[FOM] Characterization of the real numbers

[JoeShipman@aol.com]

That's a really nice theorem, which is just what I was looking for; just 2 questions:

1) what's the simplest way to formally define "order continuous" for a function from X^2 to X where X is an ordered space? 
2) How do you prove this?

-- JS

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THEOREM. Let X be a linear ordering without endpoints. Then X is order
isomorphic to the real line if and only if

i) X has the least upper bound property;
ii) there is an order continuous F:X^2 into X such that for all x,y, x < y
implies x < F(x,y) < y.

Harvey Friedman