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Re: Mathematics in brief
Posted:
Dec 8, 2012 11:55 PM
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On Dec 8, 10:46 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
> On 8 Dez., 18:55, Zuhair <zaljo...@gmail.com> wrote:
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> > On Dec 8, 1:49 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
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> > > On 8 Dez., 10:41, Zuhair <zaljo...@gmail.com> wrote:
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> > > I will give two answers, the first one depends on the finity of the
> > > universe, the second one does not.
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> > > > Mathematics is "discourse
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> > > and discourse needs the tools available to us. They all are finite.
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> > > > How those forms are known to us? the answer is through their
> > > > exemplification as part of the discourse of consistent theories about
> > > > form.
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> > > Consistent means that discourse about all elements is possible.
> > > We know that all finite words belong to a countable set. We know that
> > > no infinite word can be mentioned without having a finite name.
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> > > Therefore this silly argument is really silly:
> > > "I can decide for a real number x whether a real number y deviates in
> > > its decimal (or any other) expansion from that of x."
> > > The complete infinite expansion of x is never known, not even in an
> > > infinite universe, but only the finite formula allowing expansion to
> > > any required level.
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> > > So the argument is silly that there are uncountably many x because x
> > > has an infinite expansion. No x is known without a finite formula,
> > > name, word, ...
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> > There is no known inconsistency with
> > uncountability
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> I know an inconsistency. And many others know it too. Only people who
> refuse to know it will never know it.
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> For all non-matheologians this is clear: All reals of the unit
> interval are represented as paths in the Binary Tree. All
> distibguishable paths of the Binary Tree belong to a countable set
> because the complete tree can be covered by countably many paths. So
> there is no chance to distinguish further paths (like the famous 1/3)
> by nodes. It can only be distinguished by a finite deifnition like
> "1/3" or "0.010101...". This makes uncountability inconsistent - for
> thinkers who do not refuse to take notice.
You need to prove that the set of all paths is countable, and so far
you didn't present a proof of that. I think using Cantor's argument
one can prove that for any countably many paths of the Binary Tree
there is a path that is not among them and thus establish a proof of
uncountability of those paths.
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> > Mathematics only
> > needs to speak about possible forms.
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> Uncountable sets are impossible forms.
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> > Now whether it is TRUE that we
> > have uncountably many reals in the real world, this is another matter
> > that belong to applied mathematics and not to pure mathematics.
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> Uncountability is already impossible in the ideal world. You can
> believe that you can think of uncountable sets. But you cannot
> distinguish so many elements in principle. So your belief is unfounded
> and non-mathematical. That has nothing to do with reality but with the
> countability of all finite words. In the real world you could not even
> distinguish more than 10^100 elements.
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> Regards, WM
Math is discourse about "possible" form. Uncountability of the reals
is PROVED in very weak fragments of ZFC, actually in PREDICATIVE
fragments of second order arithemtic, which are PROVED to be
consistent. This provide a discourse about form, thus uncountability
of the reals is a possibility! thus it is mathematical, since reals
can be interpreted as forms in the set hierarchy.
Zuhair
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