Re: Epistemology 201: The Science of Science

From: Neil W Rickert (rickert+nn_at_cs.niu.edu)
Date: 02/21/05 

Date: Mon, 21 Feb 2005 22:45:46 +0000 (UTC)

Tony Orlow (aeo6) <aeo6@cornell.edu> writes:

>Thank you Allan. You are obviously no less than a first rate intellect
>as far as I can tell. Probably the only way to see beyond Cantor is to
>NOT be a mathematician by trade. I guess my problem here is that I HAVE
>been trying to say infinity+infinity=2*infinity, as long as you're
>talking about the same infinity consistently.

I see that you are still confused.

Let's talk about counting.

Suppose I have a finite set, and I want to count to find it size.
Then I pick an element, and assign the number 1 to that. Next I pick
another element of the set, and assign the number 2 to that. I keep
going. The last number that I assign is the cardinality of the set.

The particular numbers that I am assigning should be considered
ordinal numbers, since their order is significant.

You may notice that I could do this counting in different ways, by
just choosing the elements of the set in a different order. As it
happens, the final number I get will always be the same, independent
of the order of my selections.

Now suppose we try to count an infinite set in the same way. We
start by selecting elements, and assigning them to ordinal numbers.
Doing this is a bit tricky, but it can be done with the aid of
transfinite induction. Then we pick the last ordinal number we used
in our counting procedure.

The trouble is that, with infinite sets, if we do it again me might
get a different answer. The order in which we pick the elements does
turn out to matter.

To avoid this ambiguity, the cardinality of the set is defined to be
the smallest possible ordinal number that you could get with all of
the different ways of counting.

 --------

What has been troubling you is that it seems obvious that if you
count all integers you should get a bigger count than if you count
just the even numbers. But this observation deals only with
particular ways of counting. Since the answer for particular ways of
counting is ambiguous, it doesn't tell us much. Yes, you could count
the integers and get a bigger answer than with just counting the
evens. But then we could try counting the evens in a different order
and get a bigger answer still.

Cardinality is defined as the smallest possible answer you could get,
so as to avoid this confusing ambiguity.

I hope this explanation clarifies things a little.

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