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.
> 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. > > 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.
> 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. And if the identity of numbers defined by different definitions cannot be proved (or if nobody is interested like in case of 1+1+1 and 3, for instance), then this fact will not increase the number of definable numbers. Then there are simply less numbers than different definitions, finite definitions, of course. That is a result of logic.