
Re: The Distinguishability argument of the Reals.
Posted:
Jan 4, 2013 4:36 PM


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 COUNTERINTUITIVE 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

