fom
Posts:
1,031
Registered:
12/4/12
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Re: The Distinguishability argument of the Reals.
Posted:
Jan 4, 2013 1:33 PM
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On 1/4/2013 10:29 AM, WM wrote:
> On 4 Jan., 01:35, fom <fomJ...@nyms.net> wrote:
>
>> Dedekind cuts define all reals.
>
> Every cut is defined by a finite word. The set of definable cuts is
> the set of cuts and is countable.
>
>> Cantor fundamental sequences define all reals.
>
> No infinite definition defines anything.
No infinite definition is finitely realizable.
The problem is the use and interpretation of "all".
Dedekind and Cantor speak of "systems." It was
Russell and Wittgenstein who tried to ground
systems so that "all" had a more definite conception.
Russell did not confine his logic by the introduction
of names (it was, in fact, designed that way so that
one could speak of non-existents without presupposition
failure).
Wittgenstein was a finitist. To my knowledge, he is the
earliest author to point out that Cantor's proof was as
much an indictment of the use of "all" as it was a
proof of an uncountable infinity.
Neither Russell or Wittgenstein (or Skolem, for that
matter) has given a system that is useful for the
exercise of empirical science. Computational models
are obscuring that fact, but even a modest introduction
to numerical analysis explains the role of classical
mathematics behind those models.
That is the pragmatic problem. The theoretical problem
is that mathematicians are confronted with the science
of mathematics as a logical system. If a completed
infinity is ground for a system of names reflecting
geometric completeness, then its investigation is an issue.
>
>> You may, as WM does, deny uses of a completed infinity.
>
> I do not deny it, but show that it is self-contradictory.
That may be. Your proofs, however, lie with the nature
of models and not with the nature of how a deductive
calculus relates to definitions and axioms. In that
sense you are not speaking of self-contradiction. Rather,
you speak of the ill-foundedness of trees having
infinite branches.
To be honest, I prefer your contemptuousness for
it over the kind of crap that was published in the
popular book "Goedel, Escher, Bach"
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