thomas (12/05/82)
Further explanation.
1. Yes, the sets used in the proof are quite obnoxious. What he is
trying to say is that you can't define an area function which works
for ALL sets. Thus "Lebesgue measurable" and friends.
2. You can't do this paradox on the infinite plane. I think that Banach
proved this back in the 20's, you have to be imbedded in 3-space or
worse.
3. When I get out from under moving our machine, I'll write up the
"Banach-Tarski" paradox, the culmination of which is:
"Given two sets in 3-space, both of which have some interior points, one
of them can be cut into a finite number of (quite nasty looking) pieces
and reassembled via rigid motions into the other."
(You can cut a pea up and reassemble it into the sun!)
=Spencer