This is great!I'd love to read a similar explanation about surreal numbers. Are surreal numbers in N or not? How do we know? (In my limited understanding, they're not easily excluded.)

 The surreals have no relation to any of this. I'm guessing you've confused them with nonstandard natural numbers? In which case the answer is, well, what are we assuming? From the point of view of ZFC, of course there are no nonstandard naturals -- but PA can't prove this. Note that this is from the point of ZFC that PA can't prove this, since PA itself can't even formalize the notion of a nonstandard natural number (if it could, it could prove they don't exist).In any case, surreal numbers are an entirely different system of numbers that exist in ordinary mathematics. As opposed to nonstandard naturals, which are a "what if we look at other ways of doing math?" thing.
 Yes, nonstandard natural numbers are what I meant. Sorry!
 What do you mean by "in N"?If you mean the natural numbers then they definitely aren't "in N" since the reals already have a larger cardinality than the naturals, and the reals are a subset of the surreals.

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