Message from discussion Epistemology 201: The Science of Science

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From: Tony Orlow (aeo6) <a...@cornell.edu>
Newsgroups: comp.ai.philosophy,sci.physics,sci.math,sci.cognitive,sci.philosophy.meta
Subject: Re: Epistemology 201: The Science of Science
Date: Thu, 10 Feb 2005 10:18:28 -0500
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step...@nomail.com said:
> In sci.math Tony Orlow (aeo6) <a...@cornell.edu> wrote:
> : step...@nomail.com said:
> :> In sci.math Tony Orlow (aeo6) <a...@cornell.edu> wrote:
> :> : If you say one set has the same "cardinality" but half the "measure", 
> :> : then you are agreeing that Cantor's measure of cardinality is not a true 
> :> : measure of the "size" of the set. By basic set theory it is a 
> :> : contradiction to say one set is a proper subset containing half of all 
> :> : the elements of another set, and containing the same number of members. 
> :> 
> :> Where does basic set theory say this?
> : Hmmm....maybe not in "basic set theory". It logically follows, though, 
> : that if set A is a proper subset of set B, then set B contains all 
> : members of A and additional members by definition. Those additonal 
> : members constitute a greater cardinality.  
> 
> Not according to the definition of cardinality.  Cardinality is well
> defined, and it logically follows that |A + B| can equal |A| even
> when |B|!=0.  You are not appealing to logic, but to intuition.
> What does your intuition think infinity+1 equals?
Infinity+1 is (1+0)*infinity. Like the zero is nothing compared to the 
one, the one is nothing compared to the infinity. It is therefore 
impossible to distinguish between infinity and infinity+1. Given a 
certain infinity, though, I would say infinity is distinguishable from 2
*infinity or infinity^2. I understand how cardinality is defined pretty 
much. I just think the definition needs to be refined, because some of 
the implications are not just counterintuitive, but patently wrong, in 
my opinion.

> 
> :> 
> :> : There are half as many even integers as there are integers, even both 
> :> : sets are infinite, with the same "cardinality". Cantor's cardinality 
> :> : measure simply indicates a level of infinitude, not an exact size of the 
> :> : infinite set. When the cardinality is the same, it means there is a 
> :> : finite ratio between the sizes of the infinite sets. 
> :> 
> :> You have to define what you mean by the "size of an infinite set" before
> :> you start talking about it.  Apparently you have some definition in mind.
> :> Do you care to share it?  Cardinality is well defined.  As far as I know,
> :> "size of an infinite set" is not well defined.  Until you define that,
> :> how do you know that cardinality is not the exact size of an infinite set?
> :> How can you possibly know anything about a statement containing undefined
> :> terms?
> 
> : Okay, that's fair. Let's say |N| is Aleph(0). In my mind, |E|=|O|=Aleph
> : (0)/2 (evens and odds). The size of the set of all integral multiples of 
> : n would be Aleph(0)/n. I am simply talking about finite ratios between 
> : infinities, as opposed to the infinite ratio Aleph(1)/Aleph(0). We 
> : should be able to talk about a basic infinite equivalent to |N|, and 
> : then finite numbers of infinite partitions of N without losing the fact 
> : that they are proper subset and do not contain all members of the 
> : superset. 
> 
> Well as has been pointed out not all sets with the same cardinality
> have a "finite ratio", and when we look at sets of strings this
> "finite ratio" depends on how we interpret the strings.
> 
> : We should be able to talk about the difference between the 
> : infinity of points in a unit of line vs. the infinity of points in a 
> : square or cubic unit, which are infinitely greater. 
> 
> We can if we want to.  But that is  not what cardinality is about.
> 
> <snip>
> 
> : Am I insane? I don't think so....;)
> 
> No, but you are confusing logic with intuition and size with cardinality.
> 
> Stephen
> 
When people say there are the same number of odd integers as integers in 
general, maybe they are the ones that are confused. If this is the 
conclusion, there is something wrong with the system. I think I have a 
valid point, but you are free to toe the party line as you see fit.
-- 
Smiles,

Tony