FOM: RE: A Puzzle

[Insall]

On 23 May 2002, Dean Buckner wrote:

``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.''

and

``Then (N) will not contain a subset designating a similarly-numbered
collection.  But of course it will, since (N) must contain (E).''

and

``I am not (necessarily) denying that the *names* in (N) and (E) are
equinumerous.  It is what they name, I am worried about.''


The first of these claims is misleading.  The word ``need'' should be
replaced by ``want'', because you, for some reason ``want'' the condition
you expressed as an axiom.  I do not want it, and I do not ``need'' it.  It
is not a ``necessary condition'' either.  It is, however, a sufficient
condition for what you do in the second claim, namely, you claim to derive a
contradiction from reasonable mathematical assumptions about infinite
collections.  Of course, by assuming the axiom expressed in your first
claim, you have denied one form of the axiom of infinity, so of course you
are able to eventually derive a contradiction by conjoining a denial of the
axiom of infinity with a form of the axiom of infinity itself.  Curiosly,
while you have attempted to deny the axiom of infinity for collections of
objects in the universe of discourse in your first claim, your third claim
demonstrates that you are willing to accept the axiom of infinity for
collections of names for the objects in the universe of discourse.  While
this is not actually contradictory, so long as you do not allow (at least
some) names to be objects in the universe of discourse, it is also not
``necessary''.  From a mathematical standpoint, it just depends on what you
want to do with the language and its related logic systems and models.  From
a philosophical standpoint, it would seem to me to be a bit odd to try to
use the above three claims to draw some kind of metaphysical conclusions
about what we ``need''.  (On the other hand, you are certainly free to
``want'' whatever you choose to ``want''.  In some cases, you just may not
get it.)


Matt Insall