Re: Epistemology 201: The Science of Science

From: Neil W Rickert (rickert+nn_at_cs.niu.edu)
Date: 03/04/05 

Date: Fri, 4 Mar 2005 20:47:07 +0000 (UTC)

Tony Orlow (aeo6) <aeo6@cornell.edu> writes:
>Neil W Rickert said:

>> 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?

This involves defining cardinality in terms of mapping. Set A has
the same cardinality as set B if there is a one-to-one mapping of A
onto B.

Schroeder Bernstein shows that if there is a one-to-one mapping of A
into a subset of B, and if there is a one-to-one mapping of B into a
subset of A, then there is a one-to-one mapping of A onto B.

In the case of integers and rationals, we can map the integers into
the rationals in the obvious way. That is, the integer n maps to the
fraction n/1. To map rationals to integers, we proceed as follows:
given a positive rational, express as a fraction a/b where a and b
are integers, and the fraction is in simplest form (no common divisor
of a and b). We map this rational to the integer (2^a)*(3^b). And
we map -a/b to the negative of this. We map 0/1 to 0.

Then Schroeder-Bernstein shows how to construct a one-to-one mapping
of the integers onto the rationals.

Sorry if the above is a little technical.

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