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From: Richard Damon <richard@damon-family.org>
Newsgroups: comp.theory
Subject: Re: Peano Axioms anchored in First Grade Arithmetic on ASCII Digit
 String pairs
Date: Sat, 2 Nov 2024 11:44:44 -0400
Organization: i2pn2 (i2pn.org)
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On 11/2/24 7:05 AM, olcott wrote:
> On 11/2/2024 3:29 AM, Mikko wrote:
>> On 2024-11-01 11:53:00 +0000, olcott said:
>>
>>> On 11/1/2024 3:47 AM, Mikko wrote:
>>>> On 2024-10-31 14:18:40 +0000, olcott said:
>>>>
>>>>> On 10/31/2024 8:58 AM, joes wrote:
>>>>>> Am Thu, 31 Oct 2024 07:19:18 -0500 schrieb olcott:
>>>>>>> On 10/31/2024 5:34 AM, Mikko wrote:
>>>>>>>> On 2024-10-30 12:16:02 +0000, olcott said:
>>>>>>>>> On 10/30/2024 5:02 AM, Mikko wrote:
>>>>>>>>>> On 2024-10-27 14:21:25 +0000, olcott said:
>>>>>>>>>>> On 10/27/2024 3:37 AM, Mikko wrote:
>>>>>>>>>>>> On 2024-10-26 13:17:52 +0000, olcott said:
>>>>>>>>>>>>
>>>>>>>>>>>>> Just imagine c functions that have enough memory to compute 
>>>>>>>>>>>>> sums
>>>>>>>>>>>>> and products of ASCII strings of digits using the same 
>>>>>>>>>>>>> method that
>>>>>>>>>>>>> people do.
>>>>>>>>>>>> Why just imagein? That is fairly easy to make. In some other
>>>>>>>>>>>> lanugages (e.g. Python, Javascript) it is alread in the 
>>>>>>>>>>>> library or
>>>>>>>>>>>> as a built-in feature.
>>>>>>>>>>> OK next I want to see the actual Godel numbers and the 
>>>>>>>>>>> arithmetic
>>>>>>>>>>> steps used to derive them.
>>>>>>>>>> They can be found in any textbook of logic that discusses
>>>>>>>>>> undecidability.
>>>>>>>>>> If you need to ask about details tell us which book you are 
>>>>>>>>>> using.
>>>>>>>>> Every single digit of the entire natural numbers not any 
>>>>>>>>> symbolic name
>>>>>>>>> for such a number.
>>>>>>>> Just evaluate the expressions shown in the books.
>>>>>>> To me they are all nonsense gibberish. How one can convert a 
>>>>>>> proof about
>>>>>>> arithmetic into a proof about provability seems to be flatly false.
>>>>>
>>>>>> The key is selfreference. There is a number that encodes the sentence
>>>>>> "the sentence with the number [the number that this sentence 
>>>>>> encodes to]
>>>>>> is not provable".
>>>>>
>>>>> Can you please hit return before you reply?
>>>>> Your reply is always buried too close to what you are replying to.
>>>>>
>>>>> We simply reject pathological self-reference lie
>>>>> ZFC did and the issue ends.
>>>>
>>>> You cannot reject any number from atrithmetic. If you do the result is
>>>> not arithmetic anymore.
>>>
>>> I claims that his whole proof is nonsense until you
>>> provide 1200% concrete proof otherwise.
>>
>> Crackpots claim all all sorts of things. There is no way to change that
>> so there is no point to try.
>>
>>> All of arithmetic is inherently computable and
>>> any non-arithmetic operation on a number is a type
>>> mismatch error.
>>
>> There are arithmetic functions and predicates that are not Turing
>> computable. For example, Busy Beaver.
>>
> 
> Not computable because of self-reference is a different class
> than not computable for other reasons. The Goldbach conjecture
> seems not computable only because it seems to require an infinite
> number of steps.
> 

No, because the ability to form self-reference is totally inherent in 
the nature of Turing Complete computation systems.

The essential fact that infinity is not finite is the core reason for 
both non-computable natures.

Note, we don't know if Goldbach is non-computable, and in fact, if we 
did, we would have the answer, and thus it can not be proven to be 
non-computable, as the proof of truth of Goldback being non-computable 
would be a proof that it was true, as if the Goldback conjecture is 
false, there is a compuation that proves that.