[FOM] What are sets?

Zuhair Abdul Ghafoor Al-Johar zaljohar at yahoo.com
Fri May 20 12:46:10 EDT 2011

Dear FOMers.

The following reflect the same idea of my previous post, but using units
instead of atoms. The number of primitives is reduced to two binary
relations only.

Language: FOL + Primitive binary relations: "is a part of" denoted by p
and "is collected by" denoted by @.

Axiom: (for all x. x p y -> x p z) -> y p z

Axiom: x p y & y p z -> x p z

Define (=): x=y <-> x p y & y p x

Define (pp): x pp y <-> x p y & ~ y p x

Axiom: x @ y & z p x -> z @ y

Define (unit): unit(x) <-> (exist c. for all y. y @ c <-> y p x) &
(~ exist c,z. z pp x & for all y. y @ c <-> y p z)

Axiom: unit(x) & unit(y) & ~ x=y -> ~ exist z. z p x & z p y

A particle is a proper part of a unit.

Define (collection): collection(x) <->
(for all y. y p x -> exist c,z. z p y & z p c & unit(c) & c p x)

Define (e): x e y <-> unit(x) & x p y & collection(y)

e is read as: is a trivial member of.

Axiom schema: if phi is a formula in which x is not free, then
(exist z. unit(z)phi -> (exist x. for all y. y e x <-> unit(y)phi)) is an axiom.

Define [|]: y=[x|phi] <-> collection(y)&(for all x. x e y <-> unit(x)phi)

Define (precollector): precollector(x) <-> exist y. y @ x

Define (collector): collector(x) <-> unit(x) & exist y. y @ x

A proper precollector is a precollector that is not a unit.

A class is a collection of collectors where distinct collectors do collect
distinct collections.

Define (class): class(x) <-> collection(x) &(for all y. y e x -> collector(y)) & 
(for all y,z. y e x & z e x & ~ y=z -> exist u,w. u=[k| k @ y] & w=[k| k @ z] & ~ u=w)

For every collector x the collection [y| y @ x] is called
the *extension* of x, and x is its *exclusive* collector.
When this extension is a class, it's called
a *class extension* of x.

A set is a class extension of a collector.

An Ur-element is a unit that is not a collector, which is an extension of
a collector.

Epsilon membership is defined as:

x epsilon y <->  class(y) & exist z. collector(z) & x=[u| u @ z] & z e y.

This explains those terms in standard set\class theories.

However in set theories it is sufficient to interpret sets as collectors
and epsilon membership as "is a unit collected by the collector".

Zuhair Al-Johar

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