Date: May 18, 2013 6:47 PM Author: fom Subject: Re: A logically motivated theory 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.