[FOM] Discriminated Part-hood

[Zuhair Abdul Ghafoor Al-Johar]

Dear FOMers

This come as a continuation to prior posts by the title
"Multi-Discrimination Theory" posted to FOM at February 2012.
The theory presented was inconsistent because it used an equality
relation over all discrimination levels which is too strong.

The following is a trial to remedy that.

Multi-Discrimination Theory

Language: First order logic

Primitives: P_i for each i=1,2,3,..., each P_i denotes a part-hood binary
relation at discrimination level i; a constant symbol C_j for each
j=1,2,3,...

Define (=_i): x =_i y iff  x P_i y & y P_i x

Axioms Per i Per j: i=1,2,3,...; j=1,2,3,....

I. Part-hood: (for all z. z P_i x -> z P_i y) -> x P_i y

Def.) x is i_atom <-> for all y. y P_i x ->  y =_i x

Def.) x is i_atom of y <-> x is i_atom & x P_i y

II. Atomicity: (~ x P_i y) -> Exist z. z is i_atom of x & ~ z P_i y

III. Comprehension: [(Exist z. z is i_atom & phi(z)) ->
Exist x for all y ( y is i_atom of x <-> y is i_atom & phi(y))]
is an axiom.

IV. Discrimination: for all x. Exist x*. x* =_i x  & x* is i+1_atom

V. Infinity: C_i is 1_atom & ~ C_i =_1 C_i+j

/

An axiom that might be added to the above is:

VI: Reduction: x P_i+1 y -> x P_i y

Now this theory involves working with very weak kinds of part-hood 
relation and after them are defined very weak kinds of equality. 
Axiom scheme III builds aggregates from atoms at the respective level
of discrimination. So it is limited in the sense that it cannot build
aggregates of non atomic aggregates at the same level, that's why
axiom IV is stipulated! it allows those aggregates to become atoms at
higher levels (of indiscrimination actually) and thereby they can be
gathered to form aggregates of them, this hierarchy if not inconsistent
could provide the necessary milieu for second order arithmetic to be
implemented in, thereby reducing most of mathematics to very weak 
part-hood relations.

Best Regards

Zuhair Al-Johar