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From: Mikko <mikko.levanto@iki.fi>
Newsgroups: sci.logic
Subject: Re: Mathematical incompleteness has always been a misconception --- Ultimate Foundation of True(L,x)
Date: Fri, 28 Feb 2025 13:17:36 +0200
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On 2025-02-25 17:41:44 +0000, olcott said:

> On 2/25/2025 9:46 AM, Mikko wrote:
>> On 2025-02-24 22:53:06 +0000, olcott said:
>> 
>>> On 2/24/2025 3:13 AM, Mikko wrote:
>>>> On 2025-02-22 18:27:00 +0000, olcott said:
>>>> 
>>>>> On 2/22/2025 3:18 AM, Mikko wrote:
>>>>>> On 2025-02-21 23:19:10 +0000, olcott said:
>>>>>> 
>>>>>>> On 2/20/2025 2:54 AM, Mikko wrote:
>>>>>>>> On 2025-02-18 03:59:08 +0000, olcott said:
>>>>>>>> 
>>>>>>>>> On 2/12/2025 4:21 AM, Mikko wrote:
>>>>>>>>>> On 2025-02-11 14:07:11 +0000, olcott said:
>>>>>>>>>> 
>>>>>>>>>>> On 2/11/2025 3:50 AM, Mikko wrote:
>>>>>>>>>>>> On 2025-02-10 11:48:16 +0000, olcott said:
>>>>>>>>>>>> 
>>>>>>>>>>>>> On 2/10/2025 2:55 AM, Mikko wrote:
>>>>>>>>>>>>>> On 2025-02-09 13:10:37 +0000, Richard Damon said:
>>>>>>>>>>>>>> 
>>>>>>>>>>>>>>> On 2/9/25 5:33 AM, Mikko wrote:
>>>>>>>>>>>>>>>> Of course, completness can be achieved if language is sufficiently
>>>>>>>>>>>>>>>> restricted so that sufficiently many arithemtic truths become inexpressible.
>>>>>>>>>>>>>>>> 
>>>>>>>>>>>>>>>> It is far from clear that a theory of that kind can express all arithmetic
>>>>>>>>>>>>>>>> truths that Peano arithmetic can and avoid its incompletness.
>>>>>>>>>>>>>>> 
>>>>>>>>>>>>>>> WHich, it seems, are the only type of logic system that Peter can understand.
>>>>>>>>>>>>>>> 
>>>>>>>>>>>>>>> He can only think in primitive logic systems that can't reach the 
>>>>>>>>>>>>>>> complexity needed for the proofs he talks about, but can't see the 
>>>>>>>>>>>>>>> problem, as he just doesn't understand the needed concepts.
>>>>>>>>>>>>>> 
>>>>>>>>>>>>>> That would be OK if he wouldn't try to solve problems that cannot even
>>>>>>>>>>>>>> exist in those systems.
>>>>>>>>>>>>> 
>>>>>>>>>>>>> There are no problems than cannot be solved in a system
>>>>>>>>>>>>> that can also reject semantically incorrect expressions.
>>>>>>>>>>>> 
>>>>>>>>>>>> The topic of the discussion is completeness. Is there a complete system
>>>>>>>>>>>> that can solve all solvable problems?
>>>>>>>>>>> 
>>>>>>>>>>> When the essence of the change is to simply reject expressions
>>>>>>>>>>> that specify semantic nonsense there is no reduction in the
>>>>>>>>>>> expressive power of such a system.
>>>>>>>>>> 
>>>>>>>>>> The essence of the change is not sufficient to determine that.
>>>>>>>>> 
>>>>>>>>> In the same way that 3 > 2 is stipulated the essence of the
>>>>>>>>> change is that semantically incorrect expressions are rejected.
>>>>>>>>> Disagreeing with this is the same as disagreeing that 3 > 2.
>>>>>>>> 
>>>>>>>> That 3 > 2 need not be (and therefore usually isn't) stripualted.
>>>>>>> 
>>>>>>> The defintion of the set of natural numbers stipulates this.
>>>>> 
>>>>> If NOTHING ever stipulates that 3 > 2 then NO ONE can
>>>>> possibly know that 3 > 2 leaving the finite string
>>>>> "3 > 2" merely random gibberish.
>>>> 
>>>> A formal language of a theory of natural numbers needn't define "2" or
>>>> "3". Those concepts can be expressed as "1+1" and "1+1+1" or as "SS0"
>>>> and "SSS0" depending on which symbols the language has.
>>> 
>>> If nothing anywhere specifies that "3>2" then no one
>>> ever has any way of knowing that 3>2.
>> 
>> Of course there is. From definitions and psotulates one can prove
>> that 3 > 2, at least in some formulations. Or that 1+1+1 > 1+1 if
>> the language does not contaion "3" and "2".
> 
> In other words you don't know what "nothing anywhere" means.

Irrelevant. Whether anything anywhere specifies or not that 3 > 2 that
can be determined from the meanings of "3", ">" adn "2". The knowledge
of those meanings need not come from the same source.

-- 
Mikko