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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: Mon, 24 Feb 2025 11:13:25 +0200
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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.

It is not possible to stipulate separately for every pair of natural
numbers whether the first one is grater than the second one. Instead,
a theory of natural numbers may have postulates that permit the proofs
of all true claims about the order of two natural numbers but no proof
of a false claim.

-- 
Mikko