Re: Epistemology 201: The Science of Science

From: Daryl McCullough (stevendaryl3016_at_yahoo.com)
Date: 03/17/05 

Date: 17 Mar 2005 12:32:01 -0800

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.

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? 

--
Daryl McCullough
Ithaca, NY

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