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))))