The
philosophy of the
infinite is infinitely
hard to grasp. So the philosophy of infinity
is infinitely inadequate, but here goes... What
is infinity like exactly? It must be a strange
place to visit. If you go to infinity you will
arrive at a land where parallel lines meet.
Baffling? Infinity is baffling. But some questions
about infinity can be solved (chiefly by mathematicians)
and pondering the answers is often like having
one's skull drilled out and stuffed with writhing
trilobites.
Some infinities are bigger than others. In fact
we do know what the smallest infinite number
is. It is the total of all of the integers,
i.e.
0,1,2,3...
infinity.
This infinite number is called
Aleph0.
When you add
1 to
Aleph0
you do not end up with a larger infinite number.
You get the same infinite number. Honestly,
this isn't humbug. All this has been proved
mathematically to be true. Is the number of
positive integers smaller than the combined
number of positive and negative integers? No,
it is the same size,
Aleph0!
There are
Aleph0 even
integers,
Aleph0 odd
integers and
Aleph0 odd
and even integers combined.
2
multiplied by
Aleph0 equals
Aleph0. Indeed
Aleph0
multiplied by
Aleph0 equals
the same number,
Aleph0! There
are an infinite number of prime numbers too:
there are
Aleph0 of them altogether.
But, remember,
Aleph0 is the
smallest infinite number. Two to the
power of
Aleph0 is actually
larger than
Aleph0 and is called
Aleph1. This is surmised to
be the total number of real (decimal) numbers.
(Examples of real numbers are
0.1,
-1097.0768, the
square-root
of two, and
pi).
The number of points on a line is equal to the
number of real numbers, conjectured to be the
infinite number
Aleph1. Bizarrely,
there is a mathematical proof that shows that
this conjecture can never be proved! Incidentally
there are as many points on a line as short
as the width of an atomic nucleus as on a line
spanning the width of the entire universe. This
is equivalent to the fact that there are the
same number of real numbers between
0
and
1 as between
0
and
1,000,000,000,000,000,000,000,000.
Any line contains the same number of points
as any other line. This isn't speculation, it
is a mathematically proven fact. It was proven
by the great Russian Mathematician Georg Cantor.
(To digress, mathematically proven facts are
true, incorruptible, absolute and unchallengeable).
Cantor was the first to discover that some infinities
are larger than others. His ingenious proof
is famous in its own right. Cantor also showed
that the number of points on a line is equal
to the number of points contained in any volume,
so for example, the number of points along a
pubic hair is equal to the number of points
inside the entire Universe.
If some infinities are larger than others then
what is the largest infinity of them all? There
is another paradox here. Any given set can never
be the largest set. This is because the set
of all subsets of a set is larger than that
set, even if that set contains an infinite number
of members. So any given infinite number must
be smaller than another. Mathematics forbids
the concept of a
largest infinite number.
The largest Infinity is like the perfect
woman, unobtainable, inconceivable, non-existent,
tantalising and... a paradox.
Add your comment to this page
 | | | | |
| From: |
Hugh Jassovich | Subject: | 2001-01-24 18:08:35 |
| | | | |
| From: |
Rod Gaskins | Subject: | 2001-12-20 19:40:45 |
| | | | |
| From: |
Wu Li | Subject: | 2003-02-24 08:40:51 |
| | | | |
| From: |
liverbean | Subject: | 2003-03-02 14:35:01 |
| | | | |
| From: |
jamie | Subject: | 2003-07-18 15:43:06 |
| | | | |
| From: |
Clayton Carter | Subject: | 2003-09-05 18:12:59 |
| | | | |
| From: |
ivan | Subject: | 2004-02-01 01:31:05 |
| | | | |
| From: |
Dakota | Subject: | 2004-03-08 02:11:01 |
| | | | |
| From: |
MadPole | Subject: | 2004-04-24 09:47:26 |
| | | | |
| From: |
Clayton Carter | Subject: | 2004-08-06 02:34:28 |
| | | | |
| From: |
Starhawk | Subject: | 2004-08-21 01:04:56 |
| | | | |
| From: |
Benedict Linus | Subject: | 2005-03-13 02:12:46 |
| | | | |
| From: |
al | Subject: | 2005-03-27 15:10:53 |
| | | | |
| From: |
Harry J. Burns | Subject: | 2006-12-06 22:48:39 |
| | | | |
| From: |
Harry J. Burns | Subject: | 2006-12-06 23:14:54 |
| | | | |
| From: |
Harry J. Burns | Subject: | 2006-12-06 23:20:51 |
| | | | |
| From: |
Harry J. Burns | Subject: | 2006-12-07 00:07:09 |
| | | | |
| From: |
Rod Gaskins | Subject: | 2007-02-23 16:31:43 |
| | | | |
| From: |
E+L D=aleph0 | Subject: | 2008-03-19 19:12:58 |
| | | | |
| From: |
E+L,D=aleph0 | Subject: | 2008-03-19 19:38:41 |
| | | | |
| From: |
E+L,D=Aleph0 | Subject: | 2008-03-25 05:18:03 |
| | | | |
help: how to add your comment Page hits: 6887
