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"