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. Zuhair Al-Johar March 2 2012