Re: Some things I'm confused on in math

From: Arturo Magidin (magidin_at_math.berkeley.edu)
Date: 10/17/04 

Date: Sun, 17 Oct 2004 19:47:29 +0000 (UTC)

In article <e62610ea.0410171143.3ea052a6@posting.google.com>,
Joe <jhelfand@umd.edu> wrote:
>Okay, thanks for the responses. I think you made some good points.
>But here are some more attempts at slipperiness:
>
>Many of you said that there are an infinite number of finite sets.
>Okay. First let's list them from one to n:
>
>{1}
>{1,2}
>{1,2,3}
>.
>.
>.
>{1,2,..,n}
>
>This seems obvious. Now let's list them ALL:
>
>{1}
>{1,2}
>{1,2,3}
>.
>.
>.
>{1,2,3,..,Infinity}
>
>It's this last line that I'm worried about.

You should be. There is no "last line", so writing one is sheer
nonsense. Your list should with the elipsis "..."

> Now, this seems like a
>cheap trick. So okay, let's do another thing. List them all in
>incresaing size. We agree there are an infinite number of these sets.
> Let's have the next larger one swallow the smaller one. So for the
>pair:
>
>{1}
>{1,2}
>
>it becomes:
>{1,2}
>
>because the {1} is a subset of {1,2}. Let's do this in pairs,
>starting from the bottom going up. Notice that the larger one, gains
>nothing. It doesn't increase at all in size or the number of
>elements. It has the same quantity of elements. And it is still just
>as finite. Now let's go up to n. Again no problem. The final set is
>just {1,2,3,...,n} and all the bigger ones are still listed after it.
>But now let's do it for ALL the sets, all the ones lined up. Now
>something weird happens, you end up with a set of infinite numbers,
>{1,...,Infinity}.

No: this is the set of all finite number, and it does not contain any
number called "infinity". Like the first list, it has no "last"
element. It just keeps going.

> And yet, where did it go wrong? At which n was the
>mistake made?

The mistake was not made at any n. It was made, by you, after all the
n's were listed and you decided to introduce a new item, which you
called "infinity". There was no reason to introduce it in the first place.

>As for the length of [0,1], I would like to know more about Lebesgue
>Measure, or just Measure, or what ever it's called that deals with
>this problem. Can you point me to some sources? I might actually be
>able to follow it. (I have heard of this stuff before, but I don't
>understand it.) Maybe I should go bother a professor to explain.

Any book called "Measure Theory" will do. However, I suspect it will
be well beyond you right now. Instead, try some book on Mathematical
Analysis as a first step; Rudin's, perhaps, which will cover Riemann
and Riemann-Stieljes integration, might be a good first step. 

-- 
======================================================================
"It's not denial. I'm just very selective about
 what I accept as reality."
    --- Calvin ("Calvin and Hobbes")
======================================================================
Arturo Magidin
magidin@math.berkeley.edu

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