Re: Epistemology 201: The Science of Science

stephen_at_nomail.com
Date: 02/28/05 

Date: 28 Feb 2005 03:50:03 GMT

In sci.math Allan C Cybulskie <allan.c.cybulskie@yahoo.ca> wrote:

: <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.

Left over at the end? So you think there is some integer x out there
such that 2*x is not an integer, or that 2*x is not even? And
what end are you talking about? Do you think there is a last
integer out there?

Stephen

Relevant Pages

  • Re: Epistemology 201: The Science of Science
    ... :>:>> works for infinite sets, ... :>:> it seems like you get the evens by taking half of the whole numbers. ... :>:> of wholes as evens; take any whole number, and you can produce an even ... :>:> intuition we should go with. ...
    (sci.math)
  • Re: Epistemology 201: The Science of Science
    ... :>:>> works for infinite sets, ... :>:> it seems like you get the evens by taking half of the whole numbers. ... :>:> of wholes as evens; take any whole number, and you can produce an even ... :>:> intuition we should go with. ...
    (sci.physics)
  • Re: Epistemology 201: The Science of Science
    ... :>> works for infinite sets, ... :> it seems like you get the evens by taking half of the whole numbers. ... :> of wholes as evens; take any whole number, and you can produce an even ... :> intuition we should go with. ...
    (sci.math)
  • Re: Epistemology 201: The Science of Science
    ... :>> works for infinite sets, ... :> it seems like you get the evens by taking half of the whole numbers. ... :> of wholes as evens; take any whole number, and you can produce an even ... :> intuition we should go with. ...
    (sci.cognitive)
  • Re: Epistemology 201: The Science of Science
    ... :>> works for infinite sets, ... :> it seems like you get the evens by taking half of the whole numbers. ... :> of wholes as evens; take any whole number, and you can produce an even ... :> intuition we should go with. ...
    (sci.physics)
Quantcast