[FOM] characterization of Real numbers? (by Saeed Salehi)

[Jesse Alama]

Hi Saeed,

Your ordered set X has the least upper bound property (this is  
inherited from the least upper bound property of R) and a countable  
dense subset (namely, Q x ([0,1] & Q)).  If the standard  
characterization of R (an ordered set that has the least upper bound  
property and a countable dense subset) is accurate , then it would seem  
that your X is order-isomorphic to the real line.  Perhaps I'm missing  
something; I would be grateful if you would discuss your counterexample  
in more detail.

Warm regards,



Jesse

On Feb 10, 2005, at 12:14, Martin Davis wrote:

> I'm posting this message for Saeed Salehi <saeed at cs.utu.fi> because  
> his attempts to post it contained unacceptable formatting codes.
>
> Martin
> ----------------------------------------------------------------------- 
> ----------------------------------------------------
> Hi everybody.
>
> I tried to prove the below theorem posted by Prof. Friedman for  
> answering the second question of Prof. Shipman to myself, but came up  
> with a counter example. I might be wrong in some point, can anybody  
> please tell me where I am making a mistake?
> Let X=R x [0,1] be the direct product of the reals with the closed  
> interval [0,1] equipped with the lexicographical order. It has the  
> properties mentioned in the theorem with the F function defined by F(  
> (a,i) , (b,j) ) = ( (a+b)/2 , (i+j)/2 ). But X is not order isomorphic  
> to R.
> ****************
> 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
>
> *****************
> 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
>
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