Re: Epistemology 201: The Science of Science

From: aeo6 (aeo6_at_cornell.edu)
Date: 03/07/05 

Date: Mon, 7 Mar 2005 09:41:40 -0500

Neil W Rickert said:
> 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.
>
>
Thanks for the explanation. However, this doesm't seem different in a
significant way from ordering the sets to achieve your mapping. Whereas
with integers and evens we had a symmetrical mapping based on an
arithmetic function, E(i)=i*2 and I(e)=e/2, here you are simply using
two different arithmetic functions to map the rationals and integers. I
am not sure how you would express the Cantorian diagonal approach to
mapping integers onto rationals as a function to achieve 1-1
correspondence, and it's probably not possible, so in this case you are
using two different functions neither of which provide 1-1
correspondence, but together show a mapping in each direction, because
this achieves what Cantor's counting approach does, which as I've said I
find lacking.

Sure, you can use a mapping of A/B(i)=R(i)=i/1 and I(a/b)=2^a*3^b, but
since this mapping is not symmetrical, it describes even less precisely
the relationship between the sets. You are still using the values of the
members, which are ordered cardinalities, to create functions for your
mappings, so this is not an example of comparing infinite sets without
ordering the set members. It just hides the ordering in terms of an
arithmetic mapping. Does this seem like a valid objection?

Aain, I ask, can you compare infinite sets WITHOUT ordering the members
in some fashion? If not, doesn't the method of ordering say anything
about the relationship between the sets? 

-- 
Smiles,
Tony

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