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

On 10/23/24 9:15 AM, olcott wrote:

On 10/23/2024 2:28 AM, Mikko wrote:

On 2024-10-22 14:02:01 +0000, olcott said:
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On 10/22/2024 2:13 AM, Mikko wrote:

On 2024-10-21 13:52:28 +0000, olcott said:
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On 10/21/2024 3:41 AM, Mikko wrote:

On 2024-10-20 15:32:45 +0000, olcott said:
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The actual barest essence for formal systems and computations
is finite string transformation rules applied to finite strings.

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Before you can start from that you need a formal theory that
can be interpreted as a theory of finite strings.

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Not at all. The only theory needed are the operations
that can be performed on finite strings:
concatenation, substring, relational operator ...

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You may try with an informal foundation but you need to make sure
that it is sufficicently well defined and that is easier with a
formal theory.
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The minimal complete theory that I can think of computes
the sum of pairs of ASCII digit strings.

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That is easily extended to Peano arithmetic.
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As a bottom layer you need some sort of logic. There must be unambifuous
rules about syntax and inference.
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I already wrote this in C a long time ago.
It simply computes the sum the same way
that a first grader would compute the sum.
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I have no idea how the first grade arithmetic
algorithm could be extended to PA.

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Basically you define that the successor of X is X + 1. The only
primitive function of Peano arithmetic is the successor. Addition
and multiplication are recursively defined from the successor
function. Equality is often included in the underlying logic but
can be defined recursively from the successor function and the
order relation is defined similarly.
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Anyway, the details are not important, only that it can be done.
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 First grade arithmetic can define a successor function
by merely applying first grade arithmetic to the pair
of ASCII digits strings of [0-1]+ and "1".
https://en.wikipedia.org/wiki/Peano_axioms

But the problem is that "First Grade Arithmetic" doesn't PROVE that fact, but ASSUMES it.
Just like your logic ASSUMES your resuls, but doesn't prove it, just in your case it isn't true.

 The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an effective procedure (i.e. an algorithm) is capable of proving all truths about the arithmetic of natural numbers. For any such consistent formal system, there will always be statements about natural numbers that are true, but that are unprovable within the system.
https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems
 When we boil this down to its first-grade arithmetic foundation
this would seem to mean that there are some cases where the
sum of a pair of ASCII digit strings cannot be computed.
 

Nope, because some truths of arithmetic aren't just "summing".
Godel's Primitive Recursive Relationship is just a 'simple' arithmetical operation, and the existance of numbers that satisfies it are  truths about that arithemtic.
He shows that there can not be a Natural number that satisfies that relationship, and that there can not be a proof in that system of axioms to prove that fact, as in the meta-theory that the relationship was developed in (but using only operations available in that theory) that if such a number existed, (and only if such a number existed) it would BE a proof of the statement, but that number would also make the statement false.
Since you can not prove a false statement in a consistent system of axioms, there can not be a Natural Number that satisfies it, and since that also means a proof can not ezist (since the existance of the proof creates a number that would satisfy that relationship).
Note, all of this is based on the ideas you are trying to form, but with understanding and looking at there consequence.
CHECKING a proof is a computable operation, as Godel Proved.
Encoding that into the mathematics of Natural Numbers is a possible operation, as Godel Proved.
Forming that statement, is possible, as Godel proved, and thus, he shows that he can make a statement that MUST be True, but also can not be proven in the system.
Sorry, your logic just falls appart when put against the facts. Your problem is you just don't understand how to formalize you statement's so your logic isn't precise enough to do what you want, and that leads you down false alleys.