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, 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. Indeed, in most constructions ω₁ and ℵ₁ are 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. (See Hartogs number.)

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 [0,ω₁) to emphasize that it is the space consisting of all ordinals smaller than ω₁.

Every increasing ω-sequence of elements of [0,ω₁) converges to a limit in [0,ω₁). The reason is that the union (=supremum) of every countable set of countable ordinals is another countable ordinal.

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