Re: Zuhair's set theory

On Oct 27, 11:20 am, Zaljo...@xxxxxxxxx wrote:
Hi all,

I will present a set theory here, the intention beyond which is that
this theory would clarify our intutive notions beyond the ad hoc
axiomatic system of ZF ( with inaccessible cardinals ).

Zuhair's set theory "Zh":

Zh is the collection of all sentences entailed from ( by first order
logic with identity and the primitive constants V1,V2,V3,...... were
for each i,j such that ~i=j we have ~Vi=Vj ) the following axioms:

1) Axiom of Extensionality: Az( zex <-> zey ) -> x=y

2) Axiom of Transitive closure:

AxE!y( Am(mex->mey) & y is transitive &
Am( (mey & ~mex) -> Ez( zey & mez ) ) ).

were: y is transitive <-> Amn( (mey & nem) -> ney )

Definition:

y=Tc(x) <-> ( Am(mex->mey) & y is transitive &
Am( (mey & ~mex) -> Ez( zey & mez ) ) ).

3) Axiom of Uniformity: Ax ( ~xeTc(x) ).

4) Completeness axiom schemas: For every i

(xeVi & mex) -> meVi
(xeVi & y subset_of x ) -> yeVi

were: y subset_of x <-> Az( zey -> zex ).

5) Axiom schema of Comprehension: For every i, if P is a formula in
which x is not free , then all closures of

Ex( xeVi+1 & Ay( yex <-> ( yeVi & P(y))))

are axioms.

6) Axiom of schema of inclusion: For every i, if P is a formula in
which x is not free , then all closures of

( Ay(yex<->(yeVi & P(y))) & Ay(xeTc(y) ->~P(y)) ) -> xeVi

are axioms.

/

Theory definition finished.

************

Important theorems:

Ex Ay ~yex

Proof: Let P<->~y=y

from comprehension theorem is proved.

E!x Ay ~yex

Proof: Extensionality.

Definition: x=0 <-> Ay ~yex

******

0eV1

Proof: Let P<-> ~y=y and Let i=1

it is obviouse that Ay( 0eTc(y) -> y=y ) is always true
since every set is equal to itself ( identity theory )
then from Inclusion 0eV1.

******

Theorem schema of Pairing : For every i

AreVi AseVi Ex( xeVi & Ay( yex<-> ( y=s v y=r ) ) )

is a theorem.

Proof: Since reVi and seVi
then (yeVi & (y=s v y=r)) <-> ( y=s v y=r )

From axiom schema 6 we have:

For seVi , reVi
( Ay(yex<-> (y=s v y=r)) & Ay(xeTc(y) ->~(y=s v y=r)) ) -> xeVi

It is clear that Ay(yex<-> (y=s v y=r)) -> Ay(xeTc(y) ->~(y=s v y=r))

is a theorem of this theory, otherwise we will violate Uniformity.

Theorem proved.

In exactly a similar manner to the proof of pairing we can prove Union
per Vi , power per Vi , separation per Vi , Replacement per Vi.

Infinity can be proved easily: Since the set x of all hereditarily
finite sets satisfy the condition Ay( xeTc(y) -> ~ y is hereditarily
finite ) , then xeV1, since all hereditarily finite sets are in V1.

Now existence of Omega is proved easily from Separation schema
on the set of all hereditarily finite sets using the predicate
finite ordinal.

********

Reflection schema of Ackermann's is also proved from 6.
In a similar way to that of Pairing.

So all the axioms of ZF ( other than Regularity and choice ) are
proved from this theory.

*********

The main difference between this theory and ZF+Regularity is that this
theory is NOT well founded, in a sense that there is nothing to
suggest that there cannot exist a set with infinite descending epsilon
membership.

I call this theory Semi-Founded theory or Uniform theory
since from axiom of uniformity we cannot have circular membership.

In another post I said that ZF is based on Regularity,
More preciselly I wanted to say that it is based on what I call here
as Uniformity.

I would like to see objections to this theory.

Zuhair

I'm a bit worried about your definition of "transitive closure". I
suspect that in ZF we can prove that transitive closure as you define
it is not unique. If this is right, and if your theory contains ZF as
you claim, then your theory is inconsistent.

.

Relevant Pages

  • Re: Zh set theory
    ... Axiom of Transitive closure: ... Axiom schema of Transitiveness: ... subset of a member of V is a member of V. ... x is within the transitive closure of y, ...
    (sci.logic)
  • Re: Zuhairs set theory
    ... Completeness axiom schemas: For every i ... Axiom schema of Comprehension: For every i, if P is a formula in ... is a theorem of this theory, otherwise we will violate Uniformity. ... I'm a bit worried about your definition of transitive closure. ...
    (sci.logic)
  • Re: Zuhairs set theory
    ... Completeness axiom schemas: For every i ... Axiom schema of Comprehension: For every i, if P is a formula in ... is a theorem of this theory, otherwise we will violate Uniformity. ... on the set of all hereditarily finite sets using the predicate ...
    (sci.logic)
  • Zuhairs set theory
    ... Completeness axiom schemas: For every i ... Axiom schema of Comprehension: For every i, if P is a formula in ... is a theorem of this theory, otherwise we will violate Uniformity. ... on the set of all hereditarily finite sets using the predicate ...
    (sci.logic)
  • Re: Zh set theory with some detail.
    ... Axiom schema of Comprehension: For every n, ... Proof: Power and Transitiveness. ...
    (sci.logic)