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I know that the standard way of proving that the set of all countable ordinals is uncountable is by stating that if the set is countable, then it incurs Burali-Forti paradox.

Is there other ways of proving this?

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    Not Russell's paradox so much as Burali-Forti's paradox, but they're closely related.2012-10-18
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    Some proofs are given in this question: [Uncountability of countable ordinals](http://math.stackexchange.com/questions/71726/uncountability-of-countable-ordinals). (And you might want to have a look at linked questions, too.)2012-10-18
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    I think this question already has an answer at https://math.stackexchange.com/questions/38468/no-uncountable-ordinals-without-the-axiom-of-choice.2017-10-26
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    This question has nothing to do with the axiom of choice, so I've removed that tag.2017-11-12

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