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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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Jesse F. Hughes

Posts: 9,776
Registered: 12/6/04
Re: The Distinguishability argument of the Reals.
Posted: Jan 4, 2013 4:36 PM
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Zuhair <zaljohar@gmail.com> writes:

> On Jan 4, 8:13 pm, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
>> Zuhair <zaljo...@gmail.com> writes:
>> > Dear fom I'm not against Uncountability, I'm not against Cantor's
>> > argument. I'm saying that Cantor's argument is CORRECT. All what I'm
>> > saying is that it is COUNTER-INTUITIVE as it violates the
>> > Distinguishability argument which is an argument that comes from
>> > intuition excerised in the FINITE world. That's all.

>>
>> But you've neither explained the meaning of your second premise nor
>> given any indication why it is plausible.
>>

> I did but you just missed it.
>
> My second premise is that finite distinguishability is countable.
>
> What I meant by that is that we can only have countably many
> distinguishable finite initial segments of reals. And this has already
> been proved. There is no plausibility here, this is a matter that is
> agreed upon.


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.

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,

(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 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.

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.


--
"Philosophy, as a part of education, is an excellent thing, and there
is no disgrace to a man while he is young in pursuing such a study;
but when he is more advanced in years, the thing becomes ridiculous
[like] those who lisp and imitate children." -- Callicles, in Gorgias



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