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


Re: A logically motivated theory
Posted:
May 18, 2013 6:47 PM


On 5/18/2013 2:52 PM, Zuhair wrote: > 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. > > Zuhair >
I will refrain from making a long posting based on my earlier mistaken impressions.
However, your remarks here suggest that you should take a look at Quine's "New Foundations" and the interpretation of stratified formulas. If you have Quine's book "Set Theory and Its Logic" available to you, a couple of hours reading the appropriate chapters and flipping forward to the definitions in earlier chapters should give you some sense of the matter as he saw it.
I believe it is Thomas Forster who is making his book available online concerning NF, if you should become more interested in Quine's theory.

