CENTROHD - Huge Dictionaries
Frasi che contengono la parola ordinals
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. 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:
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. 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).
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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
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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, this puts it in class-bijection with the class of all ordinals). So we can freely speak of the
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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
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. 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.
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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
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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
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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.)
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Ultimo aggiornamento pagina:
11 Gennaio 2022
21:56:19