X-Message-Number: 21979
Date: Sat, 14 Jun 2003 17:40:29 EDT
Subject: Re: infinites

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> 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?
> =>Cheers,
> =>Matthew

I would say the second, because it has the power of the continuous. Infinites 
are sometime counterintuitives. Most think that infinite must be a very large 
number, it is not always so. Think for example about countable infinite: it 
is the next number after the last defined one. For example in modulo numbers, 
the modulo is infinite. Here is an example: The alowed number in modulo 3 are: 
0 1 2. After that you get back to 0 so the endless number suite is: 0 1 2 0 1 
2 0 1 2 0..... in this numbering system 3 is the infinite.

At a fundamental level, the universe may be a spontaneous quantum computer 
unable to define number beyond 0, so here, 1 is the infinite! J. H Conway has 
shown that such a system can give a number field, he called ON2, so you can do 
any computing with this rather limited system. All numbers larger than one are 
Cantor's transfinites here.  ON2 has some strange properties, for example the 
square of a quantity L is not L x L, it is: L x (L + 1), something found as an 
(arbitrary) law in quantum mechanics.

In the set of all integers, the next one after N is N+1, because this process 
can be used again and again, it seems we can't get a number after the last 
integer. If we are at step N, it suffice to write N+0.5 to get the infinite. 

This fractional number can't be defined in the frame of the whole number rule. 
is so one of the possible infinite of the whole numbers. What if we include 
in the number definition the fractional one such 2.5, the irrationals such the 
square root of 2, the transcendantals such pi? Then i, the square root of 

minus one and the basic component of complex numbers is the infinite. Well, 
into account complexes, quaternions, octonions,... what is the infinite? 

Octonions are 8 x 8 matrixes with non associative product: (ab)c is different 
a(bc). Loops have this property with only 5 x 5 matrixes, so loops are the 
infinites of hypercomplexe numbering systems.

The infinite has some specific properties, for example in differential 

geometry, if you start from a vector space E, you can define the dual space E* 
the differential operators. If you do this again, you get E**. Now, if E was 
a flat space with a finite number of dimensions, then E**=E. On the contrairy, 
if E was infinite dimensional, then E** is different from E, E*** is 

different from E*, and so on. If E was the ordinary euclidean space it had a 
number of dimensions This is equaly true of the 4 dimensional relativistic 

space, but that same relativistic space put in the restricted frame of ordinary
euclidean space is infinite because we can't count the dimensions up to four 
here.  Strange it is not?

Yvan Bozzonetti.

"Engineers aren't boring people, we just get excited about boring things."


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