First uncountable ordinal

The first uncountable ordinal

ω_{1}

is the set of all countable ordinals.

Table of contents
- First uncountable ordinal is regular First uncountable ordinal
- Tail of the first uncountable ordinal First uncountable ordinal

First uncountable ordinal is regular

The first uncountable ordinal has cofinality

ω_{1}

. Indeed, the supremum of any countable set of countable ordinals is still a countable ordinal, so no countable sequence is cofinal in

ω_{1}

.

Tail of the first uncountable ordinal

Deleting any countable initial segment from

ω_{1}

leaves a well-order of type

ω_{1}

.

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The first uncountable ordinal is denoted by the symbol \(\omega_1\). In the context of set theory and ordinal numbers, \(\omega_1\) represents the smallest ordinal number that is not countable, meaning that it cannot be put into a one-to-one correspondence with the natural numbers (the set of all finite ordinals is denoted by \(\omega\)).

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