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From: Richard Damon <richard@damon-family.org>
Newsgroups: comp.theory
Subject: =?UTF-8?Q?Re=3A_Tarski_/_G=C3=B6del_and_redefining_the_Foundation_o?=
 =?UTF-8?Q?f_Logic?=
Date: Tue, 23 Jul 2024 22:16:34 -0400
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On 7/23/24 10:53 AM, olcott wrote:
> On 7/23/2024 3:07 AM, Mikko wrote:
>> On 2024-07-22 14:40:41 +0000, olcott said:
>>
>>> On 7/22/2024 3:14 AM, Mikko wrote:
>>>> On 2024-07-21 13:20:04 +0000, olcott said:
>>>>
>>>>> On 7/21/2024 4:27 AM, Mikko wrote:
>>>>>> On 2024-07-20 13:22:31 +0000, olcott said:
>>>>>>
>>>>>>> On 7/20/2024 3:42 AM, Mikko wrote:
>>>>>>>> On 2024-07-19 13:48:49 +0000, olcott said:
>>>>>>>>
>>>>>>>>>
>>>>>>>>> Some undecidable expressions are only undecidable because
>>>>>>>>> they are self contradictory. In other words they are undecidable
>>>>>>>>> because there is something wrong with them.
>>>>>>>>
>>>>>>>> Being self-contradictory is a semantic property. Being 
>>>>>>>> uncdecidable is
>>>>>>>> independent of any semantics.
>>>>>>>
>>>>>>> Not it is not. When an expression is neither true nor false
>>>>>>> that makes it neither provable nor refutable.
>>>>>>
>>>>>> There is no aithmetic sentence that is neither true or false. If 
>>>>>> the sentnece
>>>>>> contains both existentia and universal quantifiers it may be hard 
>>>>>> to find out
>>>>>> whether it is true or false but there is no sentence that is neither.
>>>>>>
>>>>>>>  As Richard
>>>>>>> Montague so aptly showed Semantics can be specified syntactically.
>>>>>>>
>>>>>>>> An arithmetic sentence is always about
>>>>>>>> numbers, not about sentences.
>>>>>>>
>>>>>>> So when Gödel tried to show it could be about provability
>>>>>>> he was wrong before he even started?
>>>>>>
>>>>>> Gödel did not try to show that an arithmetic sentence is about 
>>>>>> provability.
>>>>>> He constructed a sentence about numbers that is either true and 
>>>>>> provable
>>>>>> or false and unprovable in the theory that is an extension of 
>>>>>> Peano arithmetics.
>>>>>>
>>>>>
>>>>> You just directly contradicted yourself.
>>>>
>>>> I don't, and you cant show any contradiction.
>>>>
>>>
>>> Gödel's proof had nothing what-so-ever to do with provability
>>> except that he proved that g is unprovable in PA.
>>
>> He also proved that its negation is unprovable in PA. He also proved
>> that every consistent extension of PA has a an sentence (different
>> from g) such that both it and its negation are unprovable.
>>
> 
> L is the language of a formal mathematical system.
> x is an expression of that language.
> 
> When we understand that True(L,x) means that there is a finite
> sequence of truth preserving operations in L from the semantic
> meaning of x to x in L, then mathematical incompleteness is abolished.
> 
> ~True(L,x) ∧ ~True(L,~x)
> means that x is not a truth-bearer in L.
>   It does not mean that L is incomplete

And thus you prove your system inconsistant, as you just admitted that 
the Goldbach's conjecture COULD be an Analytic Truth, and thus a 
Truth-Bearer, even without an finite sequence, but just an infinite 
sequence.

You are just showing you logic is based on the need to lie at times.

You validate the election and climate change deniers, whether you mean 
to or not, because you show that you agree with their basic method of logic.

> 
>>>>>>>> A proof is about sentences, not about
>>>>>>>> numbers.
>>>>>>>>
>>>>>>>>> The Liar Paradox: "This sentence is not true"
>>>>>>>>
>>>>>>>> cannot be said in the language of Peano arithmetic.
>>>>>>>
>>>>>>> Since Tarski anchored his whole undefinability theorem in a 
>>>>>>> self-contradictory sentence he only really showed that sentences 
>>>>>>> that
>>>>>>> are neither true nor false cannot be proven true.
>>>>>>
>>>>>> By Gödel's completeness theorem every consistent incomplete first 
>>>>>> order
>>>>>> theory has a model where at least one unprovable sentence is true.
>>>>>>
>>>>>>> https://liarparadox.org/Tarski_247_248.pdf // Tarski Liar Paradox 
>>>>>>> basis
>>>>>>> https://liarparadox.org/Tarski_275_276.pdf // Tarski proof
>>>>>
>>>>> It is very simple to redefine the foundation of logic to eliminate
>>>>> incompleteness.
>>>>
>>>> Yes, as long as you don't care whether the resulting system is useful.
>>>> Classical logic has passed practical tests for thousands of years, so
>>>> it is hard to find a sysem with better empirical support.
>>>
>>> When we show how incompleteness is eliminated then this also shows
>>> how undefinability is eliminated and this would have resulted in a
>>> chatbot that eviscerated Fascist lies about election fraud long
>>> before they could have taken hold in the minds of 45% of the electorate.
>>
>> The simplest way to elimita incompleteness is to construct a theory
>> where everytihing is provable. Of course such theory is not useful.
>>
>> The next simplest way is to construct a theory for a finite universe.
>> As the theory is complete it specifies the number of objects in the
>> universe. Then it is possible to evaluate every quantifier with a
>> simple finite loop or recursion, so the truth of every sentence is
>> computable.
>>
>> This kind of theory may have some use but its applicability is very
>> limited. In particular, a complete theory cannot be used in situations
>> where somthing is not known.
>>
>>> Because people have been arguing against my correct system of reasoning
>>> we will probably see the rise of the fourth Reich.
>>
>> Trying something impossible does not prevent anything.
>>
>