First uncountable ordinal

In mathematics, the first uncountable ordinal, traditionally denoted by ${\displaystyle \omega _{1}}$ or sometimes by ${\displaystyle \Omega }$, is the smallest ordinal number that is the order type of a uncountable well-ordered set. It is the supremum (least upper bound) of all countable ordinals. In the von Neumann representation, the elements of ${\displaystyle \omega _{1}}$ are the countable ordinals (including finite ordinals), of which there are uncountably many.

The cardinality of the set ${\displaystyle \omega _{1}}$ is the first uncountable cardinal number, ${\displaystyle \aleph _{1}}$ (aleph-one). The ordinal ${\displaystyle \omega _{1}}$ is thus the initial ordinal of ⁠${\displaystyle \aleph _{1}}$⁠. Like all other initial ordinals of infinite cardinals, ${\displaystyle \omega _{1}}$ is a limit ordinal, i.e. there is no ordinal ${\displaystyle \alpha }$ such that ⁠${\displaystyle \omega _{1}=\alpha +1}$⁠. Formally, cardinal numbers are usually represented as their initial ordinals, in which case ${\displaystyle \omega _{1}}$ and ${\displaystyle \aleph _{1}}$ are considered equal as sets. More generally, for any ordinal ⁠${\displaystyle \alpha }$⁠, ${\displaystyle \omega _{\alpha }}$ denotes the initial ordinal of the cardinal ⁠${\displaystyle \aleph _{\alpha }}$⁠.

The continuum hypothesis (CH) states that ⁠${\displaystyle \beth _{1}=\aleph _{1}}$⁠ (where ⁠${\displaystyle \beth _{1}=2^{\aleph _{0}}=\vert \mathbb {R} \vert }$⁠ is the second beth number), which implies that ⁠${\displaystyle \vert \omega _{1}\vert =\vert \mathbb {R} \vert }$⁠, i.e., the countable ordinals are equinumerous to the real numbers. If CH does not hold, but the axiom of choice (AC) does, then ⁠${\displaystyle \vert \omega _{1}\vert }$⁠, as the smallest uncountable cardinal, is strictly less than ⁠${\displaystyle \vert \mathbb {R} \vert }$⁠. If AC also does not hold then ⁠${\displaystyle \vert \omega _{1}\vert }$⁠ may be incomparable with ⁠${\displaystyle \vert \mathbb {R} \vert }$⁠, but never larger than ⁠${\displaystyle \vert \mathbb {R} \vert }$⁠.

The existence of ${\displaystyle \omega _{1}}$ does not depend on AC, as it can be constructed explicitly as the Hartogs number of ⁠${\displaystyle \omega _{0}=\mathbb {N} }$⁠. More concretely, the set of all well-orderings on ⁠${\displaystyle \mathbb {N} }$⁠ can be constructed as a subset of all binary relations on ⁠${\displaystyle \mathbb {N} }$⁠, and thus applying the axiom of replacement to replace every well-ordering with its order type will give ⁠${\displaystyle \omega _{1}}$⁠.