On May 18, 8:58 pm, fom <fomJ...@nyms.net> wrote: > On 5/18/2013 10:40 AM, Zuhair wrote: > > > In this theory Sets are nothing but object extensions of some > > predicate. This theory propose that for every first order predicate > > there is an object extending it defined after some extensional > > relation. > > > This goes in the following manner: > > > Define: E is extensional iff for all x,y: (for all z. z E x iff z E y) > > -> x=y > > > where E is a primitive binary relation symbol. > > So, > > <X,E> > > is a model of the axiom of extensionality. > > > Now sets are defined as > > > x is a set iff Exist E,P: E is extensional & for all y. y E x <-> P(y) > > So, > > xEX <-> ... > > where > > ... is a statement quantifying over relations and predicates.
No ... is a statement quantifying over objects. > > > Axioms: > > >  If E, D are primitive binary relation symbols then: > > > E,D are extensional -> (For all x,y: (for all z. z E x iff z D y) -> > > x=y). > > > is an axiom. > > So, for definiteness, let > > E be membership > > D be initial segment > > in a theory for which every limit ordinal is a model. > > >  If E, D are primitive binary relation symbols; P,Q are first order > > language formulas in which x do not occur free, then > > > E,D are extensional -> > > for all x ((for all y. y E x iff Q) & (for all y. y D x iff P)) -> > > (for all y. P<->Q) > > > is an axiom > > You are referring to relations here using > free variables. > > You probably mean 'axiom schema' here. > I mean the closure over all those free varaibles per each Q and P is an axiom. Of course  is a schema. > >  If P is first order predicate, then >
> > Exist E,x: E is extensional & for all y. y E x iff P(y) > > > is an axiom. > > > where E range over primitive binary relations only. > > You are quantifying over relations here. > Correct. > You probably mean 'axiom schema' here.
Of course, an axiom for each P. > > > / > > > It is possible that this might interpret PA? > > > The whole motivation beyond this theory is to extend any first order > > predicate by objects. > > Could you please clarify this remark? > Yes, for EVERY first order predicate P there will be an object X such that there exist an extensional relation E such that for all y. y E X iff P(y).
Now from the above axioms this X would represent an object that *uniquely* stands for P.
> > It is a purely logical motivation. > > Which logic? This is stronger than first-order > because of the quantifications. >
Yes, still it is logic, the main motivation is a logical one that of having object representatives of Predicates, for example denote P* to be the object representing predicate P, what we want is the following
P*=Q* iff P<->Q
Now this is copying a logical relation (biconditional) into equality between objects. So we are mirroring logical stuff at the object level.
According to this theory of course we have each object P* being "definable" in terms of some extensional relation E. so we have the following definition:
P*=X iff Exist E. E is extensional & for all y. y E X iff P(y)
> > > > > > > > If this does > > interpret PA and no inconsistency is shown with it, then PA can in a > > sense be seen as a kind of a logical theory. IF we extend  to allow > > infinitely long formulas, then possibly second order arithmetic would > > be provable? if so then it would be a kind of a logical theory also. > > > All of that motivates logicism.