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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: Mon, 28 Oct 2024 19:56:43 -0400
Organization: i2pn2 (i2pn.org)
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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
>>>>>>> actual integers, are you saying otherwise?
>>>>>>>
>>>>>>
>>>>>> Not at all, just that they may be very large numbers.
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