Citizendia
Your Ad Here

This article is about the mathematical concept. For ordinals in linguistics, see Ordinal number (linguistics). In linguistics ordinal numbers are the words representing the rank of a number with respect to some order in particular order or position (i

In set theory, ordinal, ordinal number, and transfinite ordinal number refer to a type of number introduced by Georg Cantor in 1897[1] to accommodate infinite sequences and to classify sets with certain kinds of order structures on them. A number is an Abstract object, tokens of which are Symbols used in Counting and measuring. Georg Ferdinand Ludwig Philipp Cantor ( – January 6 1918) was a German Mathematician, born in Russia. Year 1897 ( MDCCCXCVII) was a Common year starting on Friday (link will display the full calendar of the Gregorian Calendar (or a Common Infinity (symbolically represented with ∞) comes from the Latin infinitas or "unboundedness Order theory is a branch of Mathematics that studies various kinds of Binary relations that capture the intuitive notion of ordering providing a framework for saying [2] Ordinals are an extension of the natural numbers different from integers and from cardinals. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French This article describes cardinal numbers in mathematics For cardinals in linguistics see Names of numbers in English.

Well-ordering is total ordering with transfinite induction, where transfinite induction extends mathematical induction beyond the finite. In Mathematics, a well-order relation (or well-ordering) on a set S is a Total order on S with the property that every In Mathematics and Set theory, a total order, linear order, simple order, or (non-strict ordering is a Binary relation Transfinite induction is an extension of Mathematical induction to well-ordered sets, for instance to sets of ordinals or cardinals. Mathematical induction is a method of Mathematical proof typically used to establish that a given statement is true of all Natural numbers It is done by proving that Ordinals represent equivalence classes of well orderings with order-isomorphism being the equivalence relationship. Each ordinal is taken to be the set of smaller ordinals. Ordinals may be categorized as: zero, successor ordinals, and limit ordinals (of various cofinalities). In Mathematics, especially in Order theory, the cofinality cf( A) of a Partially ordered set A is the least of the cardinalities Given a class of ordinals, one can identify the α-th member of that class, i. e. one can index (count) them. A class is closed and unbounded if its indexing function is continuous and never stops. One can define addition, multiplication, and exponentiation on ordinals. The Cantor normal form is a standardized way of writing down ordinals. In the mathematical field of Set theory, ordinal arithmetic describes the three usual operations on Ordinal numbers addition multiplication and exponentiation There is a many to one association from ordinals to cardinals. Larger and larger ordinals can be defined, but they become more and more difficult to describe. Any ordinal number can be made into a topological space by endowing it with the order topology. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Mathematics, an order topology is a certain Topology that can be defined on any Totally ordered set.

Contents

Ordinals extend the natural numbers

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. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an When restricted to finite sets these two concepts coincide; there is only one way to put a finite set into a linear sequence, up to isomorphism. When dealing with infinite sets one has to distinguish between the notion of size, which leads to cardinal numbers, and the notion of position, which is generalized by the ordinal numbers described here. This article describes cardinal numbers in mathematics For cardinals in linguistics see Names of numbers in English. This is because, while any set has only one size (its cardinality), there are many nonisomorphic well-orderings of any infinite set, as explained below. In Mathematics, the cardinality of a set is a measure of the "number of elements of the set"

Whereas the notion of cardinal number is associated to a set with no particular structure on it, the ordinals are intimately linked with the special kind of sets which are called well-ordered (so intimately linked, in fact, that some mathematicians make no distinction between the two concepts). In Mathematics, a well-order relation (or well-ordering) on a set S is a Total order on S with the property that every A well-ordered set is a totally ordered set (given any two elements one defines a smaller and a larger one in a coherent way) in which there is no infinite decreasing sequence (however, there may be infinite increasing sequences); equivalently, every non-empty subset of the set has a least element. Ordinals may be used to label the elements of any given well-ordered set (the smallest element being labeled 0, the one after that 1, the next one 2, "and so on") and to measure the "length" of the whole set by the least ordinal which is not a label for an element of the set. This "length" is called the order type of the set.

Any ordinal is defined by the set of ordinals that precede it: in fact, the most common definition of ordinals identifies each ordinal as the set of ordinals that precede it. For example, the ordinal 42 is the order type of the ordinals less than it, i. e. , the ordinals from 0 (the smallest of all ordinals) to 41 (the immediate predecessor of 42), and it is generally identified as the set {0,1,2,…,41}. Conversely, any set of ordinals which is downward-closed—meaning that any ordinal less than an ordinal in the set is also in the set—is (or can be identified with) an ordinal.

