First uncountable ordinal

Summary

In mathematics, the first uncountable ordinal, traditionally denoted by or sometimes by , 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 are the countable ordinals (including finite ordinals),[1] of which there are uncountably many.

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

The cardinality of the set is the first uncountable cardinal number, (aleph-one). The ordinal is thus the initial ordinal of . Under the continuum hypothesis, the cardinality of is , the same as that of —the set of real numbers.[2]

In most constructions, and are considered equal as sets. To generalize: if is an arbitrary ordinal, we define as the initial ordinal of the cardinal .

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

Topological propertiesEdit

Any ordinal number can be turned into a topological space by using the order topology. When viewed as a topological space,   is often written as  , to emphasize that it is the space consisting of all ordinals smaller than  .

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

The topological space   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.

The space   is compact and not first-countable.   is used to define the long line and the Tychonoff plank—two important counterexamples in topology.

See alsoEdit

ReferencesEdit

  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.

BibliographyEdit

  • 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).