Re: Epistemology 201: The Science of Science

From: aeo6 (aeo6_at_cornell.edu)
Date: 03/08/05 

Date: Tue, 8 Mar 2005 12:50:36 -0500

Allan C Cybulskie said:
>
> <stephen@nomail.com> wrote in message
> news:d09vvs$2dm3$1@msunews.cl.msu.edu...
> > In sci.math Allan C Cybulskie <allan.c.cybulskie@yahoo.ca> wrote:
> > : I don't want to reply to all the posts right now, but really need to
> reply
> > : to this part of this here ...
> >
> > : Here is the definition (also helpfully left above) that you gave for
> "proper
> > : subset":
> >
> > :>>If A is a proper subset of B than B contains
> > :> :> EXACTLY THE SAME elements as A plus some more
> >
> > : If that is the definition of proper subset, then what it says is that B
> must
> > : have a larger number of elements than A since it has exactly the same
> > : elements (thus, the same number of elements) as A PLUS SOME MORE, as you
> > : said. Yes, it says nothing about bijection but note that for the
> infinite
> > : sets we have been talking about the bijection approach says that a
> proper
> > : subset has the same number of elements as the superset. THAT is a
> > : contradiction between the two definitions.
> >
> > No it is not. The answer to "the number of elements" is a number.
> > I already showed you several examples of numbers where x+a=x, even
> > when a is non zero. There is no contradiction.
>
> And I dealt with that by pointing out the mathematical trick that it relies
> on. For example, here's another claim of the same sort:
>
> infinity + 1 = infinity + 2 is in fact a balanced equation. But if I try to
> subtract out the infinities on both sides, I get 1 = 2, which is clearly
> ludicrous. The reason this is a trick is that we call this a balanced
> equation because infinity + 1 and infinity + 2 both get treated like
> infinity, but then we try to get rid of the infinity which is the only thing
> that made them balanced we end up with a balanced equation.
>
> Your argument was exactly like that. You rely on infinity + anything
> remaining infinity, but that does not mean that the relative number of
> elements cannot be said to be larger based on the definition of the set
> itself.
>
>
>
>
I agree fully. The added 1 is insignificant in terms of the relative
increase to infinity, but is a unit of increase nonetheless. To say
infinity+1=infinity is essentially useless. One member is the difference
between the infinite sets of natural and counting numbers, and is
significant enough to warrant a separate name for the set. 

-- 
Smiles,
Tony
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