So far we have mentioned only finite ordinals, which are the natural numbers. But there are infinite ones as well: the smallest infinite ordinal is ω, which is the order type of the natural numbers (finite ordinals) and which can even be identified with the set of natural numbers (indeed, the set of natural numbers is well-ordered—as is any set of ordinals—and since it is downward closed it can be identified with the ordinal associated to it, which is exactly how we define ω).

A graphical “matchstick” representation of the ordinal ω². Each stick corresponds to an ordinal of the form ω·m+n where m and n are natural numbers.
A graphical “matchstick” representation of the ordinal ω². Each stick corresponds to an ordinal of the form ω·m+n where m and n are natural numbers.

Perhaps a clearer intuition of ordinals can be formed by examining a first few of them: as mentioned above, they start with the natural numbers, 0, 1, 2, 3, 4, 5, … After all natural numbers comes the first infinite ordinal, ω, and after that come ω+1, ω+2, ω+3, and so on. (Exactly what addition means will be defined later on: just consider them as names. ) After all of these come ω·2 (which is ω+ω), ω·2+1, ω·2+2, and so on, then ω·3, and then later on ω·4. Now the set of ordinals which we form in this way (the ω·m+n, where m and n are natural numbers) must itself have an ordinal associated to it: and that is ω2. Further on, there will be ω3, then ω4, and so on, and ωω, then ωω², and much later on ε0 (epsilon nought) (to give a few examples of relatively small —countable—ordinals). We can go on in this way indefinitely far ("indefinitely far" is exactly what ordinals are good at: basically every time one says "and so on" when enumerating ordinals, it defines a larger ordinal). The smallest uncountable ordinal is the set of all countable ordinals, expressed as ω1.

Definitions

Well-ordered sets

A well-ordered set is an ordered set in which every non-empty subset has a least element: this is equivalent (at least in the presence of the axiom of dependent choice) to just saying that the set is totally ordered and there is no infinite decreasing sequence, something which is perhaps easier to visualize. In Mathematics, a well-order relation (or well-ordering) on a set S is a Total order on S with the property that every In Mathematics, the axiom of dependent choices, denoted DC, is a weak form of the Axiom of choice (AC which is still sufficient to develop most of In practice, the importance of well-ordering is justified by the possibility of applying transfinite induction, which says, essentially, that any property that passes on from the predecessors of an element to that element itself must be true of all elements (of the given well-ordered set). Transfinite induction is an extension of Mathematical induction to well-ordered sets, for instance to sets of ordinals or cardinals. If the states of a computation (computer program or game) can be well-ordered in such a way that each step is followed by a "lower" step, then you can be sure that the computation will terminate.

Now we don't want to distinguish between two well-ordered sets if they only differ in the "labeling of their elements", or more formally: if we can pair off the elements of the first set with the elements of the second set 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, or similar (obviously this is an equivalence relation). In the mathematical field of Order theory an order isomorphism is a special kind of Monotone function that constitutes a suitable notion of Isomorphism In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent" Provided there exists an order isomorphism between two well-ordered sets, the order isomorphism is unique: this makes it quite justifiable to consider the sets as essentially identical, and to seek a "canonical" representative of the isomorphism type (class). This is exactly what the ordinals provide, and it also provides a canonical labeling of the elements of any well-ordered set.

So we essentially wish to define an ordinal as an isomorphism class of well-ordered sets: that is, as an equivalence class for the equivalence relation of "being order-isomorphic". In Mathematics, given a set X and an Equivalence relation ~ on X, the equivalence class of an element a in X In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent" There is a technical difficulty involved, however, in the fact that the equivalence class is too large to be a set in the usual Zermelo–Fraenkel (ZF) formalization of set theory. Zermelo–Fraenkel set theory with the axiom of choice, commonly abbreviated ZFC, is the standard form of Axiomatic set theory and as such is the most common But this is not a serious difficulty. We will say that the ordinal is the order type of any set in the class. In Mathematics, especially in Set theory, two Ordered sets XY are said to have the same order type just when they are Order isomorphic

