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