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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: Sun, 3 Nov 2024 13:19:55 -0500
Organization: i2pn2 (i2pn.org)
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On 11/3/24 8:15 AM, olcott wrote:
> 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.

No, it shows that YOU don't undetstand Godels proof, and are just 
refuting a strawman.

If you did, you would point out the specific error being made.

> 
>>>>> 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.
> 

That isn't what he said.

He said he could construct a function that was effectively a proof 
checker, that could be used to show that there could not be a number to 
satisfy that function because such a number would prove that such a 
number doesn't exist.