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

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.

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.

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.
--
Mikko