Infinities

John K Clark (johnkc@well.com)
Sun, 16 Nov 1997 08:13:56 -0800 (PST)

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On Fri, 14 Nov 1997 Harvey Newstrom <harv@gate.net> Wrote:

>It seems obvious to me from a physical standpoint, that the 2 x
>infinity inches volume is twice as large as the 1 x infinity inches
>volume. It would take two of the former added together to equal the
>latter. If you superimposed the 1 x infinity inches volume inside
>the 2 x infinity inches volume, you would have 1 x infinity inches
>volume left over.

Infinite numbers do not obey the same laws of arithmetic that finite numbers
do, however, you do count them, that is determine how big they are, in
exactly the same way, by putting them in a one relationship with something
else. I know that if I can put each of the fingers on my right hand in a one
to one correspondence with some apples and have no apples or fingers left
over, then there must be 5 apples. In the same way I can put the odd integers
in a one to relationship with all the integers, both the odd AND the even,
so there must be an equal number of both.

1 -1
3- 2
5- 3
7- 4
9- 5
.
.

Or you can prove that all lines are composed of the same number of points
regardless of length. Draw 2 parallel lines, a short one and a long one below
it, pick a point midway along the short line but above it.

/\
/ \
/________\
/ \
/________________\

Draw a line from that point to any place on the short line, then continue it
until you hit the long line. You've made a one to one correspondence between
all the points in the short line and all the points in the long line, so they
must have an equal number of points.

But not all infinities are equal. Let's try to put the integers in a one to
one correspondence with all the points in the line from 0 to 1 expressed as a
decimal.

1 - 0.a1,a2,a3,a4,a5 ...
2 - 0.b1,b2,b3,b4,b5 ...
3 - 0.c1,c2,c3,c4,c5 ...
4 - 0.d1,d2,d3,d4.d5 ...
.
.

The trouble is it doesn't work, there are decimals not included, for example,
the point 0.A1,B2,C3,D4,E5 ... where A1 is any digit except a1, B2 is any
digit except b2, C3 is any digit except c3 etc. This point differs in at
least one decimal place with any point in our one to one scheme, we've used
all the integers but there are still points remaining, so there must be more
points on a line than integers.

John K Clark johnkc@well.com

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