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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
A logically motivated theory
Posted: May 18, 2013 11:40 AM
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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.

Now sets are defined as

x is a set iff Exist E,P: E is extensional & for all y. y E x <-> P(y)

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.

[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

[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.

/

It is possible that this might interpret PA?

The whole motivation beyond this theory is to extend any first order
predicate by objects. It is a purely logical motivation. 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.

Zuhair




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