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## Ordinal numberOrdinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. The mathematician Georg Cantor showed in 1897 how to extend this concept beyond the natural numbers to the infinite and how to do arithmetic with these transfinite ordinals. It is this generalization which will be explained below.
A natural number can be used for two purposes: to describe the
One can (and usually does) define the natural number - 0 = {} (empty set)
- 1 = {0} = { { } }
- 2 = {0,1} = { {}, { {} } }
- 3 = {0,1,2} =
- 4 = {0,1,2,3} = { {} , { { } }, { {}, { {} } } , }
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
This provides the motivation for the generalization: we want to construct ordinal numbers as special well-ordered sets in such a way that **A set***S*is an ordinal if and only if*S*is totally ordered with respect to set containment and every element of*S*is also a subset of*S*.
S is automatically well-ordered with respect to set containment. This relies on the axiom of well foundation: every 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 induction 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
Another consequence is that every set of ordinals has a supremum, the ordinal gotten 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 paradox).
To define the sum The first transfinite ordinal is ω, the set of all natural numbers. Let's try to visualize the ordinal ω+ω: two copies of the natural numbers ordered in the normal fashion and the second copy completely to the right of the first. If we write the second copy as {0'<1'<2',...} then ω+ω looks like - 0 < 1 < 2 < 3 < ... < 0' < 1' < 2' < ...
- 0 < 1 < 2 < 0' < 1' < 2' < ...
You should now be able to "see" that ω + 4 + ω = ω + ω for example.
To multiply the two ordinals Here's ω2: - 0
_{0}< 1_{0}< 2_{0}< 3_{0}< ... < 0_{1}< 1_{1}< 2_{1}< 3_{1}< ...
- 0
_{0}< 1_{0}< 0_{1}< 1_{1}< 0_{2}< 1_{2}< 0_{3}< 1_{3}< ...
Distributivity partially holds for ordinal arithmetic:
One can now go on to define exponentiation of ordinal numbers and explore its properties. Ordinal numbers present an extremely rich arithmetic. There are ordinal numbers which can not be reached from ω with a finite number of the arithmetical operations addition, multiplication and exponentiation. The smallest such is denoted by ε
The ordinals also carry an interesting order topology by virtue of being totally ordered.
In this topology, the sequence 0, 1, 2, 3, 4, ... has limit ω and the sequence ω, ω^ω, ω^(ω^ω), ... has limit ε Some special limit ordinals can be used to measure the size or cardinality of sets. These are the cardinal numbers. ## References | |||||

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