Re: Epistemology 201: The Science of Science

From: Wolf Kirchmeir (wwolfkir_at_sympatico.ca)
Date: 02/14/05 

Date: Mon, 14 Feb 2005 11:30:57 -0500

Tony Orlow (aeo6) wrote:
[...]
>
> I don't have a better classification for infinities ready at this point,
> but I am thinking about it. I think it may have been Albert that talked
> about seeing infinities as the same size, but with different densities.
> Perhaps the right measure would be, say, the inverse of the rate at
> which the set approaches infinity. In any case, I think the key to finer
> definition of infinities has to do with taking into account the values
> of the elements in the set.

Interesting notion: but it looks to me that you are talking about
infinite series here. I'm not sure that a series is the same as a set in
this context.

> Question: Aren't the reals equivalent to the set of all digitial, say
> decimal, numbers with unlimitied digits to either side of the decimal
> point? For instance, couldn't all real numbers between zero and one be
> represented with a decimal fraction 0.x, where x is an infinite number
> of decimal digits? If so, can't all reals be represented by a matrix of
> integers and such decimal fractions, which map to the integers, much
> like rational numbers are represented by a matrix of integer numerators
> and denominators? And if that's the case, then isn't it possible to map
> the reals onto the integers like one does with the rationals? What am I
> missing here?

Cool, you're just a step away from Cantor's proof that you can't map
reals 1-1 and onto integers.

H'm should I leave for you to work out, or shall I show you how Cantor
did it?

Oh wotthhell. Here goes.

Suppose such a matrix were possible, ie, that for each integer you have
listed a real. Have yopu left reals out of the list? Cantor showed
you've left out at least one, as follows: Take the first digit after the
decimal point from the first real in your list, the second digit from
the second real, the third digit from the thris real, and so on. You
have just constructed a real that nowhere occurs in your list, since
every one of its digits differs from another real at that position of
the digit. But you started with the assumption that you did list every
real. That assumption is contradicted. In fact you can construct as many
reals as you like, add them to the list, and construct even more reals
using the same method. Hence the reals are not countable (enumerable.)

I tell you, when I understood what Cantor had done, the hair on the back
of my neck stood up. Just like it did when I first hear the fourth
movement of Beethoven's Ninth Symphony. Whooohah!

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