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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: Sat, 26 Oct 2024 00:07:24 -0400
Organization: i2pn2 (i2pn.org)
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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. I guess you don't 
understand his beginning where he assigns initial values to each of the 
axioms and statements into numbers.

That was a key part of the proof, showing that all of logic could be 
encoded into finite strings and then into numbers, so that ultimately, a 
whole proof becomes just a single number.

And then that we can build a Primitive Recursive Relationship built on 
the numbers from the system, to see if that number represents a proof 
for a given statement (which is also a number).