Re: Epistemology 201: The Science of Science

From: aeo6 (aeo6_at_cornell.edu)
Date: 03/17/05 

Date: Thu, 17 Mar 2005 15:59:41 -0500

Daryl McCullough said:
> Tony says...
> >
> >Daryl McCullough said:
> >> So, if by
> >>
> >> Are there really infinitely many naturals between any two naturals?
> >>
> >> you mean
> >>
> >> Are there really infinitely many naturals between any two naturals,
> >> according to the usual ordering on rationals?
> >>
> >> Then of course the answer is "no".
>
> >Good. Then this should serve to illustrate that conclusions drawn from
> >cardinality using artificial orderings cannot be generalized onto the normally
> >ordered number systems.
>
> It doesn't matter whether the ordering is artificial or not.
> You cannot draw conclusions about the relative sizes of *sets*
> based on an ordering on that set, whether or not that ordering
> is "natural" or "artificial". Any conclusion you draw about size
> is a fact about the *ordering*, not about the underlying set.
Then the ordering used in cardinality comparisons is what determines the
conclusions of those comparisons. If the goal is to find ordering such that one
can create a 1-1 and onto relation between the sets, then the implied goal is
to prove that infinities are equivalent in this sense, so it is no wonder that
this method draws so few distinctions between the sizes of sets that are
obviously not the same real size. In other words, cardinality tries to prove
all infinities equal, and only becomes interesting when it fails, demonstrating
that there is a difference between infinities.
>
> Here's an illustration: Suppose I have an infinite collection
> of index cards. On each card is written a pair of naturals (x,y)
> with the constraint that y must be greater than zero, and
> x and y can have no common factors. Every pair of naturals
> meeting those constraints appears on some card. What is the size
> of my collection of index cards?
Some small amount less than the square of the number of naturals. It's
essentially the integral of the size of the set of naturals, the way i see it.
>
> --
> Daryl McCullough
> Ithaca, NY
>
> 

-- 
Smiles,
Tony

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