Re: V

On Jun 1, 4:52 am, zuhair <zaljo...@xxxxxxxxx> wrote:
On May 31, 12:20 am, Rupert <rupertmccal...@xxxxxxxxx> wrote:





On May 31, 1:40 pm, zuhair <zaljo...@xxxxxxxxx> wrote:

On May 30, 10:22 pm, Rupert <rupertmccal...@xxxxxxxxx> wrote:

On May 31, 12:28 pm, zuhair <zaljo...@xxxxxxxxx> wrote:

On May 30, 9:06 pm, Rupert <rupertmccal...@xxxxxxxxx> wrote:

On May 30, 11:07 pm, zuhair <zaljo...@xxxxxxxxx> wrote:

On May 30, 12:43 am, Rupert <rupertmccal...@xxxxxxxxx> wrote:

On May 30, 6:13 am, zuhair <zaljo...@xxxxxxxxx> wrote:

Hi all,

I would like to discuss the following theory , which might serve as an
introductory discussion to the Multilayer theories that I presented in
another thread.

Theory T is the set of all sentences entailed( from first order logic
with identity and the primitive constant V) by the following non
logical axioms:

Primitives: e,=,V

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

2) Regularity:Ax( Ey(yex) -> Ey(yex & y disjoint x)).

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

4) Pairing: ArAsExAy( (yex<->(y=r v y=s)) &
( (reV & seV) -> xeV ) ).

5) Union: ArExAy((yex<->Ez(yez&zer)) & (reV -> xeV)).

6) Power:ArExAy((yex<-> Az(zey->zer)) & (reV -> xeV)).

7) Infinity:Ex( xeV & 0ex & Ay(yex->yu{y}ex)).

8) Membership:Ax(xeV<->(Az(zex->zeV) & x subnumerous_to V)).

/

Theory definition finished.

This theory can be proved consistent in ZF.

Then ZF is inconsistent.
This theory proves ZF.
Since this theory proves Replacement within V

Remember the version I was talking about here didn't have any version
of Replacement. So no, it doesn't. Or if it does, then you're right,
quite a weak fragment of ZF is inconsistent. So show me your proof.

Theorem schema of Replacement:if F
is a formula in two variables and in which x is not
free then all closures of
AxE!y(F(x,y))Ay(F(x,y)->yeV) -> AreVEceVAy(yec<->ExerF)
are axoims.

Proof:from the left of the implication we have
Ax(xec -> xeV)
from the uniqueness on the left we have
~c supernumerous_to r.
But P(r) supernumerous_to r
Then P(r) supernumerous to c
But P(r)eV
Then ceV . (Membership).
theorem proved.

Anyhow Rupert, at the beginning of this post I posted two theories,
ONe that doesn't have replacement in it ( the MK like theory but not
MK itself) and the other one
is the ZFC like theory with the 10 axioms ( see the second message in
this thread) which has Replacement in it.
However even the first theory proves Replacement within V
but it doesn't prove it outside of V.
The ZFC like theory has replacement and separation as a axiom that
works within and outside V, and thus it is a stronger theory than the
first one,but not stronger than
the oo-Multilayer theory.

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Okay, I get it. You're right. Your theory proves the consistency of
ZF. It can be interpreted in ZF+"an inaccessible cardinal exists".

Ok, Rupert, I would like to know
the formula of 'x is inaccessible cardinal'.
can you please write it in FOL.

I've done this for you elsewhere in the thread. Here is a version of
"the cardinality of x is an inaccessible cardinal" which is
appropriate in the absence of the axiom of choice:

Ay(y subnumerous to x->(Az((there exists a surjection from P(y) to z)->z is subnumerous to x))) & (Ay(y subnumerous to x & Az(z e y -> z

subnumerous to x)) -> Az(there is a surjection from the union of y to
z -> z is subnumerous to x))

You can expand it further if you want. If we assume the axiom of
choice then it can be given a simpler formulation.

I think ZF+inaccessible cardinal is writtin in this manner.

ZF axioms
+ axiom of inacessible cardinal which is
Ex( x is inaccessible cardinal ).

Put the formula I gave above in brackets and precede it by Ex. Then
you have a version of the axiom that an inaccessible cardinal exists
which can be added to ZF.

Now my question to you , were do this inaccessible cardinal exist.
Does it exist in V
or outside V.

The cardinality of V is inaccessible. If the theory is consistent then
it cannot prove that there is in inaccessible cardinal in V.

