[FOM] Abstract Cartesian Products

[Zuhair Abdul Ghafoor Al-Johar]

Dear Sirs, 

If the following schema is added to axioms of Z, then the resulting theory is ZF. 

Schema of Abstract Cartesian Products: if F is a two place function symbol 
definable after a formula in which C does not occur free, then 

[For all a,b,c,d (F(a,b)=F(c,d) <-> a=c ^ b=d) 
-> 
For all A,B exists C for all y (y in C <-> Exist a in A, b in B (y=F(a,b)))] 

is an axiom. 

A finding attributable to Adrian Mathias, and discussed by Randall Holmes 
at: 

http://math.boisestate.edu/~best/best18/Talks/holmes_best18.pdf 

Also if we add the following conditions to the antecedent of the above formula, still 
the resulting schema would prove Replacement. 

For every set R of F pairs there is a set of all first projections of those pairs (i.e. domain of R) 

For every set R of F pairs there is a set of all second projections of those pairs (i.e. range of R) 


For every set R of F pairs there exists sets A,B such that AxB exists and R is a subset of AxB. 

where AxB={F(a,b)| a in A, b in B}. 

Informally this schema translates to saying that every ordered pair function 
enjoying eligible abstract properties for it to be used in implementing relations, 
Can be used to implement relations.  

Clearly Z fails this scheme, however just adding this scheme which addresses such 
a simple point [conceptually so to speak] resulting in blowing up the strength of 
the resultant theory to that of ZF, is in some sense striking. Informally it mounts to saying
that ZF is equal to Z + asserting abstractedness of relations by showing the immateriality 
of the choice of implemented desirable ordered pairs. Which seems to me to be a very naive 
assumption. 

My question why is this not considered generally to be an argument in support of Replacement and 
ZF in general. Although Holmes had discussed it in the referred document, but I don't see him fully 
refuting it conceptually so to speak. 

Best Regards, 

Zuhair Al-Johar