```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}, ifthat's what you mean, but I don't see what that has to do with whether Ris countable or not.I thought your error involved something else, namely the followingequivocation 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 setS.    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 totallydistinguishable.  But so what?  I see no reason at all to think that it*is* totally distinguishable.  The fact that each pair of reals isdistinguishable gives no reason to think that the set of all reals istotally distinguishable.-- "Philosophy, as a part of education, is an excellent thing, and thereis 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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