```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 acomplete whole, was, of course, the traditional, Aristotelian point ofview, which Intuitionists, more recently, have still held to. But manyothers now doubt this fact. Is there any way to show that Aristotlewas 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 bycounting. But what process will determine what the number is, in anyother case? The newly revealed independence of the ContinuumHypothesis shows there is no way to determine the number in certainwell known infinite sets. [...] The key question therefore is: ifthere is a determinate number of natural numbers, then by what processis it determined? Replacing 'the number of natural numbers' with'Aleph zero' does not make its reference any more determinate. Thenatural numbers can be put into one-one correspondence with the evennumbers, it is well known, but does that settle that they have thesame number? We have equal reason to say that they have a differentnumber, since there are more of them. So can we settle the determinatenumber in a set of discrete items just by stipulation?Indeed, if all infinite sets could be put into one-one correspondencewith each other, one would be justified in treating the classification'infinite' as an undifferentiated refusal of numerability. But givenCantor's discovery that there are infinite sets which cannot be putinto one-one correspondence with each other, this conclusion is lesscompelling.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 thesame, 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_paradoxesRegards, WM
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