Re: Epistemology 201: The Science of Science

stephen_at_nomail.com
Date: 02/20/05 

Date: 20 Feb 2005 21:51:32 GMT

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

: <stephen@nomail.com> wrote in message
: news:cut65p$17ho$1@msunews.cl.msu.edu...
:> In sci.math Allan C Cybulskie <allan.c.cybulskie@yahoo.ca> wrote:
:>
:> : <stephen@nomail.com> wrote in message
:> : news:curv26$4b1$1@msunews.cl.msu.edu...
:> :> In sci.math Allan C Cybulskie <allan.c.cybulskie@yahoo.ca> wrote:
:> :> : The problem, as I see it, is that it is clear that every single point
: in
:> : the
:> :> : range (0,1) is ALSO in the range (0,2), plus all the points you can
: get
:> : by
:> :> : generating them from (0,1). So how can there NOT be more elements in
:> : the
:> :> : range (0,2)?
:> :>
:> :> You have to define what "more" means. What is infinity + infinity?
:>
:> : I've already dealt clearly with that. It's merely a word game, since
:> : infinity is the largest number that we can talk about. But that does
: not
:> : allow us to go beyond that and draw any inferences beyond "we can't talk
:> : about the extra elements because we don't have a terminology for it).
:>
:> I have not seen you deal clearly with that.

: I dealt clearly with what I meant by "more", which is that in this case the
: set (0,2) has all of the elements that are in the set (0,1) and the elements
: in the set (1,2). By any reasonable definition of "more", the set (0,2)
: will have more elements than the set (0,1). The only way to say otherwise
: is to insist that they both have an infinite set of elements ... but that's
: a word game on "infinite".

By the way, here is a perfectly reasonable defintion of "more".
The set A has "more" elements than the set B if is possible
to match each element of B with a different element of A, but it
is not possible to match each element of A with a different element
of B. For example A={frog, dog, cat} has more elements than B={2,5}
because I can match 2->frog and 5->dog, so each element of B has
been matched with a different element of A. However if I want
to match the elements of A to elements in B, I am going to have
to reuse one of the elements in B. A more technical way of putting
this is that there exists a surjection from A to B, but there does
not exist a surjection from B to A.

This defintion of "more" is nice because it applies to all sets,
even sets that are totally disjoint. In contrast, definitions base on
subsets and supersets are rather limited. In the finite case this
definition corresponds exactly with our intuitions about what "more"
means. The definition can also be applied as is to the infinite case.

According to this definition of "more", there are not more elements
in (0,2) than in (0,1).

Stephen

Relevant Pages

  • Re: Epistemology 201: The Science of Science
    ... :> In sci.math Allan C Cybulskie wrote: ... is to insist that they both have an infinite set of elements ... ... this is that there exists a surjection from A to B, ...
    (sci.math)
  • Re: Epistemology 201: The Science of Science
    ... :> In sci.math Allan C Cybulskie wrote: ... is to insist that they both have an infinite set of elements ... ... this is that there exists a surjection from A to B, ...
    (sci.physics)
  • Re: Epistemology 201: The Science of Science
    ... :> In sci.math Allan C Cybulskie wrote: ... :>: is to insist that they both have an infinite set of elements ... ... this pretty much maps to my reasoning for considering the set to have ...
    (sci.cognitive)
  • Re: Epistemology 201: The Science of Science
    ... :> In sci.math Allan C Cybulskie wrote: ... :>: is to insist that they both have an infinite set of elements ... ... this pretty much maps to my reasoning for considering the set to have ...
    (sci.physics)
  • Re: Epistemology 201: The Science of Science
    ... :> In sci.math Allan C Cybulskie wrote: ... :>: is to insist that they both have an infinite set of elements ... ... this pretty much maps to my reasoning for considering the set to have ...
    (sci.math)
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