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From: Mikko <mikko.levanto@iki.fi>
Newsgroups: comp.theory
Subject: =?utf-8?Q?Re:_Tarski_/_G=C3=B6del_and_redefining_the_Foundation_of_Logic?=
Date: Tue, 23 Jul 2024 11:07:10 +0300
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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.

>>>>>> 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.

-- 
Mikko