Re: Peano Axioms anchored in First Grade Arithmetic on ASCII Digit String pairs

On 11/2/24 7:09 AM, olcott wrote:

On 11/2/2024 3:37 AM, Mikko wrote:

On 2024-11-01 11:50:24 +0000, olcott said:
>

On 11/1/2024 3:44 AM, Mikko wrote:

On 2024-10-31 12:19:18 +0000, olcott said:
>

On 10/31/2024 5:34 AM, Mikko wrote:

On 2024-10-30 12:16:02 +0000, olcott said:
>

On 10/30/2024 5:02 AM, Mikko wrote:

On 2024-10-27 14:21:25 +0000, olcott said:
>

On 10/27/2024 3:37 AM, Mikko wrote:

On 2024-10-26 13:17:52 +0000, olcott said:
>

Just imagine c functions that have enough memory to compute
sums and products of ASCII strings of digits using the same
method that people do.

>
Why just imagein? That is fairly easy to make. In some other lanugages
(e.g. Python, Javascript) it is alread in the library or as a built-in
feature.
>

>
OK next I want to see the actual Godel numbers and the
arithmetic steps used to derive them.

>
They can be found in any textbook of logic that discusses undecidability.
If you need to ask about details tell us which book you are using.
>

>
Every single digit of the entire natural numbers
not any symbolic name for such a number.

>
Just evaluate the expressions shown in the books.

>
To me they are all nonsense gibberish.

>
The books define everything needed in order to understand the encoding
rules.
>
Encoding nonsense gibberish as a string of digits is trivial.
>

How one
can convert a proof about arithmetic into a
proof about provability seems to be flatly false.

>
You needn't. The proof about provability is given in the books so
you needn't any comversion.
>

>
So you are saying that the Gödel sentence has nothing
to do with
>
BEGIN:(Gödel 1931:39-41)
   ...We are therefore confronted with a proposition which
   asserts its own unprovability.
END:(Gödel 1931:39-41)

>
Nothing is too strong but the connection is not arithmetic.
That "asserts its own unprovability" refers to a non-arithmetic
interpretation of an arithmetic formula.
>

 I want to know 100% concretely exactly what this means,
no hand waving allowed.

The statement:
There is no g that satisfies the relationship PRR(g).
Is not a statement that is a mathematical operation in the basic field of math, but is a statement ABOUT mathematics. It is a statement that due to the nature of mathematics MUST be a truth bearer, as either there exists a g that satisfies the relationship, or the doesn't.
In the meta-system, we can assign all axioms, operations, and possible statements a number (and can reverse and given a number, get back the statement), and then define a PRR that is a theorem proof checker of the statement given by the number it is given, and sees if it actually is a proof of that initial statement.
When we do this, we see that if a number exists that satisfies that PRR, then in the meta-system we can decode that number to the statement that it represents, and because it satisfied the proof checker, that statement *IS* a proof of the statement that no such number exist, and thus shows that it can not have been a number that satisfies the PRR.

 

Making arithmetic say anything about provability
seems like making an angel food cake out of lug nuts,
cannot possible be done.

>
Numbers have features and formulas have features. Therefore it is
possible to compare features of formulas to features of numbers.
>

 This seems to be a type mismatch error. I need to
see every tiny detail of how it is not.

I think it is just beyond your understanding.

 

It might be the case that one number fills 100 books
of 1000 pages each.

>
You fill find out when you evaluate the expressions. If you use Gödel's
original numbering you will need larger numbers than strictly necessary.
If you first encode symbols with a finite set of characters you can
encode everything with finite set of characters.

>
A book a trillion light years deep?

>
The number of digits in a Gödel number can be computed with less effort
than the Gödel number itself. Still easier to compute a rough estimate.

>
So you have no idea how to compute the Gödel numbers.

>
As I aleady told, I have an idea how to encode formulas with smaller
numbers than the numbers Gödel used.
>

 Exactly how many digits is G?
 

G isn't a number, g is.
G is a statement about the non-existance of a number to match the requirments of a particular Primitive Recursive Relationship.
g doesn't exist, as is proven, so doesn't have digits.
So, your question is incorrect.
That is like asking what is the final digit of the square-root of 2 in base 2.
The Godel Numbers allow us to represent ALL statements in the system as a number. (based on the assignements of primes to base statements in the meta-system).