
Re: Problems with Infinity?
Posted:
Feb 24, 2013 5:01 PM


On Feb 24, 12:00 am, Don Kuenz <garb...@crcomp.net> wrote:
> Let's say a Mobius strip goes to infinity "feedback style" (in layman's > terms) while a line goes to two separate but equal infinities "linear > style." How many different infinities does that make according to > Cantor? One, two, or three?
According to Cantor, the number of points on a line, or on a circle, has the cardinality of the continuum.
The length of a line is alephnull finite units of measure, on the other hand  and the length of a Mobius strip is finite, the distance it takes for you to get back where you started.
Cantor's infinities, as was noted, aren't really about things like that.
Basically, alephnull is the first kind of infinity  the number of integers in the set {1, 2, 3, 4, 5, 6... }.
It can be proven that one can pair off all the integers with that set:
1 <> 0 2 <> 1 3 <> 1 4 <> 2 ...
and so including the negative numbers doesn't really "double" that infinity. In fact, even the rational numbers can be paired off with the integers.
Cantor's diagonal proof shows, though, that there are more *real numbers* than integers in a very real and unavoidable sense.
http://www.quadibloc.com/math/infint.htm
discusses the subject in more detail than I can do in a post.
John Savard

