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. > > > 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. > > >  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. > > You probably mean 'axiom schema' here. > > > / > > > 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?
I'll expand on that. The motivation of this theory is to 'define' objects that uniquely corresponds to first order logic predicates,i.e for each first order predicates P,Q there exist objects P* and Q* such that P*=Q* iff for all x. P(x)<->Q(x)
We want to do that for EVERY first order predicate P.
The plan is to stipulate the existence of multiple PRIMITIVE binary extensional relations, each one of those would play the role of a set membership relation after which object representative of predicates are defined.
Now the first axiom ensures that no 'distinct' objects can be defined after equivalent predicates all across the membership relations, so although we have many membership relation but yet any objects X,Y that are co-extensional over relations E,D (i.e. for all z. z E X iff z D Y) are identical! The second axiom (schema of course) ensures that no object represent non-equivalent Predicates, and so although we do have 'multiple' membership relations (primitive extensional relations) however from axiom schemas 1 and 2 this would ensure that each object defined after any of those relations would stand 'uniquely' for a single predicate.
The last axiom scheme is just a statement ensuring the existence of an object that extends each first order predicate after some membership relation.
Those objects uniquely corresponding for first order predicates are to be called as: Sets.
The point is that paradoxes are eliminated because of having 'multiple' extensional relations each standing as a membership relation.
I'm not sure if that would interpret PA, but if it does, then PA can be said to be a PURELY logical theory!
Now if (and this is a big if) we allow infinitely long formulas to define first order predicates (infinitary first order languages) then second order arithmetic 'might' follow. And I think if this is the case, then second order arithmetic is also PURELY logical!
This mean that the bulk of traditional mathematics (most of which can be formulated within proper subsets of second order arithmetic) is purely logical!
However I don't think that higher mathematics can have a pure logical motivation comparable to the above, the motivation behind those can be said to be 'structural', or 'constructive' in the ideal sense that I've presented in my latest philosophical notes on my website and to this Usenet.
> > > It is a purely logical motivation. > > Which logic? This is stronger than first-order > because of the quantifications. > > > > > > > > > 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.