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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: Fri, 25 Oct 2024 08:31:16 -0500
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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. Will a terabyte be enough to store
the Gödel numbers?

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