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In Set Theory , ordinal, '''ordinal number''', and '''transfinite ordinal number''' refer to a type of Number introduced by Georg Cantor in 1897 , to accommodate Infinite sequences and to classify sets with certain kinds of Order structures on them. Ordinals are an extension of the Natural Number s different from Integer s and from Cardinal s. ). 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, but not subtraction or division. The Cantor normal form is a standardized way of writing down ordinals. 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 . ORDINALS EXTEND THE NATURAL NUMBERS A , or to describe the ''position'' of an element in a sequence. 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 Number s, and the notion of position, which is generalized by the ordinal numbers described here. 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. 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-order ed (so intimately linked, in fact, that some mathematicians make no distinction between the two concepts). 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). 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 ωω&2, 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 ) to just saying that the set is totally ordered and there is no infinite decreasing sequence, something which is perhaps easier to visualize. 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). 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 ). 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". 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. But this is not a serious difficulty. We will say that the ordinal is the '' Order Type '' of any set in the class. 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. 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. 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). Von Neumann definition of ordinals Rather than defining an ordinal as an ''equivalence class'' of well-ordered sets, we can try to define it as some particular well-ordered set which (canonically) represents the class. Thus, 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 ingenious definition suggested by John Von Neumann , and which is now taken as standard, is this: define each ordinal as a special well-ordered set, namely that of all ordinals before it: λ = [0,λ). Formally: :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''. (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 ''B'' has an element ''b'' which is disjoint from ''B''. 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 proper 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 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 , 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 Paradox ). 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 . Other definitions There are other modern formulations of the definition of ordinal. Each of these is essentially equivalent to the definition given above. One of these definitions is the following. A class ''S'' is called '' Transitive '' if each element ''x'' of ''S'' is a subset of ''S'', i.e. . An ordinal is then defined to be a transitive set whose members are also transitive. It follows from this that the members are themselves ordinals. Note that the Axiom Of Regularity (foundation) is used in showing that these ordinals are well ordered by containment (subset). 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 . An ordinary sequence corresponds to the case α = ω. TRANSFINITE INDUCTION See Also: Transfinite induction What is transfinite induction? Transfinite induction holds in any Well-order ed set, but it is so important in relation to ordinals that it is worth restating here. :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 follow by Transfinite Recursion - we use transfinite induction to prove that the result is well-defined): the formal statement is tedious to write, but the bottom line is, in order to define a (class) function on the ordinals ''α'', one can assume that it is already defined for all smaller ''β''<''α''. One proves 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 induction on the ordinals (more will be given later): define a function ''F'' by letting ''F''(''α'') be the smallest ordinal not in the set of ''F''(''β'') for all ''β''<''α''. Note how we assume the ''F''(''β'') known in the very process of defining ''F'': this apparent paradox is exactly what definition by transfinite induction permits. Now in fact ''F''(0) makes sense since there is no ''β''<0, so the set of all ''F''(''β'') for ''β''<0 is empty, so ''F''(0) must be 0 (the smallest ordinal of all), and now that we know ''F''(0), then ''F''(1) makes sense (and it is the smallest ordinal not equal to ''F''(0)=0), and so on (the ''and so on'' is exactly transfinite induction). Well, it turns out that this example is not very interesting since ''F''(''α'')=''α'' for all ordinals ''α'': but this can be shown, precisely, by transfinite induction. Successor and limit ordinals Any nonzero ordinal has the minimum zero. It may or may not have a maximum. 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. In the von Neumann definition of ordinals, the successor of α is since its elements are those of α and α itself. A nonzero ordinal which is ''not'' a successor is called a '' Limit Ordinal ''. One justification for this term is that a limit ordinal is indeed the Limit in a topological sense of all smaller ordinals (for the Order Topology ). |
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