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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?=
Date: Wed, 30 Oct 2024 07:13:43 -0500
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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
>>>>>>>>>>>>>> the multiply operator. Will a terabyte be enough to store
>>>>>>>>>>>>>> the Gödel numbers?
>>>>>>>>>>>>>>
>>>>>>>>>>>>>
>>>>>>>>>>>>> Likely depends on how big of a system you are making F.
>>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>> I am proposing actually doing Gödel's actual proof and
>>>>>>>>>>>> deriving all of the digits of the actual Gödel numbers.
>>>>>>>>>>>>
>>>>>>>>>>>
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