Definition of an ordinal as an equivalence class

The original definition of ordinal number, found for example in Principia Mathematica, defines the order type of a well-ordering as the set of all well-orderings similar (order-isomorphic) to that well-ordering: in other words, an ordinal number is genuinely an equivalence class of well-ordered sets. The Principia Mathematica is a 3-volume work on the Foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell This definition must be abandoned in ZF and related systems of axiomatic set theory because these equivalence classes are too large to form a set. Zermelo–Fraenkel set theory with the axiom of choice, commonly abbreviated ZFC, is the standard form of Axiomatic set theory and as such is the most common However, this definition still can be used in type theory and in Quine's set theory New Foundations and related systems (where it affords a rather surprising alternative solution to the Burali-Forti paradox of the largest ordinal). In Mathematics, Logic and Computer science, type theory is any of several Formal systems that can serve as alternatives to Naive set theory In Mathematical logic, New Foundations ( NF) is an Axiomatic set theory, conceived by Willard Van Orman Quine as a simplification of In Set theory, a field of Mathematics, the Burali-Forti paradox demonstrates that naively constructing "the set of all Ordinal numbers quot leads to

Von Neumann definition of ordinals

Rather than defining an ordinal as an equivalence class of well-ordered sets, we will define it as a particular well-ordered set which (canonically) represents the class. Thus, an ordinal number will be a well-ordered set; and every well-ordered set will be order-isomorphic to exactly one ordinal number.

The standard definition, suggested by John von Neumann, is: each ordinal is the well-ordered set of all smaller ordinals. In symbols, λ = [0,λ). [3] Formally:

A set S is an ordinal if and only if S is strictly well-ordered with respect to set membership and every element of S is also a subset of S. In Mathematics, especially Order theory, a partially ordered set (or poset) formalizes the intuitive concept of an ordering sequencing or arrangement

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, that is, there is an order preserving bijective function between them. Transfinite induction is an extension of Mathematical induction to well-ordered sets, for instance to sets of ordinals or cardinals. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property

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 proper subset of T. 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. In Mathematics and Set theory, a total order, linear order, simple order, or (non-strict ordering is a Binary relation Further, every set of ordinals is well-ordered. This generalizes the fact that every set of natural numbers is well-ordered.

Consequently, every ordinal S is a set having as elements precisely the ordinals smaller than S. For example, every set of ordinals has a supremum, the ordinal obtained by taking the union of all the ordinals in the set. This union exists regardless of the set's size, by the axiom of union). In Axiomatic set theory and the branches of Logic, Mathematics, and Computer science that use it the axiom of union is one of the Axioms

The class of all ordinals is not a set. If it were a set, one could show that it was an ordinal and thus a member of itself which would contradicts its strict ordering by membership. This is the Burali-Forti paradox. In Set theory, a field of Mathematics, the Burali-Forti paradox demonstrates that naively constructing "the set of all Ordinal numbers quot leads to The class of all ordinals is variously called "Ord", "ON", or "∞".

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 maximum. In Mathematics, a set is called finite if there is a Bijection between the set and some set of the form {1 2.

Other definitions

There are other modern formulations of the definition of ordinal. For example, assuming the axiom of regularity, the following are equivalent for a set x:

These definitions cannot be used in non-well-founded set theories. Non-well founded set theories are variants of Axiomatic set theory which allow sets to contain themselves and otherwise violate the rule of Well-foundedness. In set theories with urelements, one has to further make sure that the definition excludes urelements from appearing in ordinals. In Set theory, a branch of Mathematics, an urelement or ur-element (from the German prefix ur-, 'primordial' is an object (concrete

Transfinite sequence

If α is a limit ordinal and X is a set, an α-indexed sequence of elements of X is a function from α to X. This concept, a transfinite sequence or ordinal-indexed sequence, is a generalization of the concept of a sequence. In Mathematics, a sequence is an ordered list of objects (or events An ordinary sequence corresponds to the case α = ω.

Transfinite induction

Main article: Transfinite induction

What is transfinite induction?

Transfinite induction holds in any well-ordered set, but it is so important in relation to ordinals that it is worth restating here. Transfinite induction is an extension of Mathematical induction to well-ordered sets, for instance to sets of ordinals or cardinals. In Mathematics, a well-order relation (or well-ordering) on a set S is a Total order on S with the property that every

Any property which passes from the set of ordinals smaller than a given ordinal α to α itself, is true of all ordinals.

