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ORDINAL NUMBERS

Specialty Definition: Ordinal number

(From Wikipedia, the free Encyclopedia)

Ordinal 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 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.

One can (and usually does) define the natural number n as the set of all smaller natural numbers:

0 = {} (empty set)

1 = {0} = { { } }

2 = {0,1} = { {}, { {} } }

3 = {0,1,2} =

4 = {0,1,2,3} = { {} , { { } }, { {}, { {} } } , }

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. In this case, an equivalent definition for finite is that the opposite order is also well-ordered, or that every subset has a maximal element.

This provides the motivation for the generalization: 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:

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.

Such a set 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 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 ordered. 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 induction 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 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 S + T of two ordinal numbers S and T, one proceeds as follows: first the elements of T are relabeled so that S and T become disjoint, then the well-ordered set S is written "to the left" of the well-ordered set T, meaning one defines an order on S∪T in which every element of S is smaller than every element of T. The sets S and T themselves keep the ordering they already have. This way, a new well-ordered set is formed, and this well-ordered set is order-isomorphic to a unique ordinal, which is called S + T. This addition is associative and generalizes the addition of natural numbers.

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' < ...

This is different from ω because in ω only 0 does not have a direct predecessor while in ω+ω the two elements 0 and 0' don't have direct predecessors. Here's 3 + ω:

0 < 1 < 2 < 0' < 1' < 2' < ...

and after relabeling, this just looks like ω itself: we have 3 + ω = ω. But ω + 3 is not equal to ω since the former has a largest element and the latter doesn't. So our addition is not commutative.

You should now be able to "see" that ω + 4 + ω = ω + ω for example.

To multiply the two ordinals S and T you write down the well-ordered set T and replace each of its elements with a different copy of the well-ordered set S. This results in a well-ordered set, which defines a unique ordinal; we call it ST. Again, this operation is associative and generalizes the multiplication of natural numbers.

Here's ω2:

0₀ < 1₀ < 2₀ < 3₀ < ... < 0₁ < 1₁ < 2₁ < 3₁ < ...

and we see: ω2 = ω + ω. But 2ω looks like this:

0₀ < 1₀ < 0₁ < 1₁ < 0₂ < 1₂ < 0₃ < 1₃ < ...

and after relabeling, this looks just like ω and so we get 2ω = ω. Multiplication of ordinals is not commutative.

Distributivity partially holds for ordinal arithmetic: R(S+T) = RS + RT. One can actually "see" that. However, the other distributive law (T+U)R = TR + UR is not generally true: (1+1)ω is equal to 2ω = ω while 1ω + 1ω equals ω+ω. Therefore, the ordinal numbers do not form a ring.

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 ε₀. ε₀ is still countable, but there are also uncountable ordinals. The smallest uncountable ordinal may be identified with the set of all countable ordinals, and is usually denoted by ω₁.

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 ε₀. Ordinals which don't have an immediate predecessor can always be written as a limit like this and are called limit ordinals. The topological spaces ω₁ and its successor ω₁+1 are frequently used as the text-book examples of non-countable topologies. For example, in the topological space ω₁+1, the element ω₁ is in the closure of the subset ω₁ even though no sequence of elements in ω₁ has the element ω₁ as its limit.

Some special limit ordinals can be used to measure the size or cardinality of sets. These are the cardinal numbers.

References

• Conway, J. H. and Guy, R. K. "Cantor's Ordinal Numbers." In The Book of Numbers. New York: Springer-Verlag, pp. 266-267 and 274, 1996.

  Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Ordinal number."

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Use of ordinals by monarchs

(From Wikipedia, the free Encyclopedia)

Ordinal numbers or regnal numbers are used to distinguish between persons with the same name who held the same office. Most importantly, they are used to distinguish monarchs. The tradition of numbering monarchs dates back at least as early as the reign of King Edward III of England.

This article is purely for the purpose of information. Wikipedia has strict naming conventions on the use of ordinals in Wikipedia articles, and these should be respected by anyone writing an article for Wikipedia.

It is common to start counting either since the beginning of the monarchy, or since the beginning of a particular line of dynastic succession. For example, Boris III of Bulgaria and his son Simeon II were given their regnal numbers because it was decided that the medieval kings (between 679-1018 and 1186-1393) would be included, even though their dynasty only dated back to 1887 and had no connection to the previous monarchies. On the other hand, the kings of England were counted starting with the Norman Conquest. That's why the son of Henry III of England is counted as Edward I, even though there were three Edwards before the Conquest.

In any case, it is usual to count only the monarchs or heads of the family, and to numbering them sequentially up to the end of the dynasty. Sometimes, such as in the case of the Swedish kings, mythical or semi-mythical perons are included. A notable exception to this rule is the German House of Reuss. This family has the particularity that every male member during the last centuries was named Heinrich, and all of them - not only the head of the family - were numbered. While the member of the elder branch were numbered in order of birth until the extinction of the branch in 1927, the members of the younger line were (and still are) numbered in sequences which began and ended roughly as centuries began and ended. This explains why the current head of the Reuss family is called Heinrich IV, his son Heinrich XIV and his son Heinrich XXIX.

The first

In some monarchies it is costumary not to use an ordinal when there has been one holder of that name. For example, Queen Victoria of the United Kingdom is not called Victoria I. This tradition is applied in the United Kingdom, the Netherlands, Luxembourg, Sweden and Norway. It was also applied in in most of the German monarchies.

Other monarchies do assign ordinals to monarchs who are the only ones of their name. This is a more recent invention and appears to be done for the first time when King Francis I of France issued testoons (silver coins) bearing the legend FRANCISCVS I DE. GR. FRANCORV. REX . This currently is the regular practice in Belgium, Spain and Monaco (at least for Prince Albert I, as Princess Louise Hippolyte, who reigned 150 years earlier, doesn't appear to have used an ordinal). It was also applied in Albania during the reign of King Zog, Brazil, Italy, Mexico, Montenegro, Portugal, Russia and by the Papacy under Pope John Paul I.

It should be noted that there are cases when the national tradition were not respected. For example, when Mary I of England acceeded to the throne, her regnal style was announced as "the most high, most puissant, and most excellent Princess Mary the First, by the Grace of God Queen of England, France, and Ireland, Defender of the Faith, and of the Church of England and Ireland Supreme Head".

Elizabeth Queen of Scots?

It is sometimes said that the present queen of the United Kingdom is not called Elizabeth II in Scotland, but simply Elizabeth, because she is the first monarch of her name to reign over Scotland. This is incorrect.

Between 1603, when the crowns were united in the person of James VI, and the union in 1707, the monarchs where numbered separately. After that, a single ordinal was used throughout Great Britain, and this has always been consistent with the English sequence of sovereigns. Hence, Edward VII of the United Kingdom was called Edward VII throughout the entire United Kingdom, even though he was only the second of that name to reign in Scotland.

In order to avoid controversy, it was announced after the accession of Elizabeth II that, in the future, the highest numeral from each sequence would be used. So any future British King Edward would be given the number IX, even though there have only been three previous Edwards in Scotland, but any future King Robert would be given the number IV, even though he would be the first Robert to reign in England.

Pretenders

It is traditional amongst monarchists to continue to number their pretenders, even though they have never reigned. Hence, a supporter of the Comte de Paris would call him Henri VII, even though only four Henri's have been King of France. This also explains why there was a Louis XVIII of France, even though France had never experienced the reign of Louis XVII.

Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Use of ordinals by monarchs."

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