In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, nth, etc.) 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 (set or proper class) of ordinals has a least 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 number ω {\displaystyle \omega } (omega) to be the least element that is greater than every natural number, along with ordinal numbers ω + 1 {\displaystyle \omega +1} , ω + 2 {\displaystyle \omega +2} , etc., which are even greater than ω {\displaystyle \omega } . 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?" (the Burali-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 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 (one can go from one to the other just by counting labels), they are very different in the infinite case, where different infinite ordinals can correspond to sets having the same cardinal. Like other kinds of numbers, ordinals can be added, multiplied, and exponentiated, although none of these operations are commutative. Ordinals were introduced by Georg Cantor 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 (which, in this context, includes the number 0) 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 n {\displaystyle n} implies its truth for n + 1 {\displaystyle n+1} , then it holds for all natural numbers. This process corresponds to the first infinite ordinal, ω {\displaystyle \omega } .
Mathematical contexts often require iterating beyond a single infinite limit. The ordinal ω 2 {\displaystyle \omega ^{2}} (represented in the figure) 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 ω 2 {\displaystyle \omega ^{2}} , one performs an "inner" induction (counting through 0 , 1 , 2 , … {\displaystyle 0,1,2,\dots } ), establishes the limit at ω {\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)} ordered lexicographically). Ordinals allow the definition of processes of arbitrary complexity, such as ω 3 {\displaystyle \omega ^{3}} (triple nesting) or ω ω {\displaystyle \omega ^{\omega }} (induction over the depth of nested induction). 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
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