That is, if P(α) is true whenever P(β) is true for all β<α, then P(α) is true for all α. Or, more practically: in order to prove a property P for all ordinals α, one can assume that it is already known for all smaller β<α.

Transfinite recursion

Transfinite induction can be used not only to prove things, but also to define them. Such a definition is normally said to be by transfinite recursion – the proof that the result is well-defined uses transfinite induction. Transfinite induction is an extension of Mathematical induction to well-ordered sets, for instance to sets of ordinals or cardinals. Let F denote a (class) function F to be defined on the ordinals. The idea now is that, in defining F(α) for an unspecified ordinal α, one may assume that F(β) is already defined for all β < α and thus give a formula for F(α) in terms of these F(β). It then follows by transfinite induction that there is one and only one function satisfying the recursion formula up to and including α.

Here is an example of definition by transfinite recursion on the ordinals (more will be given later): define function F by letting F(α) be the smallest ordinal not in the class {F(β) | β < α}, that is, the class consisting of all F(β) for β < α. This definition assumes the F(β) known in the very process of defining F; this apparent vicious circle is exactly what definition by transfinite recursion permits. In fact, F(0) makes sense since there is no ordinal β < 0, and the class {F(β) | β < 0} is empty. So F(0) is equal to 0 (the smallest ordinal of all). Now that F(0) is known, the definition applied to F(1) makes sense (it is the smallest ordinal not in the singleton class {F(0)} = {0}), and so on (the and so on is exactly transfinite induction). It turns out that this example is not very exciting, since provably F(α) = α for all ordinals α, which can be shown, precisely, by transfinite induction.

Successor and limit ordinals

Any nonzero ordinal has the minimum element, zero. It may or may not have a maximum element. For example, 42 has maximum 41 and ω+6 has maximum ω+5. On the other hand, ω does not have a maximum since there is no largest natural number. If an ordinal has a maximum α, then it is the next ordinal after α, and it is called a successor ordinal, namely the successor of α, written α+1. When defining the ordinal numbers, an absolutely fundamental operation that we can perform on them is a successor operation S to get the next higher one In the von Neumann definition of ordinals, the successor of α is \alpha\cup\{\alpha\} since its elements are those of α and α itself.

A nonzero ordinal which is not a successor is called a limit ordinal. A limit ordinal is an Ordinal number which is neither zero nor a Successor ordinal. One justification for this term is that a limit ordinal is indeed the limit in a topological sense of all smaller ordinals (under the order topology). In Mathematics, informally speaking a limit point of a set S in a Topological space X is a point x in X that can be "approximated" In Mathematics, an order topology is a certain Topology that can be defined on any Totally ordered set.

When \langle \alpha_{\iota} | \iota < \gamma \rangle is an ordinal-indexed sequence, indexed by a limit γ and the sequence is increasing, i. e. \alpha_{\iota} < \alpha_{\rho}\! whenever \iota < \rho,\! we define its limit to be the least upper bound of the set \{ \alpha_{\iota} | \iota < \gamma \},\! that is, the smallest ordinal (it always exists) greater than any term of the sequence. In this sense, a limit ordinal is the limit of all smaller ordinals (indexed by itself). Put more directly, it is the supremum of the set of smaller ordinals.

Another way of defining a limit ordinal is to say that α is a limit ordinal if and only if:

There is an ordinal less than α and whenever ζ is an ordinal less than α, then there exists an ordinal ξ such that ζ < ξ < α.

So in the following sequence:

0, 1, 2, . . . , ω, ω+1

ω is a limit ordinal because for any ordinal (in this example, a natural number) we can find another ordinal (natural number) larger than it, but still less than ω.

Thus, every ordinal is either zero, or a successor (of a well-defined predecessor), or a limit. This distinction is important, because many definitions by transfinite induction rely upon it. Very often, when defining a function F by transfinite induction on all ordinals, one defines F(0), and F(α+1) assuming F(α) is defined, and then, for limit ordinals δ one defines F(δ) as the limit of the F(β) for all β<δ (either in the sense of ordinal limits, as we have just explained, or for some other notion of limit if F does not take ordinal values). Thus, the interesting step in the definition is the successor step, not the limit ordinals. Such functions (especially for F nondecreasing and taking ordinal values) are called continuous. We will see that ordinal addition, multiplication and exponentiation are continuous as functions of their second argument.

