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 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 previous 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 ⁠ 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. 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. Consequently, ordinal numbers are defined as the representative forms of these isomorphism classes. == Definitions == 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"). 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 ⁠ ω α {\displaystyle \omega _{\alpha }} ⁠, it is always a limit ordinal. Its cardinality is written ⁠ ℵ α {\displaystyle \aleph _{\alpha }} ⁠. For example, the cardinality of ω0 = ω is ⁠ ℵ 0 {\displaystyle \aleph _{0}} ⁠, which is also the cardinality of ω2 or ε0 (all are countable ordinals). 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) ⊇ ···. The superscripts containing ∞ are just indices defined by the derivation process. Cantor used these sets in the theorems: These theorems are proved by partitioning P′ into pairwise disjoint sets: P′ = (P′\ P(2)) ∪ (P(2) \ P(3)) ∪ ··· ∪ (P(∞) \ P(∞ + 1)) ∪ ··· ∪ P(α). For β < α: since P(β + 1) contains the limit points of P(β), the sets P(β) \ P(β + 1) have no limit points. Hence, they are discrete sets, so they are countable. Proof of first theorem: If P(α) = ∅ for some index α, then P′ is the countable union of countable sets. Therefore, P′ is countable. The second theorem requires proving the existence of an α such that P(α) = ∅. To prove this, Cantor considered the set of all α having countably many predecessors. To 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. Cantor called the set of finite ordinals the first number class. The second number class is the set of ordinals whose predecessors form a countably infinite set.