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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: Fri, 25 Oct 2024 18:06:53 -0500
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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?

-- 
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer