Re: How to do magic with infinity
From: Eray Ozkural exa (erayo_at_bilkent.edu.tr)
Date: 10/07/04
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Date: 7 Oct 2004 12:06:48 -0700
Han de Bruijn <Han.deBruijn@DTO.TUDelft.NL> wrote in message news:<ck36ts$fmi$1@news.tudelft.nl>...
> In message <aa503d8.0410060859.4e6f36bb@posting.google.com>
> Leonard Blackburn:
>
> > Also, facts like that the set of even naturals has the same
> > cardinality as the set of naturals are all quite intuitive.
>
> These are not "facts" and they are not "quite intuitive". See:
>
> http://huizen.dto.tudelft.nl/deBruijn/programs/collatz.htm#CFD
Hmm. You argue rather fiercely against Cantor's way of measuring the
size of infinite sets. Why not calm down a little?
Let me put it in a more acceptable form.
Arguing for or against the existence of a bijection (in the realist
sense) is not the only possible way of comparing the size of two sets:
http://ls.poly.edu/~jbain/philmath/philmathlectures/M05.Cantor.pdf
http://www.mlahanas.de/Greeks/Infinite.htm
http://www.mathacademy.com/pr/minitext/infinity/index.asp
You are referring to the "paradox of even numbers", or "Galileo's
paradox" in philosophy of mathematics. Of course many people must have
felt that this result is paradoxical, just like Russell's paradox. The
funny part: nobody cares to fix this paradox.
The truth is that both methods are valid, and measure different
aspects of multiplicity. However, I too feel that a more complete
account of this matter should be provided. (I will not present my
version before it's complete. It's just a very small draft at the
moment, and I haven't yet proven some key lemmas. So, I don't really
have a position, but I do not believe that we have a satisfactory
answer, either.)
As to what it means that even natural numbers and natural numbers have
the same _cardinality_, I would rather ask you first, because I think
we should find the same answer. (since we are both not realists) There
is a proof. The proof is correct. What does this proof mean to you
(regardless of how you think the size of infinite sets must be
measured)?
Regards,
-- Eray
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