CENTROHD - Huge Dictionaries
Frasi che contengono la parola ordinal
|
, 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).
|
|
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
|
|
. 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:
|
|
. 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).
|
|
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
|
|
; 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).
|
|
Each ordinal has an associated cardinal, its cardinality, obtained by simply forgetting the order. 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.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 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
|
|
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
|
|
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). Considerably large ordinals can be defined below
|
|
Any ordinal can be made into a topological space by endowing it with the order topology (since, being well-ordered, an ordinal is in particular totally ordered): in the absence of indication to the contrary, it is always that order topology which is meant when an ordinal is thought of as a topological space. (Note that if we are willing to accept a proper class as a topological space, then the class of all ordinals is also a topological space for the order topology.)
|
Tutti i dati sono automaticamente, anche se accuratamente raccolti da fonti di pubblico accesso. Le frasi vengono selezionate automaticamente e non sono destinate ad esprimere le nostre opinioni. Il contenuto e le opinioni espresse sono esclusivamente a nome degli autori delle frasi.
Ultimo aggiornamento pagina:
11 Gennaio 2022
21:49:32