What happens if I add the following axiom to theory
I posted in this thread( the one with the 10 axioms,
what I called the ZFC like thoery )

Ex(xeV & x is inaccessible cardinal)

Would this addition of this axiom result in an inconsistent theory.
Or it will result in a theory that is stronger
than ZF+inaccessible cardinal exists.

Did you include any form of choice? I can't remember. If you take the
theory you gave in the first post and add the axiom that there exists
an inaccessible cardinal in V, as well as all the axioms of ZFC in
unrestricted form, you get a theory which is equi-interpretable with
ZFC+"two inaccessible cardinals exist". This theory is generally
believed to be consistent, but it has stronger consistency strength
than ZFC+"one inaccessible cardinal exist", in the following sense.
Let Con(T) be the sentence asserting that theory T is consistent, and
let IC be sentence asserting that an inaccessible cardinal exists.
Then if ZFC+Con(ZFC) is consistent, it cannot prove Con(ZFC+IC), and
if ZFC+Con(ZFC+IC) is consistent, it cannot prove Con(ZFC+"there exist
two inaccessible cardinals"). And so on. But most set theorists
believe that ZFC+"there exist n inaccessible cardinals") is consistent
for any n and that many much more powerful large-cardinal axioms are
consistent.

OK, Rupert. I read your post, also I read some about inaccessible
cardinals and their consistency, although I didn't read it enough to
have a full idea about them, but I think I grapsed the main idea.

You are saying that oo-Multilayer theory ( not the one in this post,
the one in this post is not multilayered theory) is interpretable in
ZF/ZFC + one inaccessible cardinal and I think you mean the strongly
inaccessible cardinals.

However It appears to me that oo-ML is Equi-interpretable with
ZF/ZFC + inaccessible cardinal exist.

You cannot prove the consistency of oo-ML from ZF/ZFC+IC, since they
are equi-interpretable.

Perhaps I am wrong ( quite likely) but if so then I would be glad to
see were I am wrong in.

So what I am claiming here is that oo-ML is not a subtheory of ZF/ZFC
+IC.
I am claiming that oo-ML is equi-interpretable with ZF/ZFC+IC.

Personally I don't think that we can have an inaccessible ordinal in
ZF/ZFC+IC and not in oo-ML and vice a versa.


I'll have to have a look at your definition of the theory. I suspect
that there is an interpretation of your theory on which "There is an
inaccessible cardinal" comes out true, but I doubt that all the
Replacement axioms in unrestricted form come out true.

However I can extend oo-ML to be two dimensional.
The theory of oo-ML I presented in my website at
http://zaljohar.tripod.com/index.html#multilayer
was a one dimentional oo-ML.
Two dimensional oo-ML would have the
the list of primitive constants Vi.j
were for each i and j , i=1,2,3,...
j=1,2,3,..... there is a primitive constant Vi.j
so we have a two dimensional matrix of Vij.
so the V1,V2,V3,.... primitive constants in the
one dimensional oo-ML would be
V1.1,V1.2,V1.3,.....................
so Vi.j were i=1 are inaccessible cardinals
present in ZF/ZFC+IC.
But Vi.j were i>1 are inacessible cardinals
presnet in ZF/ZFC+iIC
i.e the i inaccessible cardinals.

OF course to do that I should add an axiom schema
of dimensionas to
oo-ML that states that

Axiom scheam of Dimensions:
For every i,j,m,k were k>i
Ax(xeVi.j -> x subnumerous_to Vk.m)
is an axiom.

the other axioms needs some workable modifications,
were Pairing schema for example would be
For the same i
AreVi.jAseVi.j ExeVi.j Ay(yex<->(y=r v y=s))
In a similar way union, power,membership
and comprehension should be modified.

so the i-th dimensional oo-ML is equi-interpretable with
ZF/ZFC + i inaccessible cardinals exist.

Example of an i-th dimensional oo-ML is the theory which
has Vi.j.k is called the three dimensional oo-ML
we can have omega-dimensional oo-ML which will
be equi-interpretable with ZF/ZFC + w IC.

That's why I like the formulation of oo-ML
it can be extended to accomodate any theory
and its formulation is simple
and it follows the same STRUCTURE.
It is a more displained theory.
IT is not vage.
It is simple.

I will try to work out the w-dimensional oo-ML
thoery so that things would be clearer and one
can see clearly weather it is equi-interpretable
with ZF/ZFC+ wIC.

Zuhair





Zuhair

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Relevant Pages

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