Re: Epistemology 201: The Science of Science

From: aeo6 (aeo6_at_cornell.edu)
Date: 02/23/05 

Date: Wed, 23 Feb 2005 10:16:10 -0500

Neil W Rickert said:
> 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.
>
>
Thanks, Neil. that makes sense, yet I don't think it contradicts my
point. Perhaps measure theory has already done what I am proposing, and
if so, then I'd be interested in learning more. I don't really find
Cantorian cardinality of infinite sets to be very useful beyond
demonstrating that there are indeed at least a few varieties of
infinity, which had never been demonstrated rigorously before. That was
quite an achievement, an excellent start, but not the final word. Do you
disagree with that? 

-- 
Smiles,
Tony

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