fom
Posts:
1,968
Registered:
12/4/12


Re: A logically motivated theory
Posted:
May 18, 2013 3:38 PM


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))))

