Re: Epistemology 201: The Science of Science

From: Wolf Kirchmeir (wwolfkir_at_sympatico.ca)
Date: 02/26/05 

Date: Sat, 26 Feb 2005 13:25:24 -0500

Allan C Cybulskie wrote:
> <stephen@nomail.com> wrote in message
[...]
>>You have not defined what "twice as many" means when applied to
>>infinite numbers. Until you do that you cannot make any argument.
>
>
> But I'm not TALKING about infinite numbers, but am talking directly about
> the sets and ignoring what the number actually is. And my analysis shows
> that the set (0,2) BY DEFINITION OF THE SET has more elements than the set
> (0,1), since it has all of the elements that are in the set (0,1) AND an
> exactly equal set of elements extra (1,2). The same proof holds for the set
> (0,3) with respect to (0,1) and the set (0,1.5) with respect to (0,1)
> (although the actual ratio of extra elements changes).
>
> So I don't need to define "twice as many" as applied to infinite numbers,
> since I avoid your whole word game by simply not talking about the actual
> number.
[...]

Your problem is almost exactly what Cantor noticed. He noticed that you
can match every natural number {1, 2, 3, 4,....} with one and exactly
one even number {2, 4, 6, 8, ...}. No matter how large a natural number
you name, there is an even number to match it. Sinct one-to-one matching
is one way of determining whether two sets are the same size, he
realised there was problem with the concept of "set."

Unlike you, Cantor was willing to reconsider the concept, and concluded
that finite and infinite sets are different. That is, certain
consequences of reasoning about infinite sets as if they were finite
sets led to paradoxes (such as, the set {0,....2} contains as many real
numbers as the set {0,....1}. He resolved the paradox by refining the
concept of "matching" (or "mapping"). Non-technically, he said that if
it can be shown that a matching procedure will uniquely pair up every
possible element of one set with every possible element of the another
set, then the two sets match: ie, we say they are the same size. This
refinement of "mapping" applies both to finite and infinite sets, and it
resolves the paradox over which you stumble. Set theory has worked a
number of consequences of that insight. For some reason, you are
unwilling to accept that the consequences are perfectly logical.

To prove that the set N ={1, 2, 3, 4...} does in fact uniquely match the
set E = {2, 4, 6, 8, ...}:
Suppose this is not the case. Then there must be a number in N that does
not match any number in E. Call that number n. According to our rule of
matching, there should be a number e such that 2* = n. But we have
assumed there is no match for n. That means that e cannot be an even
number, else it would be a member of e. Is e an odd number, then? No,
for any number multiplied by 2 is an even number. Therefore e must be
even after all, and our assumption that there is an n in N without a
match in E is contradicted. Therefore every number if N has a match in
E. A similar argument proves the converse, that for every number in E
there is a number in N. Therefore, fo every N there is one and only one
match in E and vice versa. By the mathcing rule, the two sets are the
same size.

However, by your definition of "set", N should have twice as many
members as E, by the same reasoning by which you infer that {0,...2} has
twice as many members as {0,...1}. Hence your definition of "set" is
flawed. More precisely, your notion of the "size" of a set is
self-contradictory.

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