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

On 11/1/24 7:50 AM, olcott wrote:

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

On 2024-10-31 12:19:18 +0000, olcott said:
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On 10/31/2024 5:34 AM, Mikko wrote:

On 2024-10-30 12:16:02 +0000, olcott said:
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On 10/30/2024 5:02 AM, Mikko wrote:

On 2024-10-27 14:21:25 +0000, olcott said:
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On 10/27/2024 3:37 AM, Mikko wrote:

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

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

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

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Just evaluate the expressions shown in the books.

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To me they are all nonsense gibberish.

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The books define everything needed in order to understand the encoding
rules.
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Encoding nonsense gibberish as a string of digits is trivial.
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How one
can convert a proof about arithmetic into a
proof about provability seems to be flatly false.

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

Sort of. The Godel sentence only allows us to derive such a statement, and only in the meta-logic that was used to develop the relationship that the Godel sentence uses.

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

But the testing of a proof HAS BEEN shown to be computable, and thus subject to the laws of arithmetic.
You are just showing your ignorance.

 

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

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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.

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A book a trillion light years deep?

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

He didn't say that.
The problem with computing a Godel Number, is you first need to choose the System you want to express them about, and then the Meta-System that allocates the primes to the various elements of the system.
For the typical systems used, it could well be that the work needed to actually produce said numbers exceeds the ability for a person to do in their lifetime using a supercomputer.
But that doesn't mean that he didn't show they could be produced.
Godel's proof was never about "feasability" but possibility.
After all, we don't need the numbers to understand the proof that used those numbers is correct.