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In mathematics, the first uncountable ordinal, traditionally denoted by or sometimes by , is the smallest ordinal number that, considered as a set, is uncountable.It is the supremum (least upper bound) of all countable ordinals.
When λ = ω (the first infinite ordinal), the space [0,ω) is just N with the usual (still discrete) topology, while [0,ω] is the one-point compactification of N. Of particular interest is the case when λ = ω 1, the set of all countable ordinals, and the first uncountable ordinal.
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, n th, etc.) aimed to extend enumeration to infinite sets. [ 1 ] A finite set can be enumerated by successively labeling each element with the least natural number that has not been previously used.
For each Borel set B, there is some countable ordinal α B such that B can be obtained by iterating the operation over α B. However, as B varies over all Borel sets, α B will vary over all the countable ordinals, and thus the first ordinal at which all the Borel sets are obtained is ω 1, the first uncountable ordinal.
Also, 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 is the order type of that set), is the smallest ordinal whose cardinality is greater than , and so on, and is the limit of for natural ...
Ordinal number – Generalization of "n-th" to infinite cases (the related, but more formal and abstract, usage in mathematics) Ordinal data, in statistics; Ordinal date – Date written as number of days since first day of year; Regnal ordinal – Ordinal numbers used to distinguish among persons with the same name who held the same office
Under this definition, the first uncountable ordinal can be enumerated by the identity function on so that these two notions do not coincide. More generally, it is a theorem of ZF that any well-ordered set can be enumerated under this characterization so that it coincides up to relabeling with the generalized listing enumeration.
ℵ 1 is, by definition, the cardinality of the set of all countable ordinal numbers. This set is denoted by ω 1 (or sometimes Ω). The set ω 1 is itself an ordinal number larger than all countable ones, so it is an uncountable set. Therefore, ℵ 1 is distinct from ℵ 0.