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Maybe I'm just being a bit stupid with where I'm searching here, but I can't actually find a proper definition of these things wherever I look, books or online. The definition I find most is that it's the order type of a well-ordered set, which wouldn't be so bad if the definition of order type seemed to exist anywhere beyond simply saying two sets have the same order type when they're order isomorphic. That's great but it doesn't really say what an order type actually is, I can't help but think there must surely be a better definition for either ordinals or order type?

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    Have you seen http://en.wikipedia.org/wiki/Ordinal_number, "Von Neumann definition of ordinals"?2011-11-08
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    Wow, I'm giving up on Wolfram and stuff like that, really should have seen that coming, I feel pretty stupid for that now. God only knows how I missed that section. Thanks mate2011-11-08
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    No problem! And don't feel stupid.2011-11-08
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    I think that I have defined ordinals on this very website more than once.2011-11-08
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    Isomorphism classes of well-orderings.2011-11-08
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    I think https://math.stackexchange.com/questions/53770/defining-cardinality-in-the-absence-of-choice?rq=1 is very related to this question but it is not shown as related. It pretty much asks whether you can give a formal definition of a cardinal number in ZF.2018-01-06

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