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Topic: A logically motivated theory
Replies: 15   Last Post: May 21, 2013 8:22 AM

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Zaljohar@gmail.com

Posts: 2,665
Registered: 6/29/07
Re: A logically motivated theory
Posted: May 18, 2013 3:21 PM
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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.
>
> > Axioms:
>
> > [1] If E, D are primitive binary relation symbols then:
>
> > E,D are extensional -> (For all x,y: (for all z. z E x iff z D y) ->
> > x=y).

>
> > is an axiom.
>
> So, for definiteness, let
>
> E be membership
>
> D be initial segment
>
> in a theory for which every limit ordinal is a model.
>

> > [2] If E, D are primitive binary relation symbols; P,Q are first order
> > language formulas in which x do not occur free, then

>
> > E,D are extensional ->
> > for all x ((for all y. y E x iff Q) & (for all y. y D x iff P)) ->
> > (for all y. P<->Q)

>
> >   is an axiom
>
> You are referring to relations here using
> free variables.
>
> You probably mean 'axiom schema' here.
>

I mean the closure over all those free varaibles per each Q and P is
an axiom.
Of course [2] is a schema.
> > [3] If P is first order predicate, then
>

> > Exist E,x: E is extensional & for all y. y E x iff P(y)
>
> > is an axiom.
>
> > where E range over primitive binary relations only.
>
> You are quantifying over relations here.
>

Correct.
> You probably mean 'axiom schema' here.

Of course, an axiom for each P.
>
> > /
>
> > It is possible that this might interpret PA?
>
> > The whole motivation beyond this theory is to extend any first order
> > predicate by objects.

>
> Could you please clarify this remark?
>

Yes, for EVERY first order predicate P there will be an object X such
that there exist an extensional relation E such that for all y. y E X
iff P(y).

Now from the above axioms this X would represent an object that
*uniquely* stands for P.

> > It is a purely logical motivation.
>
> Which logic?  This is stronger than first-order
> because of the quantifications.
>


Yes, still it is logic, the main motivation is a logical one that of
having object representatives of Predicates, for example denote P* to
be the object representing predicate P, what we want is the following

P*=Q* iff P<->Q

Now this is copying a logical relation (biconditional) into equality
between objects. So we are mirroring logical stuff at the object
level.

According to this theory of course we have each object P* being
"definable" in terms of some extensional relation E. so we have the
following definition:

P*=X iff Exist E. E is extensional & for all y. y E X iff P(y)

Zuhair

>
>
>
>
>
>

> > If this does
> > interpret PA and no inconsistency is shown with it, then PA can in a
> > sense be seen as a kind of a logical theory. IF we extend [3] to allow
> > infinitely long formulas, then possibly second order arithmetic would
> > be provable? if so then it would be a kind of a logical theory also.

>
> > All of that motivates logicism.




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