Date: May 18, 2013 3:38 PM
Author: fom
Subject: Re: A logically motivated theory
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))))