Indexing classes of ordinals

We have mentioned that any well-ordered set is similar (order-isomorphic) to a unique ordinal number α, or, in other words, that its elements can be indexed in increasing fashion by the ordinals less than α. This applies, in particular, to any set of ordinals: any set of ordinals is naturally indexed by the ordinals less than some α. The same holds, with a slight modification, for classes of ordinals (a collection of ordinals, possibly too large to form a set, defined by some property): any class of ordinals can be indexed by ordinals (and, when the class is unbounded in the class of all ordinals, this puts it in class-bijection with the class of all ordinals). So we can freely speak of the γ-th element in the class (with the convention that the “0-th” is the smallest, the “1-th” is the next smallest, and so on). Formally, the definition is by transfinite induction: the γ-th element of the class is defined (provided it has already been defined for all β < γ), as the smallest element greater than the β-th element for all β < γ.

We can apply this, for example, to the class of limit ordinals: the γ-th ordinal which is either a limit or zero is \omega\cdot\gamma (see ordinal arithmetic for the definition of multiplication of ordinals). In the mathematical field of Set theory, ordinal arithmetic describes the three usual operations on Ordinal numbers addition multiplication and exponentiation Similarly, we can consider additively indecomposable ordinals (meaning a nonzero ordinal which is not the sum of two strictly smaller ordinals): the γ-th additively indecomposable ordinal is indexed as ωγ. In Set theory, a branch of Mathematics, an additively indecomposable ordinal &alpha is any Ordinal number that is not 0 such that for any \beta\gamma The technique of indexing classes of ordinals is often useful in the context of fixed points: for example, the γ-th ordinal α such that ωα = α is written \varepsilon_\gamma. These are called the "epsilon numbers".

Closed unbounded sets and classes

A class of ordinals is said to be unbounded, or cofinal, when given any ordinal, there is always some element of the class greater than it (then the class must be a proper class, i. e. , it cannot be a set). It is said to be closed when the limit of a sequence of ordinals in the class is again in the class: or, equivalently, when the indexing (class-)function F is continuous in the sense that, for δ a limit ordinal, F(δ) (the δ-th ordinal in the class) is the limit of all F(γ) for γ < δ; this is also the same as being closed, in the topological sense, for the order topology (to avoid talking of topology on proper classes, one can demand that the intersection of the class with any given ordinal is closed for the order topology on that ordinal, this is again equivalent). Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Mathematics, an order topology is a certain Topology that can be defined on any Totally ordered set.

Of particular importance are those classes of ordinals which are closed and unbounded, sometimes called clubs. In Mathematics, particularly in Mathematical logic and Set theory, a club set is a subset of a Limit ordinal which is closed under For example, the class of all limit ordinals is closed and unbounded: this translates the fact that there is always a limit ordinal greater than a given ordinal, and that a limit of limit ordinals is a limit ordinal (a fortunate fact if the terminology is to make any sense at all!). The class of additively indecomposable ordinals, or the class of \varepsilon_\cdot ordinals, or the class of cardinals, are all closed unbounded; the set of regular cardinals, however, is unbounded but not closed, and any finite set of ordinals is closed but not unbounded.

A class is stationary if it has a nonempty intersection with every closed unbounded class. All superclasses of closed unbounded classes are stationary and stationary classes are unbounded, but there are stationary classes which are not closed and there are stationary classes which have no closed unbounded subclass (such as the class of all limit ordinals with countable cofinality). Since the intersection of two closed unbounded classes is closed and unbounded, the intersection of a stationary class and a closed unbounded class is stationary. But the intersection of two stationary classes may be empty, e. g. the class of ordinals with cofinality ω with the class of ordinals with uncountable cofinality.

Rather than formulating these definitions for (proper) classes of ordinals, we can formulate them for sets of ordinals below a given ordinal α: A subset of a limit ordinal α is said to be unbounded (or cofinal) under α provided any ordinal less than α is less than some ordinal in the set. More generally, we can call a subset of any ordinal α cofinal in α provided every ordinal less than α is less than or equal to some ordinal in the set. The subset is said to be closed under α provided it is closed for the order topology in α, i. e. a limit of ordinals in the set is either in the set or equal to α itself.

