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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: Mon, 22 Jul 2024 11:14:01 +0300
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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.

>>>> 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.

> Any expression x of language L that cannot be shown
> to be true by some (possibly infinite) sequence of truth preserving 
> operations in L is simply untrue in L: True(L, x).

That does not help much if you cannot determine whether a particular
string can be shown to be true.

> Tarski showed that True(Tarski_Theory, Liar_Paradox) cannot be defined
> never understanding that Liar_Paradox is not a truth bearer.

However, every arithmetic sentence is either true or false.

-- 
Mikko