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I just read that there are an uncountable number of countable ordinals.

How the hell is that possible?

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    ... and there is an infinte number of finite ordinals. So what? – 2017-02-25
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    Hagen's comment here is really the thing to think about - if you're comfortable with there being infinitely many finite ordinals, then that should suggest that the situation may be the same with countable ordinals. Hagen's answer then constitutes a proof that this is indeed the case (modulo the claim that the set of countable ordinals is itself an ordinal, which is a good exercise). (Also, the language of your question could be a bit more polite.) – 2018-01-27

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The set(!) of countable ordinals is itself an ordinal (well-ordered and transitive). If this ordinal were countable, it would be an element of itself.

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    contradicting the [axiom of regularity](https://en.wikipedia.org/wiki/Axiom_of_regularity) – 2017-02-25