FOM: Transfinite Logic

[Dean Buckner]

We have to prise apart the notion of there being *no other* things, i.e.
there
being nothing "remaining or not included, one or ones distinct from
that or those first mentioned", from the notion of there being *no more*
things.  Otherwise we risk Setek's fallacy, of defining a proper subset so
that it cannot be equinumerous with its parent.  This is not easy.  We need
the following to be true

(i) there are some B's
(ii) there are no more B's
(iii) but there are other B's besides

(i) and (iii) must be true, in order that the parent set (the B's and the
other B's) has members (the other B's) different from or "other than" those
in the proper subset ("some B's").  (ii) must be true, in order that the
parent have the same number of members as (i.e. no more than) the subset.

But how *can* they all be true? For (surely) if there are no more B's, then
there are no other B's either.  If a proper subset is by definition a set
that contains fewer members than its parent, how can there be such sets -
even infinite ones - that contain the same number of members as their
parent?  If there are no more  B's, there are no other Bs.  If there are no
other  B's, there are no more Bs.  Show me how this follows from anything
other than the meaning of "more" and "other".

As to the argument that they fail to hold "only for finite sets".  This
presumes we have the idea of an infinite set.  But we don't: we have to
define it as a set of objects of which (i) - (iii) are all true.  But there
is no such set.

You could define proper subset in terms of "other", and equinumerousness in
terms of one-one correspondence, and thus split the ideas of "other" and
"more" that way.  But the idea of correspondence involves that of there
being no remainders or residuals or "other things", given the
correspondence.  It is quite easy to set up a 1-1 correspondence between a
proper subset and itself by means of the identity relation, and then show
that there are residuals or remainders that define thr parent set.  For
example, assume the following terms occur:

"Some B's ... these B's ... the other B's"

Then what "some B's" apply to must be identical with what "these B's" apply
to, so we have one-one correspondence between some B's, and those very same
B's.  What "these B's" and "the other B's" apply to is (by definition) a
parent set.  Yet since there is a remainder (the other B's), the parent set
cannot be equinumerous with its subset.  If there is some way this may fail
to hold, then we have by definition an infinite set.  But as I can see, and
by definition, there is not.