Quote:
Originally posted by Grothendieck
Nope KnifeMissle. The "trichotomy" theorem is the definition of a total order. It can and is and must be proven in both the case of set theory, and the real numbers. The Cantor-Bernstein theorem is *exactly* the proof in the case of sets.
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While I can't say for sets, please double check your assertion to the set of real numbers. Doing a google search, I can find references to the trichotomy property, or the trichotomy law, but no one refers to it as a theorem nor does anyone provide a proof. Indeed, it seems hard to construct one. I really do remember it being an axiom of the real numbers, much like the least upper bound axiom (or principle).