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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: Mon, 28 Oct 2024 19:41:04 -0500
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On 10/28/2024 6:56 PM, Richard Damon wrote:
> On 10/28/24 10:04 AM, olcott wrote:
>> 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.
>>>>>>>>>>
>>>>>>>>>
>>>>>>>>> Then try it and see.
>>>>>>>>>
>>>>>>>>> You do understand that the first step is to fully enumerate all 
>>>>>>>>> the axioms of the system, and any proofs used to generate the 
>>>>>>>>> needed properties of the mathematics that he uses.
>>>>>>>>>
>>>>>>>>
>>>>>>>> Gödel seems to propose that his numbers are
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