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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: Sat, 26 Oct 2024 08:57:58 -0500
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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. 

Are they less than one GB each? I want to see the c
code that computes them. I want to know how many bytes
of ASCII digits strings they are.

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