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From: olcott <polcott333@gmail.com>
Newsgroups: comp.theory
Subject: Re: Peano Axioms anchored in First Grade Arithmetic on ASCII Digit
 String pairs
Date: Sat, 26 Oct 2024 08:17:52 -0500
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On 10/26/2024 3:02 AM, Mikko wrote:
> On 2024-10-25 13:31:16 +0000, olcott said:
> 
>> On 10/25/2024 3:01 AM, Mikko wrote:
>>> On 2024-10-24 14:28:35 +0000, olcott said:
>>>
>>>> On 10/24/2024 8:51 AM, Mikko wrote:
>>>>> On 2024-10-23 13:15:00 +0000, olcott said:
>>>>>
>>>>>> On 10/23/2024 2:28 AM, Mikko wrote:
>>>>>>> On 2024-10-22 14:02:01 +0000, olcott said:
>>>>>>>
>>>>>>>> On 10/22/2024 2:13 AM, Mikko wrote:
>>>>>>>>> On 2024-10-21 13:52:28 +0000, olcott said:
>>>>>>>>>
>>>>>>>>>> On 10/21/2024 3:41 AM, Mikko wrote:
>>>>>>>>>>> On 2024-10-20 15:32:45 +0000, olcott said:
>>>>>>>>>>>
>>>>>>>>>>>> The actual barest essence for formal systems and computations
>>>>>>>>>>>> is finite string transformation rules applied to finite 
>>>>>>>>>>>> strings.
>>>>>>>>>>>
>>>>>>>>>>> Before you can start from that you need a formal theory that
>>>>>>>>>>> can be interpreted as a theory of finite strings.
>>>>>>>>>>
>>>>>>>>>> Not at all. The only theory needed are the operations
>>>>>>>>>> that can be performed on finite strings:
>>>>>>>>>> concatenation, substring, relational operator ...
>>>>>>>>>
>>>>>>>>> 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.
>>>>>>>>>
>>>>>>>>>> The minimal complete theory that I can think of computes
>>>>>>>>>> the sum of pairs of ASCII digit strings.
>>>>>>>>>
>>>>>>>>> That is easily extended to Peano arithmetic.
>>>>>>>>>
>>>>>>>>> As a bottom layer you need some sort of logic. There must be 
>>>>>>>>> unambifuous
>>>>>>>>> rules about syntax and inference.
>>>>>>>>>
>>>>>>>>
>>>>>>>> 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.
>>>>>>>>
>>>>>>>> I have no idea how the first grade arithmetic
>>>>>>>> algorithm could be extended to PA.
>>>>>>>
>>>>>>> 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.
>>>>>>>
>>>>>>> Anyway, the details are not important, only that it can be done.
>>>>>>>
>>>>>>
>>>>>> 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
>>>>>>
>>>>>> 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.
>>>>>
>>>>> No, it does not. Incompleteness theorem does not apply to artihmetic
>>>>> that only has addition but not multiplication.
>>>>>
>>>>> The incompleteness theorem is about theories that have quantifiers.
>>>>> A specific arithmetic expression (i.e, with no variables of any kind)
>>>>> always has a well defined value.
>>>>>
>>>>
>>>> So lets goes the next step and add multiplication to the algorithm:
>>>> (just like first grade arithmetic we perform multiplication
>>>> on arbitrary length ASCII digit strings just like someone would
>>>> do with pencil and paper).
>>>>
>>>> Incompleteness cannot be defined. until we add variables and
>>>> quantification: There exists an X such that X * 11 = 132.
>>>> Every detail of every step until we get G is unprovable in F.
>>>
>>> Incompleteness is easier to define if you also add the power operator
>>> to the arithmetic. Otherwise the expressions of provability and
>>> incompleteness are more complicated. They become much simpler if
>>> instead of arithmetic the fundamental theory is a theory of finite
>>> strings. As you already observed, arithmetic is easy to do with
>>> finite strings. The opposite is possible but much more complicated.
>>
>> The power operator can be built from repeated operations of
>> the multiply operator.
> 
> It is possible but to say that x is the z'th power of y is overly
> complicated with a first order formula using just addition and
> multiplication.
> 

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.
char* sum(char* x, char *y);
char* product(char* x, char *y);
char* difference(char* x, char *y); // never returns < 0

char* power_of(char* base, char * power);
I am not going to bother with the rest of the steps.
We simply multiply  base times itself power times.

-- 
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer