First uncountable ordinal

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 is the order type of an uncountable well-ordered set. It is the supremum (least upper bound) of all countable ordinals. In the von Neumann representation, the elements of ω 1 {\displaystyle \omega _{1}} are the countable ordinals (including finite ordinals), of which there are uncountably many. The cardinality of the set ω 1 {\displaystyle \omega _{1}} is the first uncountable cardinal number, ℵ 1 {\displaystyle \aleph _{1}} (aleph-one). The ordinal ω 1 {\displaystyle \omega _{1}} is thus the initial ordinal of ⁠ ℵ 1 {\displaystyle \aleph _{1}} ⁠. Like all other initial ordinals of infinite cardinals, 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} ⁠. Formally, cardinal numbers are usually represented as their initial ordinals, in which case ω 1 {\displaystyle \omega _{1}} and ℵ 1 {\displaystyle \aleph _{1}} are considered equal as sets. More generally, for any ordinal ⁠ α {\displaystyle \alpha } ⁠, ω α {\displaystyle \omega _{\alpha }} denotes the initial ordinal of the cardinal ⁠ ℵ α {\displaystyle \aleph _{\alpha }} ⁠. The continuum hypothesis (CH) states that ⁠ ℶ 1 = ℵ 1 {\displaystyle \beth _{1}=\aleph _{1}} ⁠ (where ⁠ ℶ 1 = 2 ℵ 0 = | R | {\displaystyle \beth _{1}=2^{\aleph _{0}}=\vert \mathbb {R} \vert } ⁠ is the second beth number), which implies that ⁠ | ω 1 | = | R | {\displaystyle \vert \omega _{1}\vert =\vert \mathbb {R} \vert } ⁠, i.e., the countable ordinals are equinumerous to the real numbers. If CH does not hold, but the axiom...