Re: TOE Via Cantor's Transfinite Arithmetic

From: bz (bz+sp_at_ch100-5.chem.lsu.edu)
Date: 03/21/05 

Date: Mon, 21 Mar 2005 21:09:36 +0000 (UTC)

"Randy Poe" <poespam-trap@yahoo.com> wrote in news:1111425198.661167.288600
@o13g2000cwo.googlegroups.com:

>
> bz wrote:
>> The set of points on a line is infinite but the set of points on two
> lines
>> contains twice as many points. The set of points on two lines can not
> be
>> mapped onto a single line.
>
> Uh, yes it can. The set of ordered pairs (x,y) of real
> numbers has the same cardinality as the set of real numbers,
> in the usual sense that there exists a bijection between
> the two sets.

If I start mapping points from line 1 onto the test set, call it line T,
I can't also map the points from line 2 onto the test set. There isn't room
for it. I have already used up all the points on line T.

Show me where my reasoning is wrong. If I remember the terminology
correctly, Aleph nul, Aleph one, Aleph two are three different orders of
infinity, the first for points on a line, the second for points in a plane
and the third for points in space.

I will have to dig up that book on elementry algebra (which was NOT what I
would call elementry, for sure) and see if my memory is correct. Of course
in the 30 years since I took the course, math may have changed.

>
>> The set of points on a plane is a higher order of infinity than the
> set of
>> points on a line.
>>
>> The set of points in a solid is a higher order of infinity than the
> set of
>> points in a plane.
>
> Actually, these statements are false (surprisingly).
>
>> One can NOT map the set of points in a plane to the set of points on
> a
>> line, one needs an infinite set of lines in order to map the set of
> points
>> in a plane.
>>
>> One can NOT map the set of points in space to the set of points in a
> plane,
>> one needs an infinite set of planes in order to map the set of points
> in
>> space.
>
> No, there exist bijective mappings for both of these.
>
> There are many properties of infinite sets that are
> counterintuitive.

I have no doubt of that. However, I do NOT follow your reasoning.

> That leads to anti-Cantor crackpots
> in math discussion groups in the same way that the
> counterintuitive nature of SR and GR lead to anti-Einstein
> crackpots in sci.physics.
>
> - Randy

Show me. 

-- 
bz
please pardon my infinite ignorance, the set-of-things-I-do-not-know is an 
infinite set.
bz+sp@ch100-5.chem.lsu.edu   remove ch100-5 to avoid spam trap

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