Re: Epistemology 201: The Science of Science

stephen_at_nomail.com
Date: 02/24/05 

Date: 24 Feb 2005 00:54:35 GMT

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.

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.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)
  • 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)
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