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From: Mikko <mikko.levanto@iki.fi>
Newsgroups: sci.logic
Subject: Re: Mathematical incompleteness has always been a misconception --- Tarski
Date: Sat, 22 Feb 2025 11:12:45 +0200
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On 2025-02-21 23:22:23 +0000, olcott said:

> On 2/20/2025 3:01 AM, Mikko wrote:
>> On 2025-02-18 13:50:22 +0000, olcott said:
>> 
>>> On 2/18/2025 6:25 AM, Richard Damon wrote:
>>>> On 2/17/25 10:59 PM, olcott wrote:
>>>>> 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.
>>>> 
>>>> But your logic needs to reject some of the results of your logic as 
>>>> semantically incorrect, and thus your logic is itself semantically 
>>>> incorrect.
>>>> 
>>> 
>>> There is nothing like that in the following concrete example:
>>> LP := ~True(LP)
>>> 
>>> In other words you are saying the Prolog is incorrect
>>> to reject the Liar Paradox.
>>> 
>>> Above translated to Prolog
>>> 
>>> ?- LP = not(true(LP)).
>>> LP = not(true(LP)).
>> 
>> According to Prolog rules LP = not(true(LP)) is permitted to fail.
>> If it succeeds the operations using LP may misbehave. A memory
>> leak is also possible.
>> 
>>> ?- unify_with_occurs_check(LP, not(true(LP))).
>>> false
>> 
>> This merely means that the result of unification would be that LP conains
>> itself. It could be a selmantically valid result but is not in the scope
>> of Prolog language.
>> 
> 
> It does not mean that. You are wrong.

It does in the context where it was presented. More generally,
unify_with_occurs_check also fails if the arguments are not
unfiable. But this possibility is already excluded by their
successfull unification.

> I am not going bother to quote Clocksin and Mellish
> proving that you are wrong.

You are right, a quote that does not support your claim
is not a good idea.

-- 
Mikko