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Ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. (See How to name numbers.)
In mathematics, ordinal numbers are an extension of the natural numbers to accommodate infinite sequences, introduced by Georg Cantor in 1897. It is this generalization which will be explained below.
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. While in the finite world these two concepts coincide, when dealing with infinite sets one has to distinguish between the two. The size aspect leads to cardinal numbers, which were also discovered by Cantor, while the position aspect is generalized by the ordinal numbers described here.
In set theory, the natural numbers are commonly constructed as sets, such that each natural number is the set of all smaller natural numbers:
etc.
Viewed this way, every natural number is a well-ordered set: the set 4 for instance has the elements 0, 1, 2, 3 which are of course ordered as 0 < 1 < 2 < 3 and this is a well-order. A natural number is smaller than another if and only if it is an element of the other.
We don't want to distinguish between two well-ordered sets if they only differ in the "notation for their elements", or more formally: if we can pair off the elements of the first set with the elements of the second set in a one-to-one fashion and 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.
With this convention, one can show that every finite well-ordered set is order-isomorphic to one (and only one) natural number. This provides the motivation for the generalization to infinite numbers.
We want to construct ordinal numbers as special well-ordered sets in such a way that every well-ordered set is order-isomorphic to one and only one ordinal number. The following definition improves on Cantor's approach and was first given by John von Neumann:
(Here, "set containment" is another name for the subset relationship.) Such a set S is automatically well-ordered with respect to set containment. This relies on the axiom of well foundation: every nonempty set S has an element a which is disjoint from S.
Note that the natural numbers are 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 inductionSet theory Transfinite induction is the proof technique of mathematical induction when applied to (large) well-ordered sets, for instance to sets of ordinals or cardinals, or even to the class of all ordinals. It may be regarded as one of three forms of m that every well-ordered set is order-isomorphic to exactly one of these ordinals.
Furthermore, the elements of every ordinal are ordinals themselves. Whenever you have two ordinals S and T, S is an element of T if and only if S is a subset of T, and 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 orderedIn mathematics, a total order or linear order on a set X is a binary relation that is reflexive, antisymmetric, transitive, and total . This means that, if we denote the relation by ≤, the following statements hold for all a b and c in X : a ≤ a (re. And in fact, much more is true: Every set of ordinals is well-ordered. This important result generalizes the fact that every set of natural numbers is well-ordered and it allows us to use transfinite inductionSet theory Transfinite induction is the proof technique of mathematical induction when applied to (large) well-ordered sets, for instance to sets of ordinals or cardinals, or even to the class of all ordinals. It may be regarded as one of three forms of m liberally with ordinals.
Another consequence is that every ordinal S is a set having as elements precisely the ordinals smaller than S. This statement completely determines the set-theoretic structure of every ordinal in terms of other ordinals. It's used to prove many other useful results about ordinals. One example of these is an important characterization of the order relation between ordinals: every set of ordinals has a supremumIn mathematics, the supremum of a given set is the least element which is greater than or equal to each element of the set. Consequently, it is also referred to as the least upper bound . In general, unless a set contains a greatest element, the supremum, the ordinal obtained by taking the union of all the ordinals in the set. Another example is the fact that the collection of all ordinals is not a set. Indeed, since every ordinal contains only other ordinals, it follows that every member of the collection of all ordinals is also its subset. Thus, if that collection were a set, it would have to be an ordinal itself by definition; then it would be its own member, which contradicts the axiom of regularity. (See also the Burali-Forti paradoxThe Burali-Forti paradox demonstrates that naively constructing "the set of all ordinal numbers" leads to a contradiction and therefore shows an antinomy in a system that allows its construction. The reason is that the set of all ordinal numbers carries a).
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 subsets has a greatest elementIn mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set is an element of S which is greater than or equal to any other element of S''. The term least element is defined dually. Formally, given a partially.
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