In article <firstname.lastname@example.org>, WM <email@example.com> wrote:
> On 4 Jan., 19:33, fom <fomJ...@nyms.net> wrote: > > 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. > > A definition has to be realizable.
Why, when you ignore so many of them when they are? > > > 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. > > Poincaré was also very early. > > > > 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. > > Classical mathematics has been introduced by many from Pythagoras to > Kronecker and Weierstraß, but not by Cantor.
One can only have something introduced to the world once. > > > > That is the pragmatic problem. The theoretical problem > > is that mathematicians are confronted with the science > > of mathematics as a logical system. > > Mathematics and logic as a science recognize that there are only > countably many names, or better, that there are only potentially > infinitely many names.
But a lot more things that, at least in theory, need names, but, in practice, don't. > > > If a completed > > infinity is ground for a system of names reflecting > > geometric completeness, then its investigation is an issue. > > > If this investigation shows that completed infinity is self- > contradictory, then a science should accept this result.
But mathematics is NOT a science. Its truths and values are in no way dependent on physical experimentation or scientific observations of the physical world