[FOM] Multi-level Discrimination
Zuhair Abdul Ghafoor Al-Johar
zaljohar at yahoo.com
Mon Feb 20 03:53:43 EST 2012
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
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.
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