Re: Epistemology 201: The Science of Science

From: Allan C Cybulskie (allan.c.cybulskie_at_yahoo.ca)
Date: 02/26/05 

Date: Sat, 26 Feb 2005 08:03:42 -0500

<stephen@nomail.com> wrote in message
news:cvj8kb$f38$2@msunews.cl.msu.edu...
> In sci.math Allan C Cybulskie <allan.c.cybulskie@yahoo.ca> wrote:
>
> : "The Sophist" <sophist@brown.edu> wrote in message
> : news:bOrSd.638$2s.500@lakeread06...
> :> Tony Orlow (aeo6) wrote:
> :>
> :> > The bijection rule works fine for finite sets but is unnecessary. It
> :> > works for infinite sets, but misses much. The definition of a proper
> :> > subset as a set containing only elements from a given set, but not
all
> :> > of them, is the most basic and intuitive definition, and works for
> :> > finite as well as infinite sets. The idea that a proper superset
> :> > contains more elements is by definition true, the additional
elemnents
> :> > being those not included in the proper subset. This has nothing to do
> :> > with cardinality. Cardinality misses it entirely. And, apparently, so
do
> :> > you.
> :>
> :> There are two conflicting intuitions here. In an intuitive sense, it
> :> seems like there have to be twice as many whole numbers as evens, since
> :> it seems like you get the evens by taking half of the whole numbers.
Or
> :> at least, as you put it, it seems there have to be in some way more
> :> whole numbers than evens, because the whole numbers are the evens plus
a
> :> bunch of additional numbers (the odds).
> :>
> :> But surely it is equally intuitive that there have to be the same
number
> :> of wholes as evens; take any whole number, and you can produce an even
> :> number matching it (just multiply the whole by 2). You can do this for
> :> all the whole numbers. Since each whole number has a unique matching
> :> even number, how could there be more wholes than evens? If there were
> :> more wholes than evens, we'd have to run out of evens at some point in
> :> trying to match them to the wholes.
> :>
> :> The difference is, of course, that following Cantor and accepting the
> :> latter intuition leads to fruitful mathematics, while the former view
> :> doesn't seem to be of any use whatever. So it seems clear which
> :> intuition we should go with.
>
> : I don't see how the second intuition a) is an intuition and b) is not
one
> : that we should immediately see wouldn't apply to infinite sets.
>
> : For example, since you are mapping an infinite subset onto an infinite
set,
> : doesn't that mean that you never complete the mapping? So how would you
> : know that you can map them onto each other?
>
> Let E be the set of even integers. Let I be set of integers. Define
> a mapping f: I -> E where f(n)=2n. This maps every integer to a unique
even
> integer and ever even integer to a unique integer. We know this works
> because of the definitions of integer, multiplication and even.

But don't we actually have to map all of them to know if anything is left
over at the end? For infinite sets, that can never happen.

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  • Re: Epistemology 201: The Science of Science
    ... >> I don't see how the second intuition a) is an intuition and b) is not ... >> that we should immediately see wouldn't apply to infinite sets. ... since you are mapping an infinite subset onto an infinite ... >> know that you can map them onto each other? ...
    (sci.math)