Re: Epistemology 201: The Science of Science

stephen_at_nomail.com
Date: 03/08/05 

Date: 8 Mar 2005 02:56:38 GMT

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

: <stephen@nomail.com> wrote in message
: news:d08akg$sgn$3@msunews.cl.msu.edu...
:> In sci.math Allan C Cybulskie <allan.c.cybulskie@yahoo.ca> wrote:
:>
:> : <stephen@nomail.com> wrote in message
:> : news:cvu4db$12pv$2@msunews.cl.msu.edu...
:> :> In sci.math Allan C Cybulskie <allan.c.cybulskie@yahoo.ca> wrote:
:>
:> :> : 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?
:>
:> : Take the set of integers {1, 2, 3} and the set of even integers {2, 4,
: 6,
:> : 8}. There is a mapping function that I can apply to that set of
: integers
:> : with is 2*x, just as in your example. Yet I can clearly show that if I
: map
:> : every element in the set of integers to the set of even integers that
: I've
:> : posited here that there is an extra element in the second set, and so
: they
:> : do not have the same number of elements. But the mapping worked
: perfectly
:> : fine, and it was just that the first set "ended" earlier that caused us
: to
:> : determine the outcome. Since we can never "finish" mapping the elements
: in
:> : the infinite sets of integers and even integers, we can never know if
: such a
:> : case could possibly be the case, precisely because there is no "last"
:> : integer. But then how can I be certain that my mapping would work?
: It's
:> : only because I can never "finish" that I can never -- and would never --
:> : conclude that they didn't map one onto another, since if I stopped
: mapping
:> : at any N IN the set, it would be clear that the mapping would not
: succeed.
:>
:> I can finish the mapping by describing it. f(x)=2x is a mapping
:> between the integers and even integers. In what way is it
:> not complete? Which integer has not been mapped to an even integer
:> and which even integer has not been mapped to an integer?

: We have no idea, because we don't know where the integers end and so can't
: know if there is anything left over after the mapping. If there WAS a last
: integer, then we'd almost certainly have a problem with this mapping.

By definition, there is no last integer. Every integer has
a successor, by definition, and no integer equals its successor,
by definition. I am not sure what you mean when you "integer",
but it is apparently not a well defined notion. In fact
you apparently are not even sure that the set of all integers
is in fact infinite.

Stephen

Relevant Pages

  • Re: Epistemology 201: The Science of Science
    ... Allan C Cybulskie wrote: ... which Cantor showed was NOT true for infinite sets. ... mapping one set onto another will reveal whether they are ... are bigger than others, just as some finite sets are bigger than others. ...
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  • Re: Epistemology 201: The Science of Science
    ... Allan C Cybulskie wrote: ... which Cantor showed was NOT true for infinite sets. ... mapping one set onto another will reveal whether they are ... are bigger than others, just as some finite sets are bigger than others. ...
    (sci.cognitive)
  • Re: Epistemology 201: The Science of Science
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  • Re: Epistemology 201: The Science of Science
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