Well-ordered sets

In a well-ordered set, every non-empty subset contains a distinct smallest element. Given the axiom of dependent choice, this is equivalent to saying that the set is totally ordered and there is no infinite decreasing sequence (the latter being easier to visualize). In practice, the importance of well-ordering is justified by the possibility of applying transfinite induction, which says, essentially, that any property that passes on from the predecessors of an element to that element itself must be true of all elements (of the given well-ordered set). If the states of a computation (computer program or game) can be well-ordered—in such a way that each step is followed by a "lower" step—then the computation will terminate.

It is inappropriate to distinguish between two well-ordered sets if they only differ in the "labeling of their elements", or more formally: if the elements of the first set can be paired off with the elements of the second set such that if one element is smaller than another in the first set, then the partner of the first element is smaller than the partner of the second element in the second set, and vice versa. Such a one-to-one correspondence is called an order isomorphism, and the two well-ordered sets are said to be order-isomorphic or similar (with the understanding that this is an equivalence relation).

Formally, if a partial order ≤ is defined on the set S, and a partial order ≤' is defined on the set S' , then the posets (S,≤) and (S' ,≤') are order isomorphic if there is a bijection f that preserves the ordering. That is, f(a) ≤' f(b) if and only if a ≤ b. Provided there exists an order isomorphism between two well-ordered sets, the order isomorphism is unique: this makes it quite justifiable to consider the two sets as essentially identical, and to seek a "canonical" representative of the isomorphism type (class). This is exactly what the ordinals provide, and it also provides a canonical labeling of the elements of any well-ordered set. Every well-ordered set (S,<) is order-isomorphic to the set of ordinals less than one specific ordinal number under their natural ordering. This canonical set is the order type of (S,<).

Essentially, an ordinal is intended to be defined as an isomorphism class of well-ordered sets: that is, as an equivalence class for the equivalence relation of "being order-isomorphic". There is a technical difficulty involved, however, in the fact that the equivalence class is too large to be a set in the usual Zermelo–Fraenkel (ZF) formalization of set theory. But this is not a serious difficulty. The ordinal can be said to be the order type of any set in the class.

Definition of an ordinal as an equivalence class

The original definition of ordinal numbers, found for example in the Principia Mathematica, defines the order type of a well-ordering as the set of all well-orderings similar (order-isomorphic) to that well-ordering: in other words, an ordinal number is genuinely an equivalence class of well-ordered sets. This definition must be abandoned in ZF and related systems of axiomatic set theory because these equivalence classes are too large to form a set. However, this definition still can be used in type theory and in Quine's axiomatic set theory New Foundations and related systems (where it affords a rather surprising alternative solution to the Burali-Forti paradox of the largest ordinal).

Von Neumann definition of ordinals

Rather than defining an ordinal as an equivalence class of well-ordered sets, it will be defined as a particular well-ordered set that (canonically) represents the class. Thus, an ordinal number will be a well-ordered set; and every well-ordered set will be order-isomorphic to exactly one ordinal number.

For each well-ordered set T, ${\displaystyle a\mapsto T_{<a}}$ defines an order isomorphism between T and the set of all subsets of T having the form ${\displaystyle T_{<a}:=\{x\in T\mid x<a\}}$ ordered by inclusion. This motivates the standard definition, suggested by 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". In symbols, ${\displaystyle \lambda =[0,\lambda )}$.^[7][8] Formally:

A set S is an ordinal if and only if S is strictly well-ordered with respect to set membership and every element of S is also a subset of S.

The natural numbers are thus ordinals by this definition. For instance, 2 is an element of 4 = {0, 1, 2, 3}, and 2 is equal to {0, 1} and so it is a subset of {0, 1, 2, 3}.

It can be shown by 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 between them.

Furthermore, the elements of every ordinal are ordinals themselves. Given two ordinals S and T, S is an element of T if and only if S is a proper subset of T. Moreover, either S is an element of T, or T is an element of S, or they are equal. So every set of ordinals is totally ordered. Further, every set of ordinals is well-ordered. This generalizes the fact that every set of natural numbers is well-ordered.

