Date: Jan 4, 2013 4:36 PM
Author: Jesse F. Hughes
Subject: Re: The Distinguishability argument of the Reals.
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