# Re: Infinities

carl feynman (carlf@atg.com)
Mon, 17 Nov 1997 12:48:56 -0500

At 01:19 PM 11/14/97 -0500, Harvey Newstrom wrote:

>Here is what I considered. Imagine for the sake of example that we are
>in an infinite, non-closed universe. Shooting a laserbeam into space
>would go out forever and would be considered to have an infinite length
>at the end of an infinite time.
>
>Now imagine a beam projection that is one inch wide, one inch tall, and
>of infinite length. The volume of such a beam would be 1" x 1" x
>infinity". It has one infinity of cubic inches within its volume.
>
>Now imagine a beam projection that is two inches wide, one inch tall,
>and of infinite length. The volume of such a beam would be 2" x 1" x
>infinity". It has two infinity of cubic inches within its volume.
>
>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.

You're right, almost...

You have an intuition that the 2x1 beam is 'twice as big' as the 1x1 beam.
So far so good. The you try to measure their volumes and get infinity in
both cases, and start elaborating a theory about different sizes of
infinity. I'm fairly sure that if you continue in this direction you get
all balled up, and end up without a consistent theory that lets you think
this is to give up measuring by volume. Instead, create a new kind of
measurement that you can call 'Newstrom volume'. This is a measurement of
the cross-sectional area of the thickest bundle of infinitely long lines
you can fit into a given volume. A 1x1x1 cube can't hold any infinite
lines, so it has a Newstrom volume of 0. The 1x1 beam has a Newstrom
volume of one square inch (note that Newstrom volume has units of area!).
The 2x1 beam has a Newstrom volume of 2 square inches. Then your intuition
is preserved, and there's no need to invoke different sizes of infinity.

There's a branch of geometry called 'measure theory' that gives us tools to
how big sets are. The basic concept is a 'measure'. A measure is a
function mapping chunks of a space to numbers, having these properties:

1. The measure of the union of a finite number of non-overlapping chunks
is the sum of their measures.
2. The measure of the intersection of two chunks is less than or equal to
the larger of their measures.
3. The measure of any chunk is zero or a positive number.

A measure gets to restrict what kind of chunks it applies to. For example,
asking for the length of a 1x1x1 cube is not legal. Neither is asking for
the volume of a 1x1xinfinity ray bundle.

Volume, area and Newstrom volume are all measures.

Measure theory is useful for fractal geometry, and for probability and
statistics.

--CarlF

PS. I think I've got those axioms right; it's been a few years, and I
didn't bother looking them up...