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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 11:15:06 +0200
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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.
> 
>> It follows from the traditional meanings of "3", "2", and ">".
>> Therefore the above statement is meaningless.
>> 
>>>> The
>>>> result depends on all of the change. But as long as we don't even
>>>> know whether that kind of change is possible at all the details are
>>>> impossible to determine.
>>> 
>>> LP := ~True(LP) has never been more than nonsense.
>> 
>> More specifically, your nonnsense. The symbol ":=" usually means definition
>> but requires that the symbol on the left side (in this case "LP") is not
>> used on the right side (and also that it is not used in the definition of
>> any of the symbols on the right side).
>> 
>> Usually languages of formal logic are constructed so that symbol that is
>> defined with an expression that starts with a negation operator cannot
>> be used as an argument to a function or a predicate.
>> 
>>> Tarski (although otherwise quite brilliant) had a blind spot.
>> 
>> Tarski did not use your nonsense.
>> 
> 
> Tarski anchored his whole proof in the Liar Paradox.

More specifically, to the idea that the Liar Paradox does not have a
truth value. Do you reject that idea?

-- 
Mikko