Transfinite Ordinal Number
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Transfinite Ordinal Number
Transfinite ordinal numbers are a generalization of finite ordinal numbers.
The definition of transfinite ordinal numbers is based on the concept of a well-ordered set. Every finite set can be made well ordered by arranging its elements in a definite order. A simple example of a well-ordered infinite set is the set of natural numbers in their usual order, wherein each number is smaller than the number immediately following it. If this set is arranged in the reverse order, so that each number is smaller than the number immediately preceding it, then the set is not well ordered, since none of its infinite subsets has a first element.
Two ordered sets X and Y are said to be similar, or to have the same ordinal type, if they can be put into a one-to-one correspondence that preserves the ordering. In other words, for any two elements x’ and x” of X and the corresponding elements y’ and y” of Y, y’ < y” must follow from x’ < x”, and x’ < x” must follow from y’ < y”. All well-ordered finite sets containing the same number of elements are similar. Consequently, the ordinal types of well-ordered finite sets can be identified with natural numbers, which thus play the role of ordinal numbers here. By contrast, when a natural number is used to designate the number of elements in a set, this natural number plays the role of a cardinal number.
The ordinal types of infinite well-ordered sets are called transfinite ordinal numbers. The concept of transfinite ordinal number therefore represents an extension of the concept of ordinal number to infinite sets. A similar extension of the concept of cardinal number leads to the concept of the cardinal number of an infinite set (seeCARDINALITY OF A SET). Since sets having different cardinal numbers cannot be put into one-to-one correspondence, to well-ordered sets having different transfinite cardinal numbers there correspond different transfinite ordinal numbers. In contrast to the situation with finite sets, however, well-ordered infinite sets may have the same cardinal number without being similar and may therefore be associated with different ordinal numbers.
It is possible to introduce the concepts of greater than and less than for transfinite ordinal numbers. The transfinite ordinal number α is by definition less than the transfinite ordinal number β (α < β) if some (and consequently, any) well-ordered set of ordinal type α is similar to a segment of some (and consequently, any) set of ordinal type β. (By a segment of a well-ordered set we mean the subset of elements of the set that precede some element x of the set.) It can be shown that for any two transfinite ordinal numbers α and β one, and only one, of the following is true: α < β, α = β, or α > β.
The principle of transfinite induction plays an important role in the application of transfinite ordinal numbers to various mathematical problems. Transfinite induction, which is a generalization of mathematical induction to arbitrary well-ordered sets, consists in the following: if some theorem is true for the first element of a well-ordered set X, and the theorem is true for an element x of X if it is true for each element preceding x, then the theorem is true for every element of X.