Uncountable ordinal

There exist uncountable ordinals.

Idea

If there’s an uncountable ordinal then there’s a least
Then is the set of all countable ordinals,
i.e. Order Type of Well-ordered of subsets of

Proof

Then the set
consists of all countable Ordinals.

Let .
If is countable, then is also countable and so .
Hence which is a contradiction.