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

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William Elliot

Posts: 1,226
Registered: 1/8/12
Re: What are sets? again
Posted: Dec 3, 2012 12:20 AM
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On Fri, 30 Nov 2012, Zuhair wrote:

> The following is an account about what sets are, first I'll write the
> exposition of this base theory in brief, then I'll discuss some
> related issues.
> Language: FOL + P, Rp
> P stands for "is part of"
> Rp stands for "represents"
> Axioms: Identity theory axioms +
> I. Part-hood: P partially orders the universe.
> 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.

> Def.) atom(x) <-> for all y. y P x -> x P y
> Def.) x atom of y <-> atom(x) & x P y.
> Def.) c is a collection of atoms iff for all y. y P c -> Exist z. z
> atom of y.
> Def.) c is atomless <-> ~ Exist x. x atom of c
> lll. Representation: x Rp c & y Rp d -> (x=y<->c=d)
> lV. Representatives: x Rp c -> atom(x)
> V. Null: Exist! x. (Exist c. x Rp c & c is atomless).
> A Set is an atom that uniquely represents a collection of atoms or
> absence of atoms.
> Def.) Set(x) <-> Exist c. (c is a collection of atoms or c is
> atomless) & x Rp c & atom(x)
> Here in this theory because of lV there is no need to mention atom(x)
> in the above definition.
> Set membership is being an atom of a collection of atoms that is
> uniquely represented by an atom.
> Def.) x e y iff Exist c. c is a collection of atoms & y Rp c & x atom
> of c & atom(y)
> Here in this theory because of lV there is no need to mention atom(y)
> in the above definition.
> Vl. Composition: if phi is a formula in which y is free but x not,
> then
> [Exist y. atom(y) & phi] -> [Exist x. x is a collection of atoms &
> (for all y. y atom of x <-> atom(y) & phi)] is an axiom.
> Vll. Pairing: for all atoms c,d Exist x for all y. y e x <-> y=c or
> y=d
> /
> This theory can interpret second order arithmetic. And I like to think
> of it as a base theory on top of which any stronger set theory can
> have its axioms added to it relativized to sets and with set
> membership defined as above, so for example one can add all ZFC axioms
> in this manner, and the result would be a theory that defines a model
> of ZFC, and thus proves the consistency of ZFC. Anyhow this would only
> be a representation of those theories in terms of different
> primitives, and it is justified if one think of those primitives as a
> more natural than membership, or if one think that it is useful to
> explicate the later. Moreover this method makes one see the Whole
> Ontology involved with set\class theories, thus the bigger picture
> revealed! This is not usually seen with set theories or even class
> theories as usually presented, here one can see the interplay between
> sets and classes (collections of atoms), and also one can easily add
> Ur-elements to this theory and still be able to discriminate it from
> the empty set at the same time, a simple approach is to stipulate the
> existence of atoms that do not represent any object. It is also very
> easy to explicate non well founded scenarios here in almost flawless
> manner. Even gross violation of Extensionality can be easily
> contemplated here. So most of different contexts involved with various
> maneuvering with set\class theories can be easily
> paralleled here and understood in almost naive manner.
> In simple words the above approach speaks about sets as being atomic
> representatives of collections (or absence) of atoms, the advantage is
> clearly of obtaining a hierarchy of objects. Of course an atom here
> refers to indivisible objects with respect to relation P here, and
> this is just a descriptive atom-hood that depends on discourse of this
> theory, it doesn't mean true atoms that physically have no parts, it
> only means that in the discourse of this theory there
> is no description of proper parts of them, so for example one can add
> new primitive to this theory like for example the primitive "physical"
> and stipulate that any physical object is an atom, so a city for
> example would be an atom, it means it is descriptively an atom as far
> as the discourse of this theory is concerned, so atom-hood is a
> descriptive modality here. From this one can understand that a set is
> a way to look at a collection of atoms from atomic perspective, so the
> set is the atomic representative of that collection, i.e. it is what
> one perceives when handling a collection of atoms as one descriptive
> \discursive whole, this one descriptive\discursive whole is actually
> the atom that uniquely represents that collection of atoms, and the
> current methodology is meant to capture this concept.
> Now from all of that it is clear that Set and Set membership are not
> pure mathematical concepts, they are actually reflecting a
> hierarchical interplay of the singular and the plural, which is at a
> more basic level than mathematics, it is down at the level of Logic
> actually, so it can be viewed as a powerful form of logic, even the
> added axioms to the base theory above like those of ZFC are really
> more general than being mathematical and even when mathematical
> concepts are interpreted in it still the interpretation is not
> completely faithful to those concepts. However this powerful logical
> background does provide the necessary Ontology required for
> mathematical objects to be secured and for
> their rules to be checked for consistency.
> But what constitutes mathematics? Which concepts if interpreted in the
> above powerful kind of logic would be considered as mathematical? This
> proves to be a very difficult question. I'm tending to think that
> mathematics is nothing but "Discourse about abstract structure", where
> abstract structure is a kind of free standing structural universal.
> Anyhow I'm not sure of the later. I don't think anybody really
> succeeded with carrying along such concepts.
> Zuhair

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