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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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Posts: 18,076
Registered: 1/29/05
Re: The Distinguishability argument of the Reals.
Posted: Jan 5, 2013 7:56 AM
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On 4 Jan., 22:36, "Jesse F. Hughes" <> 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.

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

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".

If these reals were not distinguishable from all other reals, then
they would not be "given". 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?

Regards, WM

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