory with the Axiom of Choice (ZFC). This is because one can find two disjoint stationary sets, which precludes the filter from deciding membership for every subset. For any regular cardinalκ>ω1{\displaystyle \kappa >\omega _{1}}, the set of ordinals withcofinalityω{\displaystyle \omega } and the set of ordinals with cofinalityω1{\displaystyle \omega _{1}} are disjoint stationary subsets ofκ{\displaystyle \kappa }. In the specific case ofκ=ω1{\displaystyle \kappa =\omega _{1}}, the non-existence of an ultrafilter relies on theAxiom of Choice. Under ZFC, the set of limit ordinals inω1{\displaystyle \omega _{1}} can be partitioned intoω1{\displaystyle \omega _{1}} disjoint stationary sets (a result related toFodor's lemma). However, in models of set theory without the Axiom of Choice, such as those satisfying theAxiom of determinacy, the club filter onω1{\displaystyle \omega _{1}} can be an ultrafilter, a property connected toω1{\displaystyle \omega _{1}} being ameasurable cardinal in those contexts.

These definitions generalize to proper classes of ordinals. A classC{\displaystyle C} of ordinals is unbounded if it contains arbitrarily large ordinals, and closed if the limit of any sequence of ordinals inC{\displaystyle C} is also inC{\displaystyle C}. This topological definition is equivalent to assuming the indexing class-function ofC{\displaystyle C} is continuous. Notable examples of closed unbounded classes include the class of all infinite cardinals, the class oflimit cardinals, and the class offixed points of theℵ{\displaystyle \aleph }-function. In contrast, the class ofregular cardinals is unbounded but not closed. A class is stationary if it intersects every closed unbounded class.

Some "large" countable ordinals

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Further information:Large countable ordinal

As mentioned above (seeCantor normal form), the ordinal ε0 is the smallest satisfying the equation⁠ωα=α{\displaystyle \omega ^{\alpha }=\alpha }⁠, so it is the limit of the sequence 0, 1,⁠ω{\displaystyle \omega }⁠,⁠ωω{\displaystyle \omega ^{\omega }}⁠,⁠ωωω{\displaystyle \omega ^{\omega ^{\omega }}}⁠, etc. Many ordinals can be defined in such a manner as fixed points of certain ordinal functions (theι{\displaystyle \iota }-th ordinal such thatωα=α{\displaystyle \omega ^{\alpha }=\alpha } is called⁠ει{\displaystyle \varepsilon _{\iota }}⁠, then one could go on trying to find theι{\displaystyle \iota }-th ordinal such that⁠εα=α{\displaystyle \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 theChurch–Kleene ordinal,ω1CK{\displaystyle \omega _{1}^{\mathrm {CK} }} (despite theω1{\displaystyle \omega _{1}} in the name, this ordinal is countable), which is the smallest ordinal that cannot in any way be represented by acomputable function (this can be made rigorous, of course). Considerably large ordinals can be defined below⁠ω1CK{\displaystyle \omega _{1}^{\mathrm {CK} }}⁠, however, which measure the "proof-theoretic strength" of certainformal systems (for example,ε0{\displaystyle \varepsilon _{0}} measures the strength ofPeano arithmetic). Large countable ordinals such as countableadmissible ordinals can also be defined above the Church-Kleene ordinal, which are of interest in various parts of logic.[citation needed]

Topology and ordinals

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Further information:Order topology

Any ordinal number can be made into atopological space by endowing it with theorder topology. This topology isdiscrete if and only if it is less than or equal to ω. In contrast, a subset of ω + 1 is open in the order topology if and only if either it iscofinite or it does not contain ω as an element.

See theTopology and ordinals section of the "Order topology" article.

History

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The transfinite ordinal numbers, which first appeared in 1883,[17] originated in Cantor's work withderived sets. IfP is a set of real numbers, the derived setP′ is the set oflimit points ofP. In 1872, Cantor generated the setsP(n) by applying the derived set operationn times toP. In 1880, he pointed out that these sets form the sequenceP'⊇ ··· ⊇P(n) ⊇P(n + 1) ⊇ ···, and he continued the derivation process by definingP(∞) 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) ⊇ ···.[18] The superscripts containing ∞ are just indices defined by the derivation process.[19]

Cantor used these sets in the theorems:

  1. IfP(α) = ∅ for some indexα, thenP′ is countable;
  2. Conversely, ifP′ is countable, then there is an indexα such thatP(α) = ∅.

