Re: Epistemology 201: The Science of Science

From: Wolf Kirchmeir (wwolfkir_at_sympatico.ca)
Date: 02/07/05 

Date: Mon, 07 Feb 2005 00:35:10 -0500

Albert wrote:
[...]
>
> Oh, I see. Mathematical jargon. I was hoping for you to tell me how to
> count an infinite number.

Consider the difference between the set of integers and the set of
reals. Both are infinite sets. Integers can be counted. That is, no
matter what integer you propose, you can count exactly one more: Given
N, you can specify that the next integer N' = N + 1 for any N. So the
set is countable. (This is not a rigorous explanation, but it will have
to do. It depends on the notion that counting is a step-by-step listing
of the elements of a set.)

You can't do this with real numbers, since you can't specify what the
next real number in the sequence will be: Given R, what is R'? Given any
R and R', there will always be an R" such that R < R" <R'. Or, no matter
how arbitrarily small you make the difference E between R and R', there
will a number R" such (R' - R") < E and (R" - R) < E. So, the set of
reals is uncountable.

See?

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