Re: V



Well, if you really had a type w^w sequence of primitive constants
(which could be indexed by all the *finite* sequences of natural
numbers with a certain well-ordering), then I think you'd get
something interpretable in ZFC+w^wIC.


hmmm......, let me try:


DEFINITION OF OMEGA DIMENSIONAL "oo-ML".


w-dimensional oo-ML: is the set of all sentences entailed (from first
order logic
with identity and the matrix of primitive constants Vi1i2i3..... were
for each vector I=i1i2i3....... we have one constant primtive VI.

Primitives: e,=, VI for each vector I=i1i2i3....
were ij in I =1,2,3,...... and j=1,2,3,.....


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 vector I=i2i3......,if F is a formula in
which x is not free
then all
closures of
ExeV((i1)+1)I Ay(yex<->(yeVi1I&F(y)))
are axioms.

Clarification V((i1)+1)I = V((i1)+1)i2i3i4...

and Vi1I=Vi1i2i3.....



For every vector I=i1i2i3........, 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)).

9) Layers: for every two vectors I=i1i2i3......, and
J=j1j2j3........, if ik in I = jk in J for every k>n
and if in in I > jn in J then all closures of

Ax(xeVJ -> xeVI)

are axioms.

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


10) Anti-choice: for every vector I=i1i2i3....
the sentence
DI subnumerous_to VI
is an axiom


with the usual meaning of 'subnumerous_to'


were x=DI<-> Ay(yex<->(yeVI & y is ordinal)).


Of course according to this theory for every vector I
the sentence DI( ~DIeVI) is a theorem.

/


Theory Definition Finished.

My claim is that this theory is equi-interpretable with
ZF/ZFC + w^w IC.

Actually I am not so sure if we have wxw matrix of VI constants
or if it is aleph-0 x aleph-0 matrix.
Since if it is the last then the theory would be in uncountable
language affecting its recursive properties.
However I think we have VI as w^w matrix and w^w is countable since it
is ordinal exponentiation here
which is countable, in contrast to aleph-0 x aleph-0 which is cardinal
exponentiation which is not countable.

One of course can go to higher inaccessible cardinals using the same
method, but that would be undersirable.




Zuhair







.

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