[FOM] Multi-level Discrimination

[Zuhair Abdul Ghafoor Al-Johar]

Dear FOMers,

The following is a first order theory that I think it would be as 
strong as second order arithmetic. However it uses the concept of 
Part-hood instead of membership. So it is a multi-extensional 
Mereological theory. 

MULTI-LEVEL DISCRIMINATION ThEORY: 

Primitives: P_i for each i=1,2,3,..., each P_i symbolize a part-hood
binary relation; i in P_i is to be denoted as the discrimination level
of the part-hood relation. A constant symbol C_i for each natural
number i=1,2,3,...; = to denote Equality relation. 

Axioms schemes Per i: 

I. Reflexive: x P_i x 

II. Anti-Symmetric: x P_i y & y P_i x -> x=y 

III. Transitive: x P_i y & y P_i z -> x P_i z 

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

Def.) x is i_atom of z iff x is i_atom & x P_i z 

IV. Atomicity: ~ x is i_atom -> Exist y. y is i_atom of x. 

V. Comprehension: ((Exist z. phi(z) & z is i_atom) ->
Exist x. For all y. y is i_atom of x iff y is i_atom & phi)
is an axiom.

Def.) x=[y| phi]^i <-> (For all y. y is i_atom of x iff y is i_atom & phi)

VI. Blurring: x P_i y -> (x P_i+1 y <-> x=y) 

VII. Infinity: for all i,j where ~i=j:  C_i is 1_atom & ~ C_i= C_j 

/ 

The idea is that discrimination would be lost at higher levels, so for
example Let X= [C_1,C_2]^1 here X is the whole of C_1 and C_2 at 
discrimination level 1, where C_1 and C_2 are atoms at level 1. Now X 
itself is not an atom at level 1 but it would be an atom at level 2 
because  C_1 and C_2 will not be recognized as parts of X at level 2, 
so X has itself as the only part so it becomes an atom at level 2, 
also each of C_1 and C_2 remain as atoms at level 2 and 
of course we can construct the set Y= [C_1,C_2]^2 which is of course 
different from the set X above since X= [X]^2. Natural numbers can be 
defined in this theory: 


I= [x| Exist y. x=[y]^1]^1

II= [x| Exist y1y2. x=[y1,y2]^1]^2 
. 
. 
n = [x| Exist y1..yn. x=[y1,..,yn]^1]^2 

Ordered pair and relations can be defined on those.

so a natural number n is a level 2 equivalence aggregate under 
relation bijection of level 1 finite aggregates. 

In a similar manner the set N of all naturals can be defined also. 

Of course this theory does not have an empty object. 

I think set membership can be interpreted meta-theoretically as
i_atomic part-hood for some i. If this theory is not clearly 
inconsistent I would tend to think it would be as strong as second 
order arithmetic, which is the consistency strength of Type theory 
with Infinity. There is a close resemblance between this multi-level
discrimination hierarchy and the typed set hierarchy. It seems that the 
Set concept could be traced to multi-level discrimination concept. 

Best Regards

Zuhair