is squeak really object oriented ?

jan ziak ziakjan at host.sk
Thu May 29 21:47:04 UTC 2003


On Thu, 29 May 2003 11:52:55 -0700, 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?"
> 

in addition to what you said above, i like to think like this:
  - infinity is a symbol
  - we people attribute to the infinity symbol a special algorithm
    which, when performed, would lead us to repeat the same steps till
    the end of our lifes (or till the end of the existence of universe, or
    something like that).

similarily, we can say that:
  - we attribute special algorithms, meanings, to the "countable set"
    and "uncountable set" symbols

if we take this point of view, then it is no miracle that "countable" can be 
put into 1:1 correspondence with "uncountable" because both are just 
individual symbols. another case is what happens when we try to interpret the 
symbols in their special ways.

i hope my explanation seems correct to you and has helped somebody. i think 
related topics to what i have written above are present in the book "godel, 
escher, bach" or in the theory of semiotics (which studies meanings of 
symbols).

> 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
> 






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