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I've been reading Knuth's Surreal Numbers recently and came up with this question about real numbers.

Is is true that among all three relationships (=, >, <), a real number must be of one, and only one relationship with another real number. If this is true, how to prove it?

wlnirvana
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1 Answers1

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Yes - this is called the trichotomy property ( http://en.wikipedia.org/wiki/Trichotomy_%28mathematics%29 ) . It can be easier to see depending on how you define the real numbers. For instance: http://en.wikipedia.org/wiki/Dedekind_cut (see ordering of cuts part way down the page)

http://en.wikipedia.org/wiki/Trichotomy_%28mathematics%29

Elle Najt
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  • Does this mean i) the trichotomy property is an axiom, thus cannot be proved, or ii) it can be derived once a clear definition of real number is given? Or maybe these two conditions are equivalent? – wlnirvana May 18 '14 at 11:34
  • Well, the integers also have trichotomy, for instance, so they are not equivalent. It can be derived from a definition of the real numbers. If you are interested in questions of this nature, you might want to read Naive Set Theory by Halmos (which deals with some foundational set theory), or the first chapter of Pugh's real analysis, which constructs the real numbers. – Elle Najt May 18 '14 at 15:54