Re: [Mid] A brief tour of inifinity

><<Well, at some level it's not important. After all, if your house burned
>down tomorrow, the cardinality of space won't help much. But to
>mathematicians, the concept of different infinities is both strange and
>enlightening, in that our ideas about the foundations of even a relatively
>simple thing as numbers gets turned upside-down.>>

> The concept of different infinities may be where we get into
>difficulty. We seem to have different concepts of infinity here. Therefore we
>have erupted into a spate of sophistry which even Occam's Razor may not be
>sharp enough to reveal the truth.

Well, I'm pretty sure the concept of different infinities has to do
with some infinities being larger than others. I am not talking about
"infinity+1" being greater than "infinity", because "infinity+1"
doesn't mean anything.

But there are infinities that are greater than others. Here is an
example.

Consider the set of counting numbers: 1, 2, 3, ... etc. That is one
infinity.

Now consider the set of rational numbers. There are an infinite
number of rational numbers between each consequtive pair of counting
numbers (for example, between the nubmers 1 and 2, there are an
infinite number of ration numbers, such as 3/2, 4/3, 5/4, 6/5, etc.,
and many many more I could think of. Therefore, the infinite set of
rational numbers is larger than the infinite set of counting numbers!

I know, it is hard for me to wrap my brain around also, which is why I
find it fascinating.

> I am prepared to continue with the discussion, however. :-) Sitting
>at home in a wheelchair is more boring than our discussion has been (and less
>enlightening).
>BTW, I do see your point and always have, but do you see my point?

I think so. As long as it is okay to think you point is wrong while
still believing I understand it, then yes, I see your point.

You are saying that (back to our old example) when you multiply
0.999... by 10, that shifts all those infinite numbers to the, um,
left, so that somewhere out to the right, at an infinite distance,
there is one less "9" in 9.999... than in 0.999... . I still maintain
that your conjecture is incorrect, based on my own reasoning and my
readings and classes in the subject. Infinity just doesn't work that
way. If it did, well, then that number wouldn't have an infinite
number of 9's. It would be called finite.

Btw, for the rest of you out there who have been following along.
This is actually a very medieval philosphical discussion! In a way,
we are arguing how many angels can dance on the head of a pin. The
angles are infinite, the pin area is finite, and we are exploring the
relationship between the two.

- Midair

From: "Charles J. Cohen" <charles@eecs.umich.edu>
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Received on Sun Jun 13 23:06:04 1999