Re: Zenkin's paper on Cantor
From: Daryl McCullough (daryl_at_atc-nycorp.com)
Date: 10/07/04
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Date: 6 Oct 2004 18:36:49 -0700
Eray Ozkural exa says...
>
>Ralph Hartley <hartley@aic.nrl.navy.mil> wrote
>> If you give me a list of real numbers, presented in that way, I can give
>> you a number not on your list.
>
>A timely observation which takes us to the heart of the matter. I will
>argue that I cannot "give you a list".
You can give me a *procedure* which, given a number n
returns the nth real number (between 0 and 1, for simplicity),
which in turn is a procedure which, given a number m, returns
the mth bit of the binary expansion of that real.
Thus a constructive list of reals in [0,1] is equivalent to
a function that takes two natural numbers and returns either 0 or 1.
A function can be finitely described by a computer program. So it
is possible to give me a list of reals.
>Every real on this list is computed by at least a 1-bit program.
>
>Since the list is *actually* infinite
No, it's not.
>*giving* you the *list*
>requires an infinite amount of information
No, it doesn't.
>I cannot send you a list.
Yes, you can. You can give a program for computing the nth
real.
>However, I think, contrary to some seemingly well written constructive
>critique in the talk section of the wikipedia page, Cantor's *first*
>proof does not require infinitary reasoning.
Neither of Cantor's proofs require infinitary reasoning. They are
both constructive.
>Of course, if you so wish, I can "give the list" to you, but I will
>argue that this requires "abstraction of actual infinity".
No, it doesn't.
>If we say that one of the main tenets of intuitionism is to reject
>"abstraction of actual infinity", then it becomes impossible for me to
>"give you a list".
No, it doesn't. Intuitionistic math has no problems dealing with
reals, lists of reals, lists of lists of reals, etc. It's just that
to *prove* the existence of a real, one must give an *algorithm*
for computing the binary expansion.
-- Daryl McCullough Ithaca, NY
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