FOM: Date: Mon, 27 May 2002 11:03:47 -0500

[Insall]

It is not a question of density, but of miscommunication on someone's part.
I saw no one on the fom list claim that (1) implies (2), where (1) and (2)
are given, as you did in your recent posting (slightly modified),

(1) the collection of names N corresponds 1-1 with the set of names E

we can derive

(2)  N and E each denote the same number of objects.

Moreover, I do not recall seeing in your earlier posting the negation of
``(1) implies (2)''.  What I recall, if I am recalling correctly, and if I
understood your intention correctly, was a claim that is equivalent to, or
at least very close to, ``(2) implies (1)''.  Let me quote from your
previous posting:

``We can place any suitable collection of names in a one-one correspondence.
For example

Clemens, Poe, Twain
Austen, Bronte, Carlyle

What is the guarantee that the things they name have the same number?  In
this case there is no such, the first series naming two authors, the other,
three.

We need as a necessary condition, that neither set of names contain a
(proper) subclass of names designating a collection with the same number as
the collection designated by the set itself.''

Your contention that we ``need'' that ``neither set of names contain a
(proper) subclass of names designating a collection with the same number as
the collection designated by the set itself'' appears to me to amount to a
belief that ``(2) implies (1)'', at least in enough cases to deny the
existence of infinite collections.  It is this contention to which I object.
Perhaps I misunderstood your intention, if all you meant was that ``(1) does
not imply (2)''.  For in fact, I agree that (1) does not imply (2).  But
your usage of the terminology ``necessary condition'', if interpreted in the
usual mathematical manner, indicates that you believe that from your (set
theoretic?) axioms the claim that ``neither set of names contain a (proper)
subclass of names designating a collection with the same number as the
collection designated by the set itself'' is derivable, and I just do not
see how this can be the case, unless you require that all sets be finite.


Matt