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In mathematics, the first uncountable ordinal, traditionally denoted by ω 1 {\displaystyle \omega _{1}} or sometimes by Ω {\displaystyle \Omega } , is the smallest ordinal number that, considered as a set, is uncountable. It is the supremum of all countable ordinals. When considered as a set, the elements of ω 1 {\displaystyle \omega _{1}} are the countable ordinals , of which there are uncountably many.
Like any ordinal number , ω 1 {\displaystyle \omega _{1}} is a well-ordered set, with set membership serving as the order relation. ω 1 {\displaystyle \omega _{1}} is a limit ordinal, i.e. there is no ordinal α {\displaystyle \alpha } such that ω 1 = α + 1 {\displaystyle \omega _{1}=\alpha +1}.
The cardinality of the set ω 1 {\displaystyle \omega _{1}} is the first uncountable cardinal number, ℵ 1 {\displaystyle \aleph _{1}} . The ordinal ω 1 {\displaystyle \omega _{1}} is thus the initial ordinal of ℵ 1 {\displaystyle \aleph _{1}}. Under the continuum hypothesis, the cardinality of ω 1 {\displaystyle \omega _{1}} is ℶ 1 {\displaystyle \beth _{1}} , the same as that of R {\displaystyle \mathbb {R} } —the set of real numbers.
In most constructions, ω 1 {\displaystyle \omega _{1}} and ℵ 1 {\displaystyle \aleph _{1}} are considered equal as sets. To generalize: if α {\displaystyle \alpha } is an arbitrary ordinal, we define ω α {\displaystyle \omega _{\alpha }} as the initial ordinal of the cardinal ℵ α {\displaystyle \aleph _{\alpha }}.