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First uncountable ordinal

Smallest ordinal number that, considered as a set, is uncountable

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In mathematics, the first uncountable ordinal, traditionally denoted by $\omega _{1}$ or sometimes by $\Omega$ , 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 $\omega _{1}$ are the countable ordinals (including finite ordinals), of which there are uncountably many.

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

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

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

01Topological properties

Any ordinal number can be turned into a topological space by using the order topology. When viewed as a topological space, $\omega _{1}$ is often written as $[0,\omega _{1})$ , to emphasize that it is the space consisting of all ordinals smaller than $\omega _{1}$ .

If the axiom of countable choice holds, every increasing ω-sequence of elements of $[0,\omega _{1}]$ converges to a limit in $[0,\omega _{1}]$ . The reason is that the union (i.e., supremum) of every countable set of countable ordinals is another countable ordinal.

The topological space $[0,\omega _{1})$ is sequentially compact, but not compact. As a consequence, it is not metrizable. It is, however, countably compact and thus not Lindelöf (a countably compact space is compact if and only if it is Lindelöf). In terms of axioms of countability, $[0,\omega _{1})$ is first-countable, but neither separable nor second-countable.

The space $[0,\omega _{1}]=\omega _{1}+1$ is compact and not first-countable. $\omega _{1}$ is used to define the long line and the Tychonoff plank, two important counterexamples in topology.

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First uncountable ordinal

01Topological properties

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Epsilon number

Large countable ordinal

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