Date: Feb 25, 2013 5:20 AM
Author: Brian M. Scott
Subject: Re: Problems with Infinity?
On Sun, 24 Feb 2013 22:23:25 -0800 (PST), "Ross A.

Finlayson" <ross.finlayson@gmail.com> wrote in

<news:c49c5ae4-834b-4791-a52e-35f9a7f6dbfa@h6g2000pbt.googlegroups.com>

in rec.arts.sf.written,sci.math:

> On Feb 24, 2:01 pm, Quadibloc <jsav...@ecn.ab.ca> wrote:

>> 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 aleph-null 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, aleph-null 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.

> Discussed the subject in somewhat more detail than a single post.

> Arrange real numbers in a line this way. For naturals n from 0 to d,

> arrange n/d, in the order of the naturals. As d (simply enough for

> denominator) diverges to infinity, there are arrayed points between

> zero and one, with a constant, infinitesimal difference from one to

> the next, uniformly between zero and one.

Good grief. Is that crackpot still around?

[...]

Brian