On May 18, 10:38 pm, fom <fomJ...@nyms.net> wrote: > On 5/18/2013 2:21 PM, Zuhair wrote: > > > > > > > > > > > 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. > > How so? The formula seems to have an > existential quantifier applying to a > relation and a subformula with the > quantified 'E' as a free variable: > > 'E is extensional' > > Using 'R' for "Relation", I read > > Ax(Set(x) <-> EREP(extensional(R) /\ Ay(yRx <-> P(y))))
I meant that P must be first order. There is no General so to say membership relation E, there are separate different membership relations all of which are 'primitive' relations.
Of course one might contemplate something like the following:
Define(E): x E X iff Exist R Exist P( extensional(R) /\ Ay(yRX <- >P(y) /\ P(x) )
This E relation would be something like a 'general' membership relation, but this is not acceptable here, because it is 'defined' membership relation and not 'primitive'. When I'm speaking about membership relations in the axioms I'm speaking about ones represented by 'primitive' symbols and not definable ones.