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From: olcott <polcott333@gmail.com>
Newsgroups: sci.logic
Subject: Re: Mathematical incompleteness has always been a misconception ---
 Ultimate Foundation of True(L,x)
Date: Tue, 25 Feb 2025 11:41:44 -0600
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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.

-- 
Copyright 2025 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer