X-Message-Number: 21966
Date: Fri, 13 Jun 2003 23:04:46 +0900 (JST)
From: "Matthew S. Malek" <>
Subject: Re: CryoNet #21955 - #21962


> For example in the simpler case of all integers vs. odd integers,
> their numbers are equal by Cantor's method of counting; but by another
> method of counting there are twice as many integers as odd integers.


If there are infinite odd integers, how can there be twice as many even

"Twice infinity" doesn't really strike me as being a well defined term.

> > it can be proved that there is no more points in a surface that there
> > are in a line.
> Yvan's speculations continue to fascinate, but the tremendous
> successes of mathematical physics must not make us forget that math
> isn't physics and that most mathematical theories, if not all, are
> approximations with limited domains of application. Whether there are
> more points in a surface than a line is a matter of definition.

Well, strictly speaking, mathematics really IS only about definitions.
Math makes up odd terms (e.g. imaginary numbers) and then sets rules that
determine how they are used.

With the surface vs. line question, Yvan is right.
Again, the issue at hand is the size of infinity.
There are infinite points in the line _and_ infinite points in the

As it turns out, there are actually different sizes of infinity, but only
two of them.  Which is the larger infinity:  The number of integers, or
the number of numbers between zero and one?


   Matthew S. Malek        |    "Judging by his outlandish attire, he's
       |     some sort of free-thinking anarchist!"

	"Many young Czechs say that their direct experience with
         communism and capitalism has taught them that the two
         systems have something in common:  they both centralize
         power in the hands of a few, and they both treat people
         as if they are less than fully human."

                                --Naomi Klein (September 2000)

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