Ordinal number


In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals aimed to extend enumeration to infinite sets. Usually Greek letters are used for ordinal number variables to help distinguish them from natural number variables.
A finite set can be enumerated by successively labeling each element with the least natural number that has not been previously used. To extend this process to various infinite sets, ordinal numbers are defined more generally as a linearly ordered class of numbers that include the natural numbers and have the property that every non-empty collection of ordinals has a least or "smallest" element. This more general definition allows us to define an ordinal number to be the least element that is greater than every natural number, along with ordinal numbers,, etc., which are even greater than.
The Zermelo–Fraenkel set theory asserts that, for any 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?" 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 a well-order. The axiom of choice implies that every set can be well-ordered. Given two well-ordered sets, one is isomorphic to an initial segment of the other, and the isomorphism is unique. This allows a unique ordinal to be associated with each well-ordered set, known as its order type.
Ordinal numbers are distinct from cardinal numbers, which measure the size of sets. Although the distinction between ordinals and cardinals is not this apparent on finite sets, they are very different in the infinite case, where different infinite ordinals can correspond to sets having the same cardinal. Like other kinds of numbers, ordinals can be added, multiplied, and exponentiated, although none of these operations are commutative.
Ordinals were introduced by Georg Cantor in 1883 to accommodate infinite sequences and classify derived sets, which he had previously introduced in 1872 while studying the uniqueness of trigonometric series.

Motivation

A natural number can be used for two purposes: to describe the size of a set, or to describe the position of an element in a sequence. When generalized to infinite sets, the notion of size leads to cardinal 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 of mathematical 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 for implies its truth for, then it holds for all natural numbers. This process corresponds to the first infinite ordinal,.
Mathematical contexts often require iterating beyond a single infinite limit. The ordinal exemplifies the concept of nested 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 than, one performs an "inner" induction, establishes the limit at, and then proceeds to the next sequence. This structure parallels a nested loop in computer programming. Ordinals allow the definition of processes of arbitrary complexity, such as or .
The validity of inductive counting rests on the property of well-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 a well-order. The existence of a "least" or minimal element in every non-empty subset of a well-ordered set grounds the principle of transfinite 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. A fundamental theorem in set theory establishes that any two well-ordered sets are comparable: given two well-orders, either they are isomorphic, or one is isomorphic to a proper initial 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

Well-ordering

The construction procedure of transfinite sequences implies that the ordinals used to label their elements are well-ordered. This means that:
Intuitively, the least ordinal in is the "next" ordinal introduced after all ordinals strictly less than every ordinal in have been used. In contrast, need not have a greatest element, for example when is the set of all natural numbers.
Being well-ordered is stronger than merely being linearly ordered. For example, the real numbers are naturally linearly ordered, but any open interval has no least element.
The importance of well-ordering is that it enables transfinite induction. If a statement about an ordinal is not universally true for all, i.e., if it has any counterexamples, then it must have a least counterexample. Conversely, if it can be proven that is true whenever is true for all, then it cannot have any least counterexample, and thus it cannot have any counterexample at all, i.e., it is indeed universally true.

Well-ordered sets

In the usual Zermelo–Fraenkel formalization of set theory, a well-ordered set is a totally ordered set such that every non-empty subset has a least element. Here “set” means a collection that is itself an object of ZF, as opposed to a proper 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: if and only if. The labels used in such an enumeration form an initial 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 the order type of the well-ordering. Thanks to 0-based indexing, the order type coincides with the cardinality for finite sets, including the empty set. However, for infinite sets, different well-order relations can have different order types.

Definition of an ordinal as an equivalence class

Order types can be defined without a preexisting notion of ordinals, by working directly with order isomorphisms between general well-ordered sets, just as cardinality can be defined through bijections between general sets. As with bijections, being order-isomorphic is an equivalence relation on well-ordered sets; its equivalence classes correspond to order types.
In the Principia Mathematica approach, the order type of a well-ordered set is identified with its isomorphism class, i.e., the set of all well-ordered sets order-isomorphic to. Since elements of are allowed to be anything, 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 in type theory and in Quine's axiomatic set theory New Foundations and related systems.
In ZF and related systems of axiomatic 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, the ideal representation is one where the abstract order relation is translated directly into the membership relation.

Von Neumann definition of ordinals

The von Neumann representation provides this canonical form. It relies on the observation that any well-founded relation satisfying certain properties can be mapped to a specific set where the relation becomes set membership. This mapping is known as the Mostowski collapse lemma.
When applied to a well-ordering, the Mostowski collapse yields a specific set where the order relation is literally true if and only if. The resulting sets have the property that they are transitive: every element of is also a subset of . In this representation, each ordinal is identified with the set of all preceding ordinals.
Thus the finite von Neumann ordinals are defined recursively as,,, etc. The first infinite ordinal is represented by the set of all finite ordinals, i.e., the set of von Neumann natural numbers. Then, and so on.
Informally, one may define an ordinal recursively as a downward closed set of ordinals. Such a recursive definition is usually justified with the transitive closure. However, the von Neumann ordinals are already transitive sets, allowing them to be formally defined by a concise statement:
The set is usually defined as the smallest inductive set. The "smallest" constraint guarantees that each element of is either zero or the successor of another element of, which allows induction to demonstrate that indeed satisfies the formal definition of ordinals stated above.

Basic properties

Defining the strict order relation as the membership relation restricted to the class of all ordinals, the recursive characterization is reduced to the following statement:
  • If and is an ordinal, then is an ordinal: transitivity of as a set is found by finding that all the relevant ordinals are elements of and then using the transitivity of the relation within ; well-orderedness of follows straightforwardly from well-orderedness of.
The non-strict order relation has an alternative characterization: if and only if for ordinals. The "if" direction follows from transitivity of, and the "only if" direction follows from:
  • If are both ordinals and , then : let. is transitive so.
This implies that is a partial order. It is in fact a total order, and a well-order:
  • If and are both ordinals, then or : is an ordinal, so or, or else, contradicting irreflexivity.
  • If is a nonempty set of ordinals, then must be in by similar logic.
So all ordinal numbers form a well-ordered class, and thus any nonempty set of ordinals equipped with is a well-ordered set.
Specific ordinals can be constructed explicitly with the following principles:
  • is an ordinal.
  • For any ordinal, is an ordinal and.
  • If A is a set of ordinals, then is an ordinal.
The explicit forms of and imply that there exists an ordinal greater than any set of ordinals. In other words,
  • The class of all ordinals is not a set; otherwise would be an ordinal not in.