First uncountable ordinal 📖 Wikipedia

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 an 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),^[1] 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 ⁠${\displaystyle \vert \mathbb {R} \vert }$⁠.^[2]

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 ⁠${\displaystyle \omega _{1}}$⁠.

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

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

The topological space ${\displaystyle [0,\omega _{1})}$ 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, ${\displaystyle [0,\omega _{1})}$ is first-countable, but neither separable nor second-countable.

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

• Epsilon numbers (mathematics)
• Large countable ordinal
• Ordinal arithmetic