Consequently, every ordinal S is a set having as elements precisely the ordinals smaller than S. For example, every set of ordinals has a supremum, the ordinal obtained by taking the union of all the ordinals in the set. This union exists regardless of the set's size, by the axiom of union.

The class of all ordinals is not a set. If it were a set, one could show that it was an ordinal and thus a member of itself, which would contradict its strict ordering by membership. This is the Burali-Forti paradox. The class of all ordinals is variously called "Ord", "ON", or "∞".

An ordinal is finite if and only if the opposite order is also well-ordered, which is the case if and only if each of its non-empty subsets has a maximum.

Other definitions

There are other modern formulations of the definition of ordinal. For example, assuming the axiom of regularity, the following are equivalent for a set x:

• x is a (von Neumann) ordinal,
• x is a transitive set, and set membership is trichotomous on x,
• x is a transitive set totally ordered by set inclusion,
• x is a transitive set of transitive sets.

These definitions cannot be used in non-well-founded set theories. In set theories with urelements, one has to further make sure that the definition excludes urelements from appearing in ordinals.

Transfinite recursion

Transfinite induction can be used not only to prove things, but also to define them. Such a definition is normally said to be by transfinite recursion – the proof that the result is well-defined uses transfinite induction. Let F denote a (class) function F to be defined on the ordinals. The idea now is that, in defining F(α) for an unspecified ordinal α, one may assume that F(β) is already defined for all β < α and thus give a formula for F(α) in terms of these F(β). It then follows by transfinite induction that there is one and only one function satisfying the recursion formula up to and including α.

Here is an example of definition by transfinite recursion on the ordinals (more will be given later): define function F by letting F(α) be the smallest ordinal not in the set {F(β) | β < α}, that is, the set consisting of all F(β) for β < α. This definition assumes the F(β) known in the very process of defining F; this apparent vicious circle is exactly what definition by transfinite recursion permits. In fact, F(0) makes sense since there is no ordinal β < 0, and the set {F(β) | β < 0} is empty. So F(0) is equal to 0 (the smallest ordinal of all). Now that F(0) is known, the definition applied to F(1) makes sense (it is the smallest ordinal not in the singleton set {F(0)} = {0}), and so on (the and so on is exactly transfinite induction). It 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.

Successor and limit ordinals

Any nonzero ordinal has the minimum element, zero. It may or may not have a maximum element. For example, 42 has maximum 41 and ω+6 has maximum ω+5. On the other hand, ω does not have a maximum since there is no largest natural number. If an ordinal has a maximum α, then it is the next ordinal after α, and it is called a successor ordinal, namely the successor of α, written α+1. In the von Neumann definition of ordinals, the successor of α is ${\displaystyle \alpha \cup \{\alpha \}}$ since its elements are those of α and α itself.^[7]

A nonzero ordinal that is not a successor is called a limit ordinal. One justification for this term is that a limit ordinal is the limit in a topological sense of all smaller ordinals (under the order topology).

When ${\displaystyle \langle \alpha _{\iota }|\iota <\gamma \rangle }$ is an ordinal-indexed sequence, indexed by a limit ${\displaystyle \gamma }$ and the sequence is increasing, i.e. ${\displaystyle \alpha _{\iota }<\alpha _{\rho }}$ whenever ${\displaystyle \iota <\rho ,}$ its limit is defined as the least upper bound of the set ${\displaystyle \{\alpha _{\iota }|\iota <\gamma \},}$ that is, the smallest ordinal (it always exists) greater than any term of the sequence. In this sense, a limit ordinal is the limit of all smaller ordinals (indexed by itself). Put more directly, it is the supremum of the set of smaller ordinals.

Another way of defining a limit ordinal is to say that α is a limit ordinal if and only if:

There is an ordinal less than α and whenever ζ is an ordinal less than α, then there exists an ordinal ξ such that ζ < ξ < α.

So in the following sequence:

0, 1, 2, ..., ω, ω+1

ω is a limit ordinal because for any smaller ordinal (in this example, a natural number) there is another ordinal (natural number) larger than it, but still less than ω.

