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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: Sun, 27 Oct 2024 09:29:22 -0500
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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.
>>
>> 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.
> 
> The memory needs are easier to estimate if you use a different
> numbering system:
> 
> 1. Encode all formulas with the 94 visible ASCII characters.
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