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From: Richard Damon <richard@damon-family.org>
Newsgroups: sci.logic
Subject: Re: Mathematical incompleteness has always been a misconception ---
 Ultimate Foundation of True(L,x)
Date: Tue, 25 Feb 2025 23:21:28 -0500
Organization: i2pn2 (i2pn.org)
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On 2/25/25 12:41 PM, olcott wrote:
> 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.
> 

The problem is you can't write your expression and have nothing anywhere.

Unless you undefine your system so that the symbol '3', the symbol '2', 
and the symbol '>' don't have meaning, from their basic definitions that 
results follow.

Just more of you not understanding how logic works, and you working on 
hypotheticals of fantasy.