[FOM] Correction/Characterization of the real numbers

[Harvey Friedman]

On 2/6/05 3:23 PM, "JoeShipman at aol.com" <JoeShipman at aol.com> wrote:

> What is the simplest characterization of the real numbers?  That is, what is
> the simplest description of a structure, any model of which is isomorphic to
> the real numbers?
> 
> A standard way of characterizing the real numbers is "ordered field with the
> least upper bound property".  But do I need to refer to field operations?
> "Dense ordering without endpoints and the least upper bound property" isn't
> sharp enough.  "Homogenous dense ordering with the least upper bound property"
> looks better, except that it doesn't rule out the "long line" (product of the
> set of countable ordinals with [0,1} in dictionary order, with intial point
> removed).  (It also doesn't rule out the reversed long line, or the symmetric
> long line.)
> 
> The best I can do without referring to relations other than the order relation
> is "dense ordering with least upper bound property, isomorphic to any of its
> nonempty open intervals". Can anyone improve on this?
> 
There is a dense linear ordering with the least upper bound property,
isomorphic to any of its nonempty open intervals, which is not separable.
Take the infinite sequences from [0,1] under the lexicographic ordering.

THEOREM. Let X be a linear ordering with left and right endpoints. Then X is
order isomorphic to the usual closed interval 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.

Order continuity can be defined nicely: for all open intervals I about
F(x,y), there exist open intervals J,K about x,y respectively, such that F[J
x K] containedin I.

Harvey Friedman