Thus, every ordinal is either zero, or a successor (of a well-defined predecessor), or a limit. This distinction is important, because many definitions by transfinite recursion rely upon it. Very often, when defining a function F by transfinite recursion on all ordinals, one defines F(0), and F(α+1) assuming F(α) is defined, and then, for limit ordinals δ one defines F(δ) as the limit of the F(β) for all β<δ (either in the sense of ordinal limits, as previously explained, or for some other notion of limit if F does not take ordinal values). Thus, the interesting step in the definition is the successor step, not the limit ordinals. Such functions (especially for F nondecreasing and taking ordinal values) are called continuous. Ordinal addition, multiplication and exponentiation are continuous as functions of their second argument (but can be defined non-recursively).

Indexing classes of ordinals

Any well-ordered set is similar (order-isomorphic) to a unique ordinal number ${\displaystyle \alpha }$; in other words, its elements can be indexed in increasing fashion by the ordinals less than ${\displaystyle \alpha }$. This applies, in particular, to any set of ordinals: any set of ordinals is naturally indexed by the ordinals less than some ${\displaystyle \alpha }$. The same holds, with a slight modification, for classes of ordinals (a collection of ordinals, possibly too large to form a set, defined by some property): any class of ordinals can be indexed by ordinals (and, when the class is unbounded in the class of all ordinals, this puts it in class-bijection with the class of all ordinals). So the ${\displaystyle \gamma }$-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: the ${\displaystyle \gamma }$-th element of the class is defined (provided it has already been defined for all ${\displaystyle \beta <\gamma }$), as the smallest element greater than the ${\displaystyle \beta }$-th element for all ${\displaystyle \beta <\gamma }$.

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

Closed unbounded sets and classes

A class ${\displaystyle C}$ of ordinals is said to be unbounded, or cofinal, when given any ordinal ${\displaystyle \alpha }$, there is a ${\displaystyle \beta }$ in ${\displaystyle C}$ such that ${\displaystyle \alpha <\beta }$ (then the class must be a proper class, i.e., it cannot be a set). It is said to be closed when the limit of a sequence of ordinals in the class is again in the class: or, equivalently, when the indexing (class-)function ${\displaystyle F}$ is continuous in the sense that, for ${\displaystyle \delta }$ a limit ordinal, ${\displaystyle F(\delta )}$ (the ${\displaystyle \delta }$-th ordinal in the class) is the limit of all ${\displaystyle F(\gamma )}$ for ${\displaystyle \gamma <\delta }$; this is also the same as being closed, in the topological sense, for the 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 on that ordinal, this is again equivalent).

Of particular importance are those classes of ordinals that are closed and unbounded, sometimes called clubs. For example, the class of all limit ordinals is closed and unbounded: this translates the fact that there is always a limit ordinal greater than a given ordinal, and that a limit of limit ordinals is a limit ordinal (a fortunate fact if the terminology is to make any sense at all!). The class of additively indecomposable ordinals, or the class of ${\displaystyle \varepsilon _{\cdot }}$ ordinals, or the class of cardinals, are all closed unbounded; the set of regular cardinals, however, is unbounded but not closed, and any finite set of ordinals is closed but not unbounded.

A class is stationary if it has a nonempty intersection with every closed unbounded class. All superclasses of closed unbounded classes are stationary, and stationary classes are unbounded, but there are stationary classes that are not closed and stationary classes that have no closed unbounded subclass (such as the class of all limit ordinals with countable cofinality). Since the intersection of two closed unbounded classes is closed and unbounded, the intersection of a stationary class and a closed unbounded class is stationary. But the intersection of two stationary classes may be empty, e.g. the class of ordinals with cofinality ω with the class of ordinals with uncountable cofinality.

Rather than formulating these definitions for (proper) classes of ordinals, one can formulate them for sets of ordinals below a given ordinal ${\displaystyle \alpha }$: A subset of a limit ordinal ${\displaystyle \alpha }$ is said to be unbounded (or cofinal) under ${\displaystyle \alpha }$ provided any ordinal less than ${\displaystyle \alpha }$ is less than some ordinal in the set. More generally, one can call a subset of any ordinal ${\displaystyle \alpha }$ cofinal in ${\displaystyle \alpha }$ provided every ordinal less than ${\displaystyle \alpha }$ is less than or equal to some ordinal in the set. The subset is said to be closed under ${\displaystyle \alpha }$ provided it is closed for the order topology in ${\displaystyle \alpha }$, i.e. a limit of ordinals in the set is either in the set or equal to ${\displaystyle \alpha }$ itself.

