Re: V

On Jun 1, 12:32 pm, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:
On Jun 1, 7:57 am, zuhair <zaljo...@xxxxxxxxx> wrote:





On Jun 1, 12:28 am, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:

On May 31, 9:28 pm, zuhair <zaljo...@xxxxxxxxx> wrote:

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

I don't know what you mean by that.

Do you intend to have an uncountable set of constants? If so, then
your theory is not recursively axiomatized.

NO, I intende to have w^w set of constants, which is countable. ( the
exponentiation in w^w is ordinal
exponentiation and not cardinal exponentiation
so w^w is a countable ordinal, i.e we have
a countable set of constants)
The constant V_i is one dimensional.
while the constant V_i_j is two dimensional
Now i is called the first dimension
while j is the second dimension
Instead of this i,j notation I use an index
on i so I symbolize V_i_j as
V_i1_i2. This way is more practical if we have
a lot of dimensions.
In general a constant with n-dimensions is writtin as
V_i1_i2_i3_......_in

You can write the constant V that is omega dimensional
as
V_i1_i2_i3_........

Now for each natural number j we have the
ij as jth dimension of V, so i1 is the first dimension
and i2 is second dimension ,now since for each
natural number j we have a dimension ij then
we have omega of dimesnions.

However for ease I symbolize i1_i2_i3_...... as I

so I = i1_i2_i3_......

so V_i1_i2_i3_........ is symbolized as VI for short.

Of course each dimension ij is equal to a natural number
so i1 for example can take the values of 1 or 2 or 3 or...
so I symbolize this as
for each ij there is a natural number n such that
ij=n
so for example we can have the constant
V_1_1_1_.... were every dimension ij is 1 in value.
the nex constant is
V_2_1_1_1_...... in this constant we have i1=2
and in=1 for every n>1
The third constant would be
V_3_1_1_1_........
so i1=3 and in=1 for every n>1

and so on for all values of i1
so we will have w of the constants
as indexed above.

These constants as indexed above are called
constants of the first degree ,because
only the first dimension is variating in them
and the rest of the dimensions has the same value
that is 1.

Then we begin with the second degree
constants were i1 and i2 values variating
and in=1 for n>2.
as above beginning with the i2=2 and in=1 for
n>2 ,so we will have

V_1_2_1_1_1_.....
V_2_2_1_1_1_.....
V_3_2_1_1_1_.....
.
.
.
.
Now we will have Omega of constants
for each i2=2
then we start varing i2 to be equal to 3
and we vary values of i1 over them
so we will have omega of i1 varying and i2=3
and so on for every value of i2
so we will have omega^2 constants of second
degree.

Similarly constants of the third degree are defined
as those in which the first three dimensions vary
and in=1 for n>3
so the total number of these would be Omega^3

So for we have Omeag^n for each n-degree constant.

so at the end continuing this process we end with the following:

Total number of VI constants is

Omega+Omega^2+Omega^3+........+Omega^n+....

Now we have a countable number of terms Omega^n, since for each n
there is Omega^n
and each term is countabel since
omega^n is countable
THEN the total sum is countable.

So we don't have an uncountable number of VI constants.
But Perhaps I was wrong when I said it was equal to
w^w, but I thought so.

I would like to know what is the ordinal number for the sum above.

Or do you intend to have a countable set of constants but each one
indexed by an ordered pair of natural numbers? In that case, all you
need to say is that for each i and j that are natural numbers, V_i_j
is a constant.

NO.

Or maybe you mean something else, in which case it would help if you'd
make it clear.

Yes , see above.

I can barely make sense any of that. You're presupposing ordinal
arithmetic just to stipulate a certain denumerable set of constants.
I'd suggest considering looking for ways to simplify, since it seems
to be quite a lot if you need ordinal arithmetic just to specify the
symbols of your language.

MoeBlee- Hide quoted text -

- Show quoted text -

I think I can simplify this list of primitives
to the following:

********************
Primitives:e,=,for every n=1,2,3,.....
there is a primitive constant V_i1_i2_..._in,
were each ij=1,2,3,..... for j=1,2,...,n-1
and in=2,3,4,........
and V1 is a primitive constant.
**********************

To claify what this means, I will go through it
stepwisely starting from n=1.

So for n=1 , every V_i1 is a primtive constant.
since n=1 then i1= 2,3,4,...., then
this means that V_2 , V_3 , V_4 ,...... are all primitive
constants.

These constants are called uni-dimensional constants.


for n=2, every V_i1_i2 is a primitive constant
so this means that
V_1_2 , V_2_2 , V_3_2,....... ,are primitive constants
V_1_3 , V_2_3 , V_3_3,........, are primitive constants

and so on.

Now these primitives are called two dimensional constants.

for n=3, every V_i1_i2_i3 is a primitive constatnt.
so this list starts with V_1_1_2,V_2_1_2, V_3_1_2,.....
and V_1_2_2 , V_2_2_2 , V_3_2_2,.....
and so on after finishing we start again
with V_1_1_3, V_2_1_3,....
and so on.

and these will be called three dimensional constants.

and so one for each n value.

since we have infinite n values.

The total number of all these n-dimensional primitive
constants would be

Omega + Omega^2 + Omega^3 + ......

and this total number of constants is countable.

I am not assuming any ordinal arithmetic here
I am only answering to the objection that
the total number of constants would be uncountable.

It wouldn't be uncountable as Rupert thought.
It would be countable.

Of course we need not assume anything about the Omegas
and this total sum when we are listing the primitives
(do you see me mentioning anything about that in my
statement of the primitives).

we just list the primitive according to the rule that
I have stated, But its final interpretation after we complete
the theory is that this number of primitives is countable.

i.e the theorem that the 'total number of primtives in this theory
is countable' can be proved in this theory. but of course
it cannot be proved at a primitive level.

Of course I should modify this theory according to this
simpler notification .

Zuhair





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