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.

Topological properties

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.

See also

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

References

1. "Set Theory > Basic Set Theory (Stanford Encyclopedia of Philosophy)". plato.stanford.edu. Retrieved 2020-08-12.
2. "first uncountable ordinal in nLab". ncatlab.org. Retrieved 2020-08-12.

Bibliography

• Thomas Jech, Set Theory, 3rd millennium ed., 2003, Springer Monographs in Mathematics, Springer, ISBN 3-540-44085-2.
• Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. ISBN 0-486-68735-X (Dover edition).