first uncountable ordinal

In mathematics, the first uncountable ordinal, traditionally denoted by ω₁ or sometimes by Ω, is the smallest ordinal number that, considered as a set, is

uncountable

. It is the

supremum

(least upper bound) of all countable ordinals. The elements of ω₁ are the countable ordinals (including finite ordinals), of which there are uncountably many.

Like any ordinal number (in von Neumann's approach), ω₁ is a well-ordered set, with set membership ("∈") serving as the order relation. ω₁ is a limit ordinal, i.e. there is no ordinal α with α + 1 = ω₁.

The cardinality of the set ω₁ is the first uncountable cardinal number, ℵ₁ ( aleph-one). The ordinal ω₁ is thus the initial ordinal of ℵ₁. Under the continuum hypothesis, the cardinality of ω₁ is the same as that of $\mathbb$—the set of real numbers.

In most constructions, ω₁ and ℵ₁ are considered equal as sets. To generalize: if α is an arbitrary ordinal, we define ω_α as the initial ordinal of the cardinal ℵ_α.

The existence of ω₁ 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, ω₁ is often written as ,ω₁),_to_emphasize_that_it_is_the_space_consisting_of_all_ordinals_smaller_than_ω₁.

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