From: scerir (scerir@libero.it)
Date: Sun Jun 16 2002 - 00:26:09 MDT
> But Hal summed up this discussion best IMO:
>
> > Most arguments cannot be reduced to simple mathematical form.
> > Often there are ambiguities relating to the use of words and
> > language. It is easy to be confused or misled.
>
> Lee
In example (adapted from a recent discussion on FOM) ...
<The theorem that the set of all natural numbers {0,1,...} is
equinumerous with the set of all even numbers (0,2,...}, is
true if it is meant that every number has a double, and every
double is the double of one, and no more than one number.
But if every number has a double, and every double is the
double of just one number, does it follow that there are "as many"
doubles (even numbers) as singles (integers), given the meaning
of "as many"? Is it correct to define "as many" or
"equinumerous" using the idea of 1-1 correspondence?
The argument usually then turns to the challenge of defining
"as many as". This brings us to argue that we can think of number
as satisfying n-place predicates such as "x is a different thing
from y", which is - in general - not satisfied.
Any two collections are equinumerous (have as many objects as
each other) when there is such a predicate they both satisfy.
Defined in this way, no proper subset can be equinumerous
with its parent. The parent, by definition, contains objects
"different" from any of those in the subset, hence the parent
cannot satisfy the same n-place predicate as the subset.>
Another deep example is at
http://community.middlebury.edu/~arthur/LeibCant.pdf
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