From: utzoo!decvax!yale-com!leichter Newsgroups: net.math Title: Re: Squares - (nf) Article-I.D.: yale-com.774 Posted: Wed Feb 2 09:30:45 1983 Received: Thu Feb 3 03:40:54 1983 References: ihlpb.284 No, that's not right; the "paradox" is really there. A one-to-one correspon- dence between the points has nothing whatsoever to do with area; any two squares in the plane, or volumes in 3-space, have Aleph-1 points. Area is a much more sensitive measure. What the "paradox" shows is that the generalized definition of area, while it accords well with intuition for finite objects, breaks down for infinite objects - actually, of course, the problem is not with the defini- tion but with our intuition! If you really want to look at it as PHYSICAL paint, then consider: Paint is made of atoms of some finite size. Hence, when you pour paint into the horn, it cannot get "below" the point where the nozzle is narrower than an atom. On the other hand, a finite thickness paint will, after a certain point, cover the outside of the horn with a constant-diameter tube (two atoms across at least). Hence, even physically, painting the outside "as far as you can go" takes an infinite amount of paint, but filling "as far as you can go" takes only a finite amount. This last argument is a pretty picture, but has NOTHING to do with area or volume. Area and volume are mathematical constructs that are useful because when applied to real-world objects - which are always finite - they provide useful values - i.e. they predict how much paint you need to cover a wall. BTW, there is a error in my original description. I said to consider the curve y = 1/x for all x>0. There is an infinite area and volume, though, in any section of the resulting horn that gets arbitrarily close to 0, though, because 1/x is blowing up there. You have to cut the curve off at some particular positive x - say, x = 1; any value will do. -- Jerry decvax!yale-comix!leichter