In mathematics, the first uncountable ordinal, 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 are the countable ordinals (including finite ordinals), of which there are uncountably many.

The cardinality is the first uncountable cardinal number (aleph-one is thus the initial ordinal of is a limit ordinal such that are considered equal as sets. More generally, for any ordinal denotes the initial ordinal of the cardinal .

The continuum hypothesis (CH) states that (where is the second beth number), which implies that , i.e., the countable ordinals are equinumerous to the real numbers. If CH does not hold, but the axiom of choice (AC) does, then , as the smallest uncountable cardinal, is strictly less than . If AC also does not hold then may be incomparable with , but never larger than .

does not depend on AC, as it can be constructed explicitly as the Hartogs number of . More concretely, the set of all well-orderings on can be constructed as a subset of all binary relations on , and thus applying the axiom of replacement to replace every well-ordering with its order type will give .

Topological properties

Any ordinal number can be turned into a topological space by using the order topology.

If the axiom of countable choice holds, every increasing ω-sequence converges to a limit. The reason is that the union (i.e., supremum) of every countable set of countable ordinals is another countable ordinal.

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 is first-countable, but neither separable nor second-countable.

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

Bibliography

• Thomas Jech, Set Theory, 3rd millennium ed., 2003, Springer Monographs in Mathematics, Springer, .

• Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. (Dover edition).