Date: Dec 4, 2012 11:02 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.
>


I'm really sorry that I didn't have the chance to look at all of your
responses. I'd do once I have time.
Anyhow for now, it is sufficient to note that my theory does prove
Weak supplementation for collections of atoms that is if x is a proper
part of y and y is a collection of atoms then there exist a part of y
that do not overlap with x.

Zuhair


> 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>