X-Message-Number: 21983
Date: Sun, 15 Jun 2003 18:36:07 +0900 (JST)
From: "Matthew S. Malek" <>
Subject: Re: CryoNet #21973 - #21982


> > If there are infinite odd integers, how can there be twice as many even
> > integers?
> First, I didn't say twice as many even integers--I said twice as many
> integers as odd integers-- although the other statement could also be
> proven by appropriate definition.

Actually, I understood your original posting; I simply made a
typographical error in writing my response.  Sorry about that...

As far as your comment about definitions go, this is basically the same as
what I said about math:  Math is mostly about definitions.  Many
mathematical concepts don't exist when strictly translated to the physical
world.  You can show me two bananas, that's for sure.  You can even show
me the square root of two bananas (or something approximate to it).  You
cannot show me the square root of negative two bananas -- or even negative
two bananas.

So, yes, math is about definition.

> How can there be twice as many integers as odd integers? By choosing
> your definition of "twice as many."

Hmmm...  The basic argument, then, for claiming that there are twice as
many integers as odd integers is you can construct a one-to-one mapping
wherein each odd integer (n) can be mapped to a pair of integers (n,n+1).

My argument for claiming that there are just as many odd integers as there
are integers comes from the one-to-one correspondence between each integer
(m) and an odd integer (2m + 1).

Of course, if you take both arguments togethers, then there are twice as
many integers as there are integers.  :)  Of course, this is somewhat
facetious, but I bring it up to note that infinities must be handled
carefully.  The proof that 1 = 2 is a common mathematical sleight-of-hand
that always involves the masked use of an infinity or similar concept
(such as a division by zero).

=>Long Life,

   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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