Re: Epistemology 201: The Science of Science

From: Allan C Cybulskie (allan.c.cybulskie_at_yahoo.ca)
Date: 02/23/05 

Date: Wed, 23 Feb 2005 17:27:27 -0500

"Tony Orlow (aeo6)" <aeo6@cornell.edu> wrote in message
news:MPG.1c83c52bbfd1a69c9897c3@newsstand.cit.cornell.edu...
> Allan C Cybulskie said:
> >
> > Stephen asked me what infinity + infinity was. Since I expect that he
would
> > not have approved of an answer of "2 * infinity", I saw the question as
only
> > possible leading to the answer "infinity", and then to the reply that
> > therefore the set (0,1) has a number of elements equalling infinity and
the
> > set (0,2) has a number of elements equalling infinity, and so there was
no
> > reason to claim that the set (0,2) had to have more elements. But
that's a
> > word game based on the fact that we won't say "2 * infinity", not an
actual
> > argument.
> >
> > Surely
> > > the sign of a less than first rate intellected. I suggest you give up
on
> > > mathematics and take up floor polishing, dish washing or even computer
> > > programming.
> >
> > This is probably more ironic than you know.
> >
> > Besides, who said I was all that interested in mathematics? I merely
> > exercise my right to not allow people playing word games to pull the
wool
> > over my eyes.
> >
> >
> >
> Thank you Allan. You are obviously no less than a first rate intellect
> as far as I can tell. Probably the only way to see beyond Cantor is to
> NOT be a mathematician by trade.

I don't really have a problem with what the mathematicians are saying, but I
think that they've taken a few basic truths and dependent facts and elevated
them above their meaning.

For example, it may well be true that for finite sets if you can map the two
sets one-to-one onto another, they are the same size (and thus cardinality
and size or number of elements match directly). But Cantor seems to have
shown that for infinite sets, subsets for certain CAN be mapped in a
"one-to-one" mapping to each other. But does this mean that these therefore
have the same size or number of elements as well. There is an inflation of
the result that fortitously allows for finite sets to show that cardinality
and equal numbers of elements are the same to simply BE what "number of
elements" means. But they never did the "if and only if" to justify the
definition.

> I guess my problem here is that I HAVE
> been trying to say infinity+infinity=2*infinity, as long as you're
> talking about the same infinity consistently.

I think that that seems odd to most people, although I have little trouble
with it. The problem is that in this sense this statement can only have
meaning with respect to a certain ... uh ... set of sets. Because if we
look at our two examples, we list both the set of integers and the set of
real (0,2) as being 2*infinity, but clearly this would not make them have
the same number of elements.

> I'm not sure why people
> have such a hard time conceiving ratios between infinite sets that are
> less than a power-set ratio. Even when there are an infinite number of
> rationals for each natural number, they are assigned the same
> cardinality, because of this counting method.

That's exactly what I'm trying to say. Yes, their number of elements is
infinite, but relatively speaking there is a ratio of number of elements
between the two sets that allow us to say that one has a larger number of
elements than the others.

Relevant Pages

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  • Re: Epistemology 201: The Science of Science
    ... Allan C Cybulskie said: ... > is to insist that they both have an infinite set of elements ... ... >> sets have the same cardinality. ... The suggestion that a proper subset by definition has fewer elements was ...
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  • Re: Epistemology 201: The Science of Science
    ... Allan C Cybulskie said: ... > is to insist that they both have an infinite set of elements ... ... >> sets have the same cardinality. ... The suggestion that a proper subset by definition has fewer elements was ...
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  • Re: Epistemology 201: The Science of Science
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