Virgil
Posts:
4,482
Registered:
1/6/11
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Re: The Distinguishability argument of the Reals.
Posted:
Jan 5, 2013 5:32 PM
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In article
<64e948d9-cb53-4798-817e-92e83e0a495f@d10g2000yqe.googlegroups.com>,
WM <mueckenh@rz.fh-augsburg.de> 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
>
> Geometry is as "incomplete" as arithmetic.
But not as incomplete as WMytheology.
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