Re: Epistemology 201: The Science of Science
From: aeo6 (aeo6_at_cornell.edu)
Date: 03/04/05
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Date: Fri, 4 Mar 2005 12:23:50 -0500
Neil W Rickert said:
> Tony Orlow (aeo6) <aeo6@cornell.edu> writes:
> >Neil W Rickert said:
>
> >> Sure. But it is not the natural order of rationals. That natural
> >> order is not suited to counting, and therefore it misleads the
> >> intuition.
>
> >What IS the "natural order" of rationals? Is it any different from the
>
> Bob Kolker answered that.
>
> >> > Can you tell ANYTHING about the cardinality of an
> >> >infintite set if you cannot order the set?
>
> >> If you assume the axiom of choice, then any set can be ordered in a
> >> way suitable for counting (i.e. can be well ordered).
>
> >Answer the question, Neil. Can you compare infinite sets WITHOUT
> >IMPOSING ANY ORDER on what you say is an unordered set.
>
> It depends on what you mean by "compare". You can define cardinality
> using the Schroeder-Bernstein theorem. In that case, you do not make
> any reference to order.
Can you give an example, say, of comparing the integers and the
rationals using this method?
>
> > How do you know
> >you are using the "natural order" of those members?
>
> If you are well-ordering, you often don't know and don't care.
>
>
-- Smiles, Tony
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