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