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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: Tue, 25 Feb 2025 17:46:40 +0200
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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".

-- 
Mikko