fom
Posts:
1,037
Registered:
12/4/12
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Re: What are sets? again
Posted:
Dec 4, 2012 9:06 PM
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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.
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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