Les Babyloniens utilisaient la base 60, pratique grâce à ses nombreux diviseurs. Les ordinateurs effectuent les calculs en base 2. Le développement en base p itérée est de même nature. Cette expression est le développement en base 2 itérée de
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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 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 with a set with no particular structure on it, the ordinals are intimately linked with the special kind of sets that are called well-ordered 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 labelled 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 that 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. So far we have mentioned only finite ordinals, which are the natural numbers. Exactly what addition means will be defined later on: just consider them as names. 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.
Definitions Further information: Ordered set In a well-ordered set, every non-empty subset has a smallest element. Given the axiom of dependent choice , this is equivalent to just saying that the set is totally ordered and there is no infinite decreasing sequence, something 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 two 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 will define it as a particular well-ordered set that 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.
Note that the natural numbers are ordinals by this definition. 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.
Furthermore, the elements of every ordinal are ordinals themselves. 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. 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.
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 contradict its strict ordering by membership. This is the Burali-Forti paradox. 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. For example, assuming the axiom of regularity , the following are equivalent for a set x: x is an ordinal, x is a transitive set , and set membership is trichotomous on x, x is a transitive set totally ordered by set inclusion, x is a transitive set of transitive sets.
These definitions cannot be used in non-well-founded set theories. In set theories with urelements , one has to further make sure that the definition excludes urelements from appearing in ordinals. This concept, a transfinite sequence or ordinal-indexed sequence, is a generalization of the concept of a sequence. 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 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.
Let F denote a class function F to be defined on the ordinals. So F 0 is equal to 0 the smallest ordinal of all. Successor and limit ordinals Any nonzero ordinal has the minimum element, zero. It may or may not have a maximum element. A nonzero ordinal that 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 under the order topology.
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.
So in the following sequence: 0, 1, 2, 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. 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 , or, in other words, that its elements can be indexed in increasing fashion by the ordinals less than.
This applies, in particular, to any set of ordinals: any set of ordinals is naturally indexed by the ordinals less than some. 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. Formally, the definition is by transfinite induction: the -th element of the class is defined provided it has already been defined for all , as the smallest element greater than the -th element for all We can apply this, for example, to the class of limit ordinals: the -th ordinal, which is either a limit or zero is see ordinal arithmetic for the definition of multiplication of ordinals.
Similarly, we can consider additively indecomposable ordinals meaning a nonzero ordinal that is not the sum of two strictly smaller ordinals : the -th additively indecomposable ordinal is indexed as.
The technique of indexing classes of ordinals is often useful in the context of fixed points: for example, the -th ordinal.
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