# [FOM] Multi-level Discrimination

Zuhair Abdul Ghafoor Al-Johar zaljohar at yahoo.com
Tue Feb 21 04:12:35 EST 2012

```Addendum:

It is more appropriate to add the following axiom scheme, preferably after
axiom scheme of Atomicity.

Atomistic Supplementation:

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

Zuhair

> On Mon, 20 Feb 2012 00:53:43 -0800 (PST)
> Zuhair Abdul Ghafoor Al-Johar <zaljohar at yahoo.com> wrote:
>
> 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
>
> /
>

```