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From: olcott <polcott333@gmail.com>
Newsgroups: comp.theory
Subject: Re: Peano Axioms anchored in First Grade Arithmetic on ASCII Digit
 String pairs
Date: Sun, 3 Nov 2024 07:15:30 -0600
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On 11/3/2024 3:16 AM, Mikko wrote:
> On 2024-11-02 11:09:06 +0000, olcott said:
> 
>> On 11/2/2024 3:37 AM, Mikko wrote:
>>> On 2024-11-01 11:50:24 +0000, olcott said:
>>>
>>>> On 11/1/2024 3:44 AM, Mikko wrote:
>>>>> On 2024-10-31 12:19:18 +0000, olcott said:
>>>>>
>>>>>> 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.
>>>>>
>>>>> The books define everything needed in order to understand the encoding
>>>>> rules.
>>>>>
>>>>> Encoding nonsense gibberish as a string of digits is trivial.
>>>>>
>>>>>> How one
>>>>>> can convert a proof about arithmetic into a
>>>>>> proof about provability seems to be flatly false.
>>>>>
>>>>> You needn't. The proof about provability is given in the books so
>>>>> you needn't any comversion.
>>>>>
>>>>
>>>> So you are saying that the Gödel sentence has nothing
>>>> to do with
>>>>
>>>> BEGIN:(Gödel 1931:39-41)
>>>>    ...We are therefore confronted with a proposition which
>>>>    asserts its own unprovability.
>>>> END:(Gödel 1931:39-41)
>>>
>>> Nothing is too strong but the connection is not arithmetic.
>>> That "asserts its own unprovability" refers to a non-arithmetic
>>> interpretation of an arithmetic formula.
>>
>> I want to know 100% concretely exactly what this means,
>> no hand waving allowed.
> 
> It means whatever Gödel wanted it to mean. As the sentence is not
> a part of a proof the only clue we have is what Gödel said.
> 

In other words you don't really understand the proof.
You are merely trusting Gödel on faith.

>>>> Making arithmetic say anything about provability
>>>> seems like making an angel food cake out of lug nuts,
>>>> cannot possible be done.
>>>
>>> Numbers have features and formulas have features. Therefore it is
>>> possible to compare features of formulas to features of numbers.
>>
>> This seems to be a type mismatch error. I need to
>> see every tiny detail of how it is not.
> 
> It is possible to compare things of different types. For example,
> chairs are not animals but we can compare the numbers of their legs.
> 

I am not convinced that Gödel proved that there
is any number that cannot be derived by arithmetic.
This seems inherently impossible. He must have
been confused.

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