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Comment by kaba0 | Hacker News Reader

It is false - real numbers fulfill the trichotomy property, which is precisely the lack of such a relationship: every two real number is either less than, equal or greater than.

But the numerical context can still be correct: (edit: ~~imaginary~~) complex numbers for example don’t have such a property.

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As I've written in another comment here, a great example of a number-y field which is not totally ordered is Games ⊂ Surreal Numbers ⊂ ℝ. There you have certain "numbers" which can can confused with (read incomparable) whole intervals of numbers. Games are really cool :)

You've got the subset relationships backwards: Reals are a subset of the field of Surreal Numbers which is a subset of the group of Games. (Probably better phrased as embeddings, rather than subsets, but the point remains...)

Note that Games do _not_ form a field: there is no general multiplication operation between arbitrary games.

i'm learning about this right here as I read, but do you mean complex numbers rather than imaginary?

Yep, that’s what I meant, sorry!

Same question.

Or more generally, vectors. They don't have a total order, because as we define "less than"/"greater than" in terms of magnitude (length), this means for any vector V (other than 0), there's an infinitely many vectors that are not equal to V, but whose length is equal to length of V.

Is this is what ≹ is talking about?

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