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

[Harvey Friedman]

On 2/10/05 3:14 PM, "Martin Davis" <martin at eipye.com> wrote:

> 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.
> ****************

I wrote:

> 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.
> 
Your F is not continuous. Suppose F is continuous.

Note that the limit of (1/2,1),(2/3,1),(3/4,1)... is (1,0).

So the limit of F((1/2,1),(1,1)),F((2/3,1),(1,1)),F((3/4,1),(1,1))... is
F((1,0),(1,1)) = (1,1/2).

However this latter limit is the limit of (3/4,1),(5/6,1),(7/8,1),..., which
is (1,0). This is definitely not (1,1/2).

Harvey Friedman