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Topic: The Distinguishability argument of the Reals.
Replies: 11   Last Post: Jan 5, 2013 10:30 PM

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Virgil

Posts: 8,833
Registered: 1/6/11
Re: The Distinguishability argument of the Reals.
Posted: Jan 5, 2013 5:17 PM
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In article
<bba43ced-8d33-44b5-b36a-6ad572bff3cb@4g2000yqv.googlegroups.com>,
WM <mueckenh@rz.fh-augsburg.de> wrote:

> On 4 Jan., 22:36, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
>

> > Sure, there's only countably many finite sequences over {0,...,9}, if
> > that's what you mean, but I don't see what that has to do with whether R
> > is countable or not.

>
> So you are blind. Let me try to make you see.


Better blind like Pontryagin then seeing like WM.

> > I thought your error involved something else, namely the following
> > equivocation on distinguishability of a set S.
> >
> > Any pair of reals is finitely distinguishable.  That is,

>
> How do you get a pait of reals?


Have them delivered!
> >
> >   (Ax)(Ay)(x != y -> (En)(x_n != y_n))
> >
> > where x_n is the n'th digit of x.
> >
> > Now, there are two possible definitions of distinguishability for a set
> > S.
> >
> >   A set S is pairwise distinguishable if each pair of (distinct)
> >   elements is finitely distinguishable.

>
> A, I see. Distinct elements must be distingusihable.


If you meant "distinguishable", he did not say that.

"If" does not require that there are any distinct elements that are
finitely distinguishable.

> >
> >   A set S is totally distinguishable if there is an n in N such that for
> >   all x, y in S, if x != y then there is an m <= n such that x_m != y_m.

>
> Nonsense. A real number need not be given by a string of digits.


Then sets containing reals which cannot be given by strings of digits
will not be "totally distinguishable".

So it is WM's nonsense in being able to understand what is being said.

> In
> most cases that is even impossible. Given is a finite definition like
> "pi". And this is distinct from all other real numbers.


Pi can be given, at need, to any finite number of digits, so can be
distinguished in this way from any other real which can be, at need,
given to any finite number of digits.
> >
> > Clearly, the set of reals is pairwise distinguishable but not totally
> > distinguishable.  But so what?  I see no reason at all to think that it
> > *is* totally distinguishable.  The fact that each pair of reals is
> > distinguishable gives no reason to think that the set of all reals is
> > totally distinguishable.


That presumes that every real can be, at need, expressed to any finite
number of digits, which need not be the case.
>
> In order to have a pair of distinct elements (reals), they must be
> finitely defined such that none of them has more than one and only one
> meaning. Already if you have only one "given real", it must be defined
> such that it is observably and provably different from *all* other
> reals, not only from a second "given real".


Since real numbers can have all sorts of extremely complicated
definitions, it may be extermely difficult, or even impossible, to prove
that two such definitions define different numbers.
>
> If these reals were not distinguishable from all other reals, then
> they would not be "given".


That assumes that all definitions easily lead to some method of
comparing, like decimal representations, but real numbers can be defined
in lots of ways which make finding their decimal approximations so
difficult as to be effectively impossible beyond a very few digits of
accuracy.

Then such real numbers cannot be distinguished from any but sufficiently
distant real numbers.




> But in order to have a given real number,
> it must have a finite definition. How else should an infinite string
> of digits be obtained if not from a finite formula?


Finite formulae may involve increasingly impractical complexity beyond
a very few digits of accuracy.
--





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