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, considered as a set, is uncountable. It is the supremum (least upper bound) of all countable ordinals. When considered as a set, the elements of ${\displaystyle \omega _{1}}$ are the countable ordinals (including finite ordinals),^[1] of which there are uncountably many.

Like any ordinal number (in von Neumann's approach), ${\displaystyle \omega _{1}}$ is a well-ordered set, with set membership serving as the order relation. ${\displaystyle \omega _{1}}$ is a limit ordinal, i.e. there is no ordinal ${\displaystyle \alpha }$ such that ${\displaystyle \omega _{1}=\alpha +1}$.

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 ${\displaystyle \aleph _{1}}$. Under the continuum hypothesis, the cardinality of ${\displaystyle \omega _{1}}$ is ${\displaystyle \beth _{1}}$, the same as that of ${\displaystyle \mathbb {R} }$—the set of real numbers.^[2]

In most constructions, ${\displaystyle \omega _{1}}$ and ${\displaystyle \aleph _{1}}$ are considered equal as sets. To generalize: if ${\displaystyle \alpha }$ is an arbitrary ordinal, we define ${\displaystyle \omega _{\alpha }}$ as the initial ordinal of the cardinal ${\displaystyle \aleph _{\alpha }}$.

The existence of ${\displaystyle \omega _{1}}$ can be proven without the axiom of choice. For more, see Hartogs number.