On 5/18/2013 2:43 PM, Zuhair wrote: > On May 18, 8:58 pm, fom <fomJ...@nyms.net> wrote: >> On 5/18/2013 10:40 AM, Zuhair wrote: >> >> >>> 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. > > Zuhair >
Once again, I think you should take a look at NF by Quine.