[FOM] Connection Set Theory

[Zuhair Abdul Ghafoor Al-Johar]

Dear FOMers:

 The following is the exposition of Connection Set Theory,
a theory that I have been lately defining with the help of 
Mr. Randall Holmes.

Exposition of Connection Set Theory "CST"

 CST is the set of all sentences entailed
(from FOL with identity "=", and membership "E")
by the following axiom scheme:

Comprehension: If phi is a well connected formula
in which y is free, and in which x do not occur,
then all closures of

Exist x (~ Exist y (for all z ( z E y iff z E x ) and ~ y=x)
and for all y ( y E x iff phi ))

are axioms.

Definition of well connected formulas:

 We say that a variable x is connected to a variable y in phi iff
any of the following formulas appear in phi

x E y , y E x , y=x , x=y

 We refer to a function s from {1,...,n} to variables in phi
as a chain of length n in phi iff for each appropriate index i:
s(i) is connected to s(i+1), and s(i),s(i+2) are two different
occurrences in phi(i.e. s(i) occur at a place in phi that is
different from the place s(i+2) occur at).

 A chain from x to y is defined as
a chain s of length n>1 with s(1)=x and s(n)=y.

 phi is a well connected formula iff
  for each variable x in phi there is no chain from x to x.

/ Theory definition finished.

 Mr. Randall Holmes pointed out that all well connected formulas
are stratified! and by then he showed that this theory
is a sub-theory of New Foundations "NF".

 Now the above theory prove Extensionality (R. Holmes), universe, empty,
absolute complements, pairing, Boolean and Set unions\intersections,
power, Cartesian products, Wiener's ordered pairs, Relations,
Functions, Injections, Surjections, Bijections, domains of relations,
ranges of relations, converses of relations, products of relations,
singleton images of relations, it also define a set extension
for the identity relation. It can define Cardinality in the
Fregian manner.

 I don't know if this theory can prove Infinity?
nor do I know the stance of this theory from Choice?

 I still don't know actually if this theory is strictly weaker
than NF? although it certainly gives this impression.

This is a logical Puzzle actually!

 So this theory is useful mathematically, and the most important
thing is that it is ***safe***. With well connected formulas defined
above, one can hardly imagine that a paradox can occur.

 This theory was discussed with Mr. Randall Holmes who assisted me
greatly at reaching to a rigorous definition of well connected
formulas, and indeed he is the one who arrived at the first
rigorous definition of the notion of connectedness mentioned
here in this theory.

 Below is Mr. Randall Holmes definition of the notion of
connectedness and well connected formulas:

 Let phi be a formula in the language of first order logic with
equality and membership as the only primitives.

 We say that a variable x is overconnected to a variable y in phi iff
more than one of the following statements is true:

(1)  x E y appears as a subformula of phi

(2)  Either or both of x=y and y=x appears as a subformula of phi

(3)  y E x appears as a subformula of phi

 It is important to note that x is overconnected to x if x E x
appears in phi, as both (1) and (3) would then be true.

 We say that a variable x is disconnected from a variable y in phi iff
none of the formulas

x E y, y E x, x = y, y = x appear in phi.

 We say that a variable x is properly connected to a variable y in phi
iff x is not disconnected from y and x is not overconnected to y.

 We refer to a function s from {1,...,n} to variables in phi
as a chain of length n in phi iff s(i) is properly connected 
to s(i+1) for each appropriate index i, 
were s(i) is distinct from s(i+2).

 A chain from x to y is defined as a chain s of
length n>1 with s(1)=x and s(n)=y.  Note that a chain from x to y is
NOT a chain from y to x; this notion has direction.

 We say that phi is a well connected formula iff 
no pair of variables (distinct or otherwise) is overconnected in phi
and for each pair of distinct variables x and y,
there is at most one chain from x to y.

/ Definition finished.

 Both definitions are not equivalent! though I think that
comprehension using either approach yields the same result.

Best Regards.

Z.Al-Johar