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From: olcott <polcott333@gmail.com>
Newsgroups: comp.theory
Subject: =?UTF-8?Q?Re=3A_G=C3=B6del=27s_actual_proof_and_deriving_all_of_the?=
 =?UTF-8?Q?_digits_of_the_actual_G=C3=B6del_numbers?=
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On 11/1/2024 3:34 AM, Mikko wrote:
> On 2024-10-31 12:15:42 +0000, olcott said:
> 
>> On 10/31/2024 4:45 AM, Mikko wrote:
>>> On 2024-10-30 12:13:43 +0000, olcott said:
>>>
>>>> On 10/30/2024 4:57 AM, Mikko wrote:
>>>>> On 2024-10-29 13:25:34 +0000, olcott said:
>>>>>
>>>>>> On 10/29/2024 2:38 AM, Mikko wrote:
>>>>>>> On 2024-10-28 14:04:24 +0000, olcott said:
>>>>>>>
>>>>>>>> On 10/28/2024 3:35 AM, Mikko wrote:
>>>>>>>>> On 2024-10-27 14:29:22 +0000, olcott said:
>>>>>>>>>
>>>>>>>>>> On 10/27/2024 4:02 AM, Mikko wrote:
>>>>>>>>>>> On 2024-10-26 13:57:58 +0000, olcott said:
>>>>>>>>>>>
>>>>>>>>>>>> On 10/25/2024 11:07 PM, Richard Damon wrote:
>>>>>>>>>>>>> On 10/25/24 7:06 PM, olcott wrote:
>>>>>>>>>>>>>> On 10/25/2024 5:17 PM, Richard Damon wrote:
>>>>>>>>>>>>>>> On 10/25/24 5:52 PM, olcott wrote:
>>>>>>>>>>>>>>>> On 10/25/2024 10:52 AM, Richard Damon wrote:
>>>>>>>>>>>>>>>>> On 10/25/24 9:31 AM, olcott wrote:
>>>>>>>>>>>>>>>>>> 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
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