Re: An infinite number question




On 1/22/06 8:41 AM, in article 147937287.00008523.008.0001@xxxxxxxxxxxxxxx,
"TomS" <TomS_member@xxxxxxxxxxx> wrote:

> "On 21 Jan 2006 21:39:17 -0800, in article
> <1137908356.983150.128730@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>, Andrew McClure
> stated..."
>>
>> The problem is that there is not just one mathematical definition of
>> "infinite". There are a couple of different mathematical concepts that
>> can be described by the word "infinity". Only two of these concepts
>> describe "numbers" such that we can use words like "infinite number" to
>> refer to them, so if you say the words "infinite number" you're
>> referring to one of these two things. The first thing is an "ordinal
>> infinity", and it refers to an infinity among numbers that are used to
>> order things or describe a position in a list. The second thing is a
>> "cardinal infinity", and it refers to an infinite quantity-- a number
>> that is used to describe the number of items in a set. Mathemeticians
>> prefer to refer to these "infinite" numbers as "transfinite numbers".
> [...snip...]
>
> I'd just mention that there is a third infinity in mathematics:
> The infinity that is an extension to the real (or complex) numbers.

Actually, that infinity is the same as the infinity of the real number line
(the uncountable infinity).

Riemann demonstrated this with something called the Riemann sphere. A
sphere, you understand, has a finite area. The sphere sets on top of an
infinite flat plane, and a line is passed from the top of the sphere to a
point on the plane.

Yes, in order to do that the line must pass through the sphere. And this
creates a one to one correspondence between the sphere and the points on the
plane itself. For you can never move the line to another point on the plane,
no matter how close, without moving the whole line and thus moving to
another point on the sphere.

Thus the points on a finite area can be mapped one to one to the points on
an infinite area, and we have the same cardinality (or number of points).

It is strange, isn't it?

Cantor did something worse. He took a line of length 1, and removed the
middle 1/3 of it with open intervals, leaving [0, 1/3], [2/3, 1] as the
remaining part of the line. (Remember, the mass of any particular point is
0, so the length of (1/3, 2/3) = the length of [1/3, 2/3]. He then continued
the process in infinite succession, removing effectively the entire length
of the line.

And yet it can be proven that the points that are left on this line segment
with no mass are as many as the points that were taken away. (I am not going
to give the proof here, but it works.)

This devilish piece of mathematics threw mathematicians into a tizzy. They
could not deny the proof. The Cantor Set was equivalent to the real numbers,
an uncountable infinity.

>
> When you're dealing with ordinal or cardinal numbers, the notion
> of division isn't fully applicable, even in the finite case. For example,
> there is no cardinal number which is the answer to 2 divided by 3.
> Cardinal and ordinal numbers are "whole numbers" only.
>
> Infinity, as an extension to the real numbers, is undoubtedly
> what the question is about. And to handle that properly, you need,
> at least, something like calculus - and, I suspect, some advanced
> topics in what are called "analysis" and "measure theory".
>

Those are "fun" courses. Your hair gradually burns away because of an
overheated brain. What doesn't burn gets pulled out trying to do the
exercises.

Regards,

Raymond E. Griffith

.

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