Arithmetic of ordinals

Main article: Ordinal arithmetic

There are three usual operations on ordinals: addition, multiplication, and (ordinal) exponentiation. In the mathematical field of Set theory, ordinal arithmetic describes the three usual operations on Ordinal numbers addition multiplication and exponentiation Each can be defined in essentially two different ways: either by constructing an explicit well-ordered set which represents the operation or by using transfinite recursion. Cantor normal form provides a standardized way of writing ordinals. The so-called "natural" arithmetical operations retain commutativity at the expense of continuity.

Ordinals and cardinals

Initial ordinal of a cardinal

Each ordinal has an associated cardinal, its cardinality, obtained by simply forgetting the order. This article describes cardinal numbers in mathematics For cardinals in linguistics see Names of numbers in English. Any well-ordered set having that ordinal as its order-type has the same cardinality. The smallest ordinal having a given cardinal as its cardinality is called the initial ordinal of that cardinal. Every finite ordinal (natural number) is initial, but most infinite ordinals are not initial. The axiom of choice is equivalent to the statement that every set can be well-ordered, i. In Mathematics, the axiom of choice, or AC, is an Axiom of Set theory. e. that every cardinal has an initial ordinal. In this case, it is traditional to identify the cardinal number with its initial ordinal, and we say that the initial ordinal is a cardinal.

The α-th infinite initial ordinal is written ωα. Its cardinality is written \aleph_\alpha. For example, the cardinality of ω0 = ω is \aleph_0, which is also the cardinality of ω² or ε0 (all are countable ordinals). So (assuming the axiom of choice) we identify ω with \aleph_0, except that the notation \aleph_0 is used when writing cardinals, and ω when writing ordinals (this is important since \aleph_0^2=\aleph_0 whereas ω2 > ω). Also, ω1 is the smallest uncountable ordinal (to see that it exists, consider the set of equivalence classes of well-orderings of the natural numbers: each such well-ordering defines a countable ordinal, and ω1 is the order type of that set), ω2 is the smallest ordinal whose cardinality is greater than \aleph_1, and so on, and ωω is the limit of the ωn for natural numbers n (any limit of cardinals is a cardinal, so this limit is indeed the first cardinal after all the ωn).

See also Von Neumann cardinal assignment. The Von Neumann cardinal assignment is a Cardinal assignment which uses Ordinal numbers For a Well-ordered set U, we define its

Cofinality

The cofinality of an ordinal α is the smallest ordinal δ which is the order type of a cofinal subset of α. In Mathematics, especially in Order theory, the cofinality cf( A) of a Partially ordered set A is the least of the cardinalities In Mathematics, a cofinal subset is a subset B of a Preordered set A such that for every a in A there is a b Notice that a number of authors define confinality or use it only for limit ordinals. The cofinality of a set of ordinals or any other well ordered set is the cofinality of the order type of that set.

Thus for a limit ordinal, there exists a δ-indexed strictly increasing sequence with limit α. For example, the cofinality of ω² is ω, because the sequence ω·m (where m ranges over the natural numbers) tends to ω²; but, more generally, any countable limit ordinal has cofinality ω. An uncountable limit ordinal may have either cofinality ω as does ωω or an uncountable cofinality.

The cofinality of 0 is 0. And the cofinality of any successor ordinal is 1. The cofinality of any limit ordinal is at least ω.

An ordinal which is equal to its cofinality is called regular and it is always an initial ordinal. Any limit of regular ordinals is a limit of initial ordinals and thus is also initial even if it is not regular which it usually is not. If the Axiom of Choice, then ωα + 1 is regular for each α. In this case, the ordinals 0, 1, ω, ω1, and ω2 are regular, whereas 2, 3, ωω, and ωω·2 are initial ordinals which are not regular.

The cofinality of any ordinal α is a regular ordinal, i. e. the cofinality of the cofinality of α is the same as the cofinality of α. So the cofinality operation is idempotent. Idempotence ˌaɪdɨmˈpoʊtəns describes the property of operations in Mathematics and Computer science which means that multiple applications of the operation

Some “large” countable ordinals

For more details on this topic, see Large countable ordinal. See also Ordinal number In the mathematical discipline of Set theory, there are many ways of describing specific countable ordinals.

