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Topic: What are sets? again
Replies: 21   Last Post: Dec 9, 2012 10:12 AM

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Posts: 2,665
Registered: 6/29/07
Re: What are sets? again
Posted: Dec 4, 2012 11:49 PM
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On Dec 5, 5:06 am, fom <> wrote:
> On 12/2/2012 11:20 PM, William Elliot wrote:

> > On Fri, 30 Nov 2012, Zuhair wrote:
> <snip>

> >> ll. Supplementation: x P y & ~ y P x -> Exist z. z P y & ~ x P z.
> > x subset y, y not subset x -> some z subset y with x not subset z.
> > x proper subset y -> some z subset y with x not subset z
> > x proper subset y -> y\x subset y, x not subset y\x

> > Oh my, no empty set.
> You have made an incorrect step here.
> In mereology there is no reason to take y\x as substantive.
> Supplementation is supposed to enforce existence of a proper part of y
> in y\x.
> In this case, z could be a proper part of x.  Then zPy and -xPz is
> satisfied.
> This is not a supplementation axiom in the classical sense.

Correct. However in this theory weak supplementation is provable for
collections of atoms.

> As for no empty set, Zuhair may have seen this axiom in a formulation of
> mereology where the axiom was intended to preclude existence of a null
> part.  This is a standard ontological position among those individuals
> who investigate and reflects a position once taken by Frege in
> criticizing the likes of Hausdorff and Cantor:
> "... a forest without trees."
> Moreover, Zuhair's construction is similar to Zermelo's 1908 paper on
> set theory.  Heijenoort's translates Zermelo's "Teil" -- that is,
> subsets of nonvoid sets -- as "parts", and, the null set is introduced
> separately.
> This is precisely what Zuhair has attempted to do.
> <snip>

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