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Topic: The Distinguishability argument of the Reals.
Replies: 83   Last Post: Jan 7, 2013 12:58 AM

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 mueckenh@rz.fh-augsburg.de Posts: 18,076 Registered: 1/29/05
Re: The Distinguishability argument of the Reals.
Posted: Jan 5, 2013 6:44 AM
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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.

Geometry is as "incomplete" as arithmetic.

Regards, WM

Date Subject Author
1/1/13 Zaljohar@gmail.com
1/2/13 mueckenh@rz.fh-augsburg.de
1/2/13 Virgil
1/2/13 Ralf Bader
1/3/13 Virgil
1/3/13 Zaljohar@gmail.com
1/3/13 gus gassmann
1/3/13 Zaljohar@gmail.com
1/3/13 gus gassmann
1/3/13 Zaljohar@gmail.com
1/3/13 mueckenh@rz.fh-augsburg.de
1/3/13 Virgil
1/3/13 fom
1/4/13 Zaljohar@gmail.com
1/4/13 fom
1/3/13 mueckenh@rz.fh-augsburg.de
1/3/13 Virgil
1/3/13 fom
1/3/13 Virgil
1/4/13 gus gassmann
1/4/13 mueckenh@rz.fh-augsburg.de
1/4/13 fom
1/5/13 mueckenh@rz.fh-augsburg.de
1/5/13 Virgil
1/5/13 fom
1/4/13 Virgil
1/5/13 mueckenh@rz.fh-augsburg.de
1/5/13 Virgil
1/4/13 Virgil
1/4/13 gus gassmann
1/4/13 ross.finlayson@gmail.com
1/5/13 Virgil
1/5/13 ross.finlayson@gmail.com
1/5/13 Virgil
1/5/13 fom
1/5/13 ross.finlayson@gmail.com
1/6/13 fom
1/6/13 ross.finlayson@gmail.com
1/6/13 Virgil
1/6/13 ross.finlayson@gmail.com
1/6/13 Virgil
1/6/13 ross.finlayson@gmail.com
1/6/13 Virgil
1/6/13 ross.finlayson@gmail.com
1/6/13 Virgil
1/6/13 ross.finlayson@gmail.com
1/6/13 Virgil
1/7/13 ross.finlayson@gmail.com
1/7/13 Virgil
1/3/13 fom
1/3/13 fom
1/4/13 mueckenh@rz.fh-augsburg.de
1/4/13 fom
1/5/13 mueckenh@rz.fh-augsburg.de
1/5/13 Virgil
1/5/13 fom
1/6/13 Virgil
1/6/13 fom
1/6/13 Virgil
1/6/13 fom
1/6/13 ross.finlayson@gmail.com
1/4/13 Virgil
1/3/13 mueckenh@rz.fh-augsburg.de
1/3/13 Virgil
1/3/13 forbisgaryg@gmail.com
1/3/13 Virgil
1/4/13 Zaljohar@gmail.com
1/4/13 Virgil
1/4/13 Zaljohar@gmail.com
1/4/13 mueckenh@rz.fh-augsburg.de
1/4/13 fom
1/5/13 mueckenh@rz.fh-augsburg.de
1/5/13 fom
1/5/13 mueckenh@rz.fh-augsburg.de
1/5/13 Virgil
1/5/13 fom
1/5/13 Virgil
1/4/13 Virgil
1/3/13 mueckenh@rz.fh-augsburg.de
1/3/13 Virgil
1/4/13 mueckenh@rz.fh-augsburg.de
1/4/13 fom
1/4/13 Virgil
1/2/13 Bill Taylor

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