We have already mentioned (see Cantor normal form) the ordinal ε0, which is the smallest satisfying the equation ωα = α, so it is the limit of the sequence 0, 1, ω, ωω, \omega^{\omega^\omega}, etc. In the mathematical field of Set theory, ordinal arithmetic describes the three usual operations on Ordinal numbers addition multiplication and exponentiation Many ordinals can be defined in such a manner as fixed points of certain ordinal functions (the ι-th ordinal such that ωα = α is called \varepsilon_\iota, then we could go on trying to find the ι-th ordinal such that \varepsilon_\alpha = \alpha, “and so on”, but all the subtlety lies in the “and so on”). We can try to do this systematically, but no matter what system is used to define and construct ordinals, there is always an ordinal that lies just above all the ordinals constructed by the system. Perhaps the most important ordinal which limits in this manner a system of construction is the Church-Kleene ordinal, \omega_1^{\mathrm{CK}} (despite the ω1 in the name, this ordinal is countable), which is the smallest ordinal which cannot in any way be represented by a computable function (this can be made rigorous, of course). Alonzo Church ( June 14, 1903 – August 11, 1995) was an American Mathematician and logician Stephen Cole Kleene ( January 5, 1909, Hartford Connecticut, USA &ndash January 25, 1994, Madison Wisconsin Computable functions are the basic objects of study in computability theory. Considerably large ordinals can be defined below \omega_1^{\mathrm{CK}}, however, which measure the “proof-theoretic strength” of certain formal systems (for example, \varepsilon_0 measures the strength of Peano arithmetic). In formal logic, a formal system (also called a logical system, a logistic system, or simply a logic Formal systems in mathematics consist In Mathematical logic, the Peano axioms, also known as the Dedekind-Peano axioms or the Peano postulates, are a set of Axioms for the Natural Large ordinals can also be defined above the Church-Kleene ordinal, which are of interest in various parts of logic.

Topology and ordinals

For more details on this topic, see Order topology. In Mathematics, an order topology is a certain Topology that can be defined on any Totally ordered set.

Any ordinal can be made into a topological space in a natural way by endowing it with the order topology. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Mathematics, an order topology is a certain Topology that can be defined on any Totally ordered set. See the Topology and ordinals section of the "Order topology" article. In Mathematics, an order topology is a certain Topology that can be defined on any Totally ordered set.

Downward closed sets of ordinals

A set is downward closed if anything less than an element of the set is also in the set. In Mathematics, an upper set, or upward set is a subset Y of a given Partially ordered set ( X,&le such that for all elements If a set of ordinals is downward closed, then that set is an ordinal — the least ordinal not in the set.
Examples:

See also

Notes

  1. ^ Georg Cantor, Beitrage zur Begrundung der transfiniten Mengenlehre. This article describes cardinal numbers in mathematics For cardinals in linguistics see Names of numbers in English. Counting is the mathematical action of repeatedly adding (or subtracting one usually to find out how many objects there are or to set aside a desired number of objects (starting See also Ordinal number In the mathematical discipline of Set theory, there are many ways of describing specific countable ordinals. A limit ordinal is an Ordinal number which is neither zero nor a Successor ordinal. In the mathematical field of Set theory, ordinal arithmetic describes the three usual operations on Ordinal numbers addition multiplication and exponentiation When defining the ordinal numbers, an absolutely fundamental operation that we can perform on them is a successor operation S to get the next higher one In Mathematics, an order topology is a certain Topology that can be defined on any Totally ordered set. Georg Ferdinand Ludwig Philipp Cantor ( – January 6 1918) was a German Mathematician, born in Russia. II (tr. : Contributions to the Founding of the Theory of Transfinite Numbers II), Mathematische Annalen 49 (1897), 207-­246. English translation. Citation from Akihiro Kanamori, Set Theory from Cantor to Cohen, to appear in: Andrew Irvine and John H. Woods (editors), The Handbook of the Philosophy of Science, volume 4, Mathematics, Cambridge University Press.
  2. ^ Thorough introductions are given by Levy (1979) and Sacks (2003).
  3. ^ Levy (1979, p. 52) attributes the idea to unpublished work of Zermelo in 1916 and several papers by von Neumann the 1920s.

References

External links

Dictionary

ordinal number

-noun

  1. (grammar) A word that expresses the location of an item in an ordered sequence; an ordinal numeral.
  2. (arithmetic) A number used to denote position in a sequence.
  3. (mathematics) A generalized kind of number to denote the size of a well-ordered set.
© 2009 citizendia.org; parts available under the terms of GNU Free Documentation License, from http://en.wikipedia.org
 Dapyx Software network: MP3 Explorer | Ebook Manager | Zenithic