Smallest ordinal number that, considered as a set, is uncountable
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.
^"Set Theory > Basic Set Theory (Stanford Encyclopedia of Philosophy)". plato.stanford.edu. Retrieved 2020-08-12.
^"first uncountable ordinal in nLab". ncatlab.org. Retrieved 2020-08-12.
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