These theorems are proved by partitioningP′ intopairwise disjoint sets:P′ = (P′\P(2)) ∪ (P(2) \P(3)) ∪ ··· ∪ (P(∞) \P(∞ + 1)) ∪ ··· ∪P(α). Forβ <α: sinceP(β + 1) contains the limit points ofP(β), the setsP(β) \P(β + 1) have no limit points. Hence, they arediscrete sets, so they are countable. Proof of first theorem: IfP(α) = ∅ for some indexα, thenP′ is the countable union of countable sets. Therefore,P′ is countable.[20]

The second theorem requires proving the existence of anα such thatP(α) = ∅. To prove this, Cantor considered the set of allα having countably many predecessors. To define this set, he defined the transfinite ordinal numbers and transformed the infinite indices into ordinals by replacing ∞ withω, the first transfinite ordinal number. Cantor called the set of finite ordinals the firstnumber class. The second number class is the set of ordinals whose predecessors form a countably infinite set. The set of allα 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.[21]

Cantor's second theorem becomes: IfP′ is countable, then there is a countable ordinalα such thatP(α) = ∅. Its proof usesproof by contradiction. LetP′ 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 countableβ. Since there are uncountably many of these pairwise disjoint sets, their union is uncountable. This union is a subset ofP′, soP' is uncountable.
  • Case 2:P(β) \P(β + 1) is empty for some countableβ. SinceP(β + 1) ⊆P(β), this impliesP(β + 1) =P(β). Thus,P(β) is aperfect set, so it is uncountable.[22] SinceP(β) ⊆P′, the setP′ is uncountable.

In both cases,P′ is uncountable, which contradictsP′ being countable. Therefore, there is a countable ordinalα such thatP(α) = ∅. Cantor's work with derived sets and ordinal numbers led to theCantor-Bendixson theorem.[23]

Using successors, limits, and cardinality, Cantor generated an unbounded sequence of ordinal numbers and number classes.[24] The(α + 1)-th number class is the set of ordinals whose predecessors form a set of the same cardinality as theα-th number class. The cardinality of the(α + 1)-th number class is the cardinality immediately following that of theα-th number class.[25] For a limit ordinalα, theα-th number class is the union of theβ-th number classes forβ <α.[26] Its cardinality is the limit of the cardinalities of these number classes.

Ifn is finite, then-th number class has cardinality⁠ℵn−1{\displaystyle \aleph _{n-1}}⁠. Ifα ≥ω, theα-th number class has cardinality⁠ℵα{\displaystyle \aleph _{\alpha }}⁠.[c] Therefore, the cardinalities of the number classes correspond one-to-one with thealeph numbers. Also, theα-th number class consists of ordinals different from those in the preceding number classes if and only ifα is a non-limit ordinal. Therefore, the non-limit number classes partition the ordinals into pairwise disjoint sets.

See also

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Notes

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  1. ^Ordinal numbers defined this way are at least two types higher than elements of the original well-ordered set, so type-raising operators may be required to "label" the original elements with them. While inconvenient, the type-raising requirement helps these systemsavoid the Burali-Forti paradox.
  2. ^Assuming theaxiom of regularity, "strictly well-ordered" can be weakened to "strictly totally ordered", as regularity prevents infinite descending chains of⁠∈{\displaystyle \in }⁠.
  3. ^The first number class has cardinality⁠ℵ0{\displaystyle \aleph _{0}}⁠.Mathematical induction proves that then-th number class has cardinality⁠ℵn−1{\displaystyle \aleph _{n-1}}⁠. Since theω-th number class is the union of then-th number classes, its cardinality is⁠ℵω{\displaystyle \aleph _{\omega }}⁠, the limit of the⁠ℵn−1{\displaystyle \aleph _{n-1}}⁠.Transfinite induction proves that ifα ≥ω, theα-th number class has cardinality⁠ℵα{\displaystyle \aleph _{\alpha }}⁠.

Citations

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  1. ^abConway & Guy 2012.
  2. ^Thorough introductions are given byLevy 1979 andJech 2003.
  3. ^Hallett 1979, footnote on p. 12.
  4. ^Cantor 1897.
  5. ^abJech 2003, Theorem 2.14.
  6. ^von Neumann 1923.
  7. ^Just & Weese 1996, p. 156.
  8. ^abJech 2003, p. 19.
  9. ^abcdJech 2003, p. 20.
  10. ^Jech 2003, Theorem 2.12.
  11. ^Holz, Steffens & Weitz 1999.
  12. ^Jech 2003, p. 23.
  13. ^Just & Weese 1996, p. 36.
  14. ^Rosenstein 1982.
  15. ^Jech 2003, Lemma 5.5.
  16. ^Evans & Hamkins 2013.
  17. ^Cantor 1883. English translation:Ewald 1996, pp. 881–920
  18. ^Ferreirós 1995, pp. 34–35;Ferreirós 2007, pp. 159, 204–5
  19. ^Ferreirós 2007, p. 269
  20. ^Ferreirós 1995, pp. 35–36;Ferreirós 2007, p. 207
  21. ^Ferreirós 1995, pp. 36–37;Ferreirós 2007, p. 271
  22. ^Dauben 1979, p. 111
  23. ^Ferreirós 2007, pp. 207–8
  24. ^Dauben 1979, pp. 97–98
  25. ^Hallett 1986, pp. 61–62
  26. ^Tait 1997, p. 5 footnote

References

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External links

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