Date: Dec 4, 2012 11:49 PM
Author: Zaljohar@gmail.com
Subject: Re: What are sets? again
On Dec 5, 5:06 am, fom <fomJ...@nyms.net> 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>