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Re: Background Theory
Posted:
Dec 8, 2012 2:23 AM


On Dec 7, 7:21 pm, fom <fomJ...@nyms.net> wrote: > On 12/7/2012 8:46 AM, Zuhair wrote: > > > One might wonder if it is easier to see matters in the opposite way > > round, i.e. interpret the above theory in set theory? the answer is > > yes it can be done but it is not the easier direction, nor does it > > have the same natural flavor of the above, > > it is just a technical formal piece of work having no natural > > motivation. Thus I can say with confidence that the case is that Set > > Theory is conceptually reducible to Representation Mereology and not > > the converse! > > I have no doubt that you are correct. In another post > in your thread I summarized the work of Lesniewski which > uses the part relation to characterize classes. His > method was specifically designed to circumvent the > grammatical form that leads to Russell's paradox. >
Yes, this is clearly resolved here. A set would be an element of itself iff it represents a collection of atoms having it among them, this is not that difficult to ponder about since indeed a representative of a group can be among that group like in for example a father representing his family. Now the collection of all sets that do not represent collections of atoms of which they are a part (i.e. sets that are not elements of themselves) can be easily composed in this theory but also this theory easily prove that such a collection cannot have a representative atom. Sethood is not about collections of atoms per se, it is about uniquely representing those by atoms. So as you see above what seems to be a counterintuitive result i.e. the non existence of the set of all sets that are not elements of themselves, is actually rendered a quite natural and intuitive result by the above line of thinking, that's why I say that the real benefit of background theory is that it makes one see the whole picture behind set theory, it reveals the whole background Ontology that is usually hidden from the customary presentation of standard set\class theories. This background theory enable one to understand NAIVELY things that otherwise would be very difficult to grasp, like non well founded sets, non definable sets, non extensional objects, the interplay between classes and sets, the paradoxes etc... so to me it aids a lot in understanding those matters.
Zuhair



