Date: Nov 30, 2012 4:27 AM
Author: mueckenh@rz.fh-augsburg.de
Subject: Matheology § 166


The fact that some discrete items might lack a determinate number,
this being connected with the possibility of them being given as a
complete whole, was, of course, the traditional, Aristotelian point of
view, which Intuitionists, more recently, have still held to. But many
others now doubt this fact. Is there any way to show that Aristotle
was right? I believe there is.

For when discrete items do clearly collect into a further individual,
and we have a finite set, then we determine the number in that set by
counting. But what process will determine what the number is, in any
other case? The newly revealed independence of the Continuum
Hypothesis shows there is no way to determine the number in certain
well known infinite sets. [...] The key question therefore is: if
there is a determinate number of natural numbers, then by what process
is it determined? Replacing 'the number of natural numbers' with
'Aleph zero' does not make its reference any more determinate. The
natural numbers can be put into one-one correspondence with the even
numbers, it is well known, but does that settle that they have the
same number? We have equal reason to say that they have a different
number, since there are more of them. So can we settle the determinate
number in a set of discrete items just by stipulation?

Indeed, if all infinite sets could be put into one-one correspondence
with each other, one would be justified in treating the classification
'infinite' as an undifferentiated refusal of numerability. But given
Cantor's discovery that there are infinite sets which cannot be put
into one-one correspondence with each other, this conclusion is less
compelling.

For Dedekind defined infinite sets as those that could be put into one-
one correlation with proper subsets of themselves, so the criteria for
'same number' bifurcate: if any two such infinite sets were numerable,
then while, because of the correlation, their numbers would be the
same, still, because there are items in the one not in the other,
their numbers would be different. Hence such 'sets' are not numerable,
and one-one correlation does not equate with equal numerosity [...]

[H. Slater: "The Uniform Solution of the Paradoxes" (2004)]
http://www.philosophy.uwa.edu.au/about/staff/hartley_slater/publications/the_uniform_solution_of_the_paradoxes

Regards, WM