Date: Jan 4, 2013 1:33 PM
Author: fom
Subject: Re: The Distinguishability argument of the Reals.
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"