fom
Posts:
1,969
Registered:
12/4/12


Re: A logically motivated theory
Posted:
May 18, 2013 1:58 PM


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.
> 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.
> [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.
You probably mean 'axiom schema' here.
> / > > 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?
> It is a purely logical motivation.
Which logic? This is stronger than firstorder because of the quantifications.
> 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. >

