Actually the upward downward Lowenheim-Skolem-Tarski theorem proves 
that any theory (the real numbers) that has a model of any infinite 
cardinality, has models of all infinite cardinality. In this case, it 
implies that if the reals can be satisfied with an uncountable model, 
then a countable model of the reals exists.

At this point you may say, "But John, how can a theory with an 
uncountable number of elements have a model of countable cardinality? 
Didn't Herr Cantor demonstrate beyond a shadow of a doubt that a 
countable set may not be put into 1:1 correspondence with an 
uncountable set?"

The answer in this case is that the real numbers in a countable model 
are represented not as atomic elements of the model, but rather as an 
iterated process expressed in terms of a countable set of atomic 
elements. In fact, the MathMorphs implementation of algebraic numbers 
work in this way. Given sufficient time, a given representation of an 
algebraic number may present a standard base 10 representation of 
arbitrary precision. But note well, that the expectation of being able 
to produce an exact representation in base 10 of a randomly chosen real 
number is zero. So in this sense, Andres is right, almost every real 
number cannot be represented in base 10 (or any other integral base 
such as 2) in a finite amount of time.
(A consequence of the fact that the measure of the rational numbers on 
any compact interval of the real line is zero.)

Mathematically yours,
:-}> John Sarkela

On Thursday, May 29, 2003, at 11:13 AM, Andres Valloud wrote:

>> NO NO NO NO NO! Come on, get your facts right, this is basic computer
>> theory, BINARY is NOT capable of expressing any number you can think
>> of. Variable BCD yes but binary NO.
>
> No computer system like we know, with countable resources whether
> infinite or not, will be able to represent an uncountable subset of the
> reals, such as [0, 1].
>
> Andres.
>