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From: Richard Damon <richard@damon-family.org>
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 19:35:34 -0400
Organization: i2pn2 (i2pn.org)
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On 10/30/24 8:13 AM, olcott wrote:
> 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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