Initial ordinal of a cardinal

Each ordinal associates with one cardinal, its cardinality. If there is a bijection between two ordinals (e.g. ω = 1 + ω and ω + 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 the initial ordinal of that cardinal. Every finite ordinal (natural number) is initial, and no other ordinal associates with its cardinal. But most infinite ordinals are not initial, as many infinite ordinals associate with the same cardinal. The axiom of choice is equivalent to the statement that every set can be well-ordered, i.e. that every cardinal has an initial ordinal. In theories with the axiom of choice, the cardinal number of any set has an initial ordinal, and one may employ the Von Neumann cardinal assignment as the cardinal's representation. (However, we must then be careful to distinguish between cardinal arithmetic and ordinal arithmetic.) In set theories without the axiom of choice, a cardinal may be represented by the set of sets with that cardinality having minimal rank (see Scott's trick).

One issue with Scott's trick is that it identifies the cardinal number ${\displaystyle 0}$ with ${\displaystyle \{\emptyset \}}$, which in some formulations is the ordinal number ${\displaystyle 1}$. 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 written ${\displaystyle \omega _{\alpha }}$, it is always a limit ordinal. Its cardinality is written ${\displaystyle \aleph _{\alpha }}$. For example, the cardinality of ω₀ = ω is ${\displaystyle \aleph _{0}}$, which is also the cardinality of ω² or ε₀ (all are countable ordinals). So ω can be identified with ${\displaystyle \aleph _{0}}$, except that the notation ${\displaystyle \aleph _{0}}$ is used when writing cardinals, and ω when writing ordinals (this is important since, for example, ${\displaystyle \aleph _{0}^{2}}$ = ${\displaystyle \aleph _{0}}$ whereas ${\displaystyle \omega ^{2}>\omega }$). Also, ${\displaystyle \omega _{1}}$ 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, and ${\displaystyle \omega _{1}}$ is the order type of that set), ${\displaystyle \omega _{2}}$ is the smallest ordinal whose cardinality is greater than ${\displaystyle \aleph _{1}}$, and so on, and ${\displaystyle \omega _{\omega }}$ is the limit of the ${\displaystyle \omega _{n}}$ for natural numbers n (any limit of cardinals is a cardinal, so this limit is indeed the first cardinal after all the ${\displaystyle \omega _{n}}$).

Cofinality

The cofinality of an ordinal ${\displaystyle \alpha }$ is the smallest ordinal ${\displaystyle \delta }$ that is the order type of a cofinal subset of ${\displaystyle \alpha }$. 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 a ${\displaystyle \delta }$-indexed strictly increasing sequence with limit ${\displaystyle \alpha }$. For example, the cofinality of ω² is ω, because the sequence ω·m (where m ranges over the natural numbers) tends to ω²; but, more generally, any countable limit ordinal has cofinality ω. An uncountable limit ordinal may have either cofinality ω as does ${\displaystyle \omega _{\omega }}$ 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 least ${\displaystyle \omega }$.

An ordinal that is equal to its cofinality is called regular and it is always an initial ordinal. Any limit of regular ordinals is a limit of initial ordinals and thus is also initial even if it is not regular, which it usually is not. If the Axiom of Choice, then ${\displaystyle \omega _{\alpha +1}}$ is regular for each α. In this case, the ordinals 0, 1, ${\displaystyle \omega }$, ${\displaystyle \omega _{1}}$, and ${\displaystyle \omega _{2}}$ are regular, whereas 2, 3, ${\displaystyle \omega _{\omega }}$, and ω_ω·2 are initial ordinals that are not regular.

The cofinality of any ordinal α is a regular ordinal, i.e. the cofinality of the cofinality of α is the same as the cofinality of α. So the cofinality operation is idempotent.