On Thursday, May 29, 2003, at 08:52 PM, John W.Sarkela wrote:

> 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.
>>
>
>