First Uncountable Ordinal

In mathematics, the first uncountable ordinal, traditionally denoted by $\omega_1$ or sometimes by $\Omega$, 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 $\omega_1$ are the countable ordinals (including finite ordinals), of which there are uncountably many.

Like any ordinal number (in von Neumann's approach), $\omega_1$ is a well-ordered set, with set membership serving as the order relation. $\omega_1$ is a limit ordinal, i.e. there is no ordinal $\alpha$ such that $\omega_1 = \alpha+1$.

The cardinality of the set $\omega_1$ is the first uncountable cardinal number, $\aleph_1$ (aleph-one). The ordinal $\omega_1$ is thus the initial ordinal of $\aleph_1$. Under the continuum hypothesis, the cardinality of $\omega_1$ is $\beth_1$, the same as that of $\mathbb$—the set of real numbers.

In most constructions, $\omega_1$ and $\aleph_1$ are considered equal as sets. To generalize: if $\alpha$ is an arbitrary ordinal, we define $\omega_\alpha$ as the initial ordinal of the cardinal $\aleph_\alpha$.

The existence of $\omega_1$ can be proven without the axiom of choice. For more, see Hartogs number.

Topological properties

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