[FOM] Multilayer Set Theories.

Mon May 28 13:08:30 EDT 2007

The MULTILAYER THEORIES

Two kinds of theories that I have lately constructed.
Both types of theories has essentially the same idea.
They are 
theories in first order logic with identity and
primitive constant 
sets V1,V2,V3,.....,Vn in the finite version and
V1,V2,V3,.......... 
( the infinite version ) , each of these sets contain
only proper 
subsets of them as members in them, so this theory has
no ur-elements; 
so Vi contain only proper subsets of Vi as members in
it. Furthermore 
not any proper subset of Vi is a member of Vi, only
those in which 
there exist a member of Vi that is supernumerous to
them are members 
of Vi, those who do not would be members of Vi+1 but
not of Vi. Every 
Vi has Vj as a proper subset of it iff j<i 
and every Vj is a member of Vi iff j<i. 
each Vi is termed as a layer. so Vi is the i-th layer 
Comprehension is only within the same layer, and it
results in the 
existence of sets within this layer and sets in the
layer above. 
Pairing, Union, Power , Separation and Replacement are
operative 
WITHIN each layer to produce sets within this layer
only, Only 
Comprehension can produce sets in its layer and the
layer above. 
These theories are well founded, although we can have
versions of them 
without regularity and by then would be not well
founded theories. 
Choice is entailed at each layer if we use the strong
version of these 
theories ( see below ) ; on the other hand if we use
the weak version 
of these theories then choice is not entailed at each
layer. In 
addition we can have versions of these theories which
has the axiom of 
negation of choice added to the weak versions at each
layer. 

So MK( Morse-Kelly ) would be n-multilayer set theory
when n=1. 

So the idea of proper classes not being members of
other classes no 
longer hold here, since they are members of sets of
higher layer, and 
some of these sets are smaller than sets in the layer
below them, so 
in this theory V1 which is the same as V in MK 
can be a member of { 0,V1}, and of course {0,V1} is
not a member of 
V1, it is a member of V2 and sets in higher layers. 

Sets in this theory can be typed after the minimal
layer they are 
members of. 

so x is Ln set <-> ~EVi ( xeVi & i<n ). 

Or equivalently 

x is Ln set <-> A Vi ( xeVi -> i>=n). 

So L2 sets do not exist as members in V1, but they are
members of 
V2,V3,.... 

So 0 is L1 set 
V1 is L2 set 
in general Vn is Ln+1 set. 

DEFINITION OF INFINITE MULTILAYER SET THEORY "oo-ML". 

oo-ML: is the set of all sentences entailed (from
first order logic 
with identity and the infinite list of primitive
constants 
V1,V2,V3,.... ) by the following non 
logical axioms: 

Primitives: e,=,V1,V2,V3,......... 

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

2)Axiom of Regularity:Ax(~x=0 -> Ey(yex & y disjoint
x)). 

3) Comprehension: For every i if F is a formula in
which x is not free 
then all 
closures of 
ExeVi+1Ay(yex<->(yeVi&F(y))) 
are axioms. 

For every i all the following sentences are axioms: 

4) Pairing: AreViAseViExeViAy(yex<->(y=r v y=s)). 

5) Union: AaeViExeViAy(yex<->Ez(zea&yez)). 

6) Power:AaeViExeViAy(yex<->Az(zey->zea)). 

7) Infinity: ExeV1( 0ex & Ay(yex->yU{y}ex)) 

8) Membership: Ax( xeVi <-> (Ay(yex->yeVi) & Ez( zeVi
and z 
supernumerous_to x))). 

with the usual meaning of 'supernumerous_to'. 

Strong version of 8) would be: 

8) Membership: Ax( xeVi <-> (Ay(yex->yeVi) & Vi
supernumerous_to x)). 

Negation of choice can be added to the theory with
weak version of 
membership, at each layer, in the following manner. 

9) Anti-choice: Di subnumerous_to Vi 

with the usual meaning of 'subnumerous_to' 

were x=Di<-> Ay(yex<->(yeVi & y is ordinal)). 

Of course according to this theory for every i 
the sentence Di( ~DieVi) is a theorem. 

Clarification: 3)4)5)6)8)both versions 9) are infinite
schemas of 
axioms were each sentence is an axiom for each i ;
while 1)2)7) are 
single axioms. 

/ 

Theory Definition Finished. 

Zuhair 

DEFINITION OF THE FINITE MULTILAYER SET THEORY "n-ML".

n-ML: is the set of all sentences entailed (from first
order logic 
with identity and the infinite list of primitive
constants 
V1,V2,V3,....,Vn ) by the following non logical
axioms: 

Primitives: e,=,V1,V2,V3,......... 

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

2)Axiom of Regularity:Ax(~x=0 -> Ey(yex & y disjoint
x)). 

3.A) Comprehension schema 1: For every i<n if F is a
formula in which 
x is not free 
then all 
closures of 
ExeVi+1Ay(yex<->(yeVi&F(y))) 
are axioms. 

3.B) Comprehension schema 1: if F is a formula in
which x is not free 
then all 
closures of 
ExAy(yex<->(yeVn&F(y))) 
are axioms 

For every i=1,2,3,..,n ; all the following sentences
are axioms: 

4) Pairing: AreViAseViExeViAy(yex<->(y=r v y=s)). 

5) Union: AaeViExeViAy(yex<->Ez(zea&yez)). 

6) Power:AaeViExeViAy(yex<->Az(zey->zea)). 

7) Infinity: ExeV1( 0ex & Ay(yex->yU{y}ex)) 

8) Membership: Ax( xeVi <-> (Ay(yex->yeVi) & Ez( zeVi
and z 
supernumerous_to x))). 

with the usual meaning of 'supernumerous_to'. 

Strong version of 8) would be: 

8) Membership: Ax( xeVi <-> (Ay(yex->yeVi) & Vi
supernumerous_to x)). 

Negation of choice can be added to the theory with
weak version of 
membership, at each layer, in the following manner. 

9) Anti-choice: Di subnumerous_to Vi 

with the usual meaning of 'subnumerous_to' 

were x=Di<-> Ay(yex<->(yeVi & y is ordinal)). 

Of course according to this theory for every i 
the sentence Di( ~DieVi) is a theorem. 

Clarification: 3.A and 3.B are infinite schemas
4)5)6)8)both versions 
9) are finite schemas of 
axioms were each sentence is an axiom for each i ;
while 1)2)7) are 
single axioms. 

/ 

Theory Definition Finished. 

Zuhair

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