Tarski / Gödel and redefining the Foundation of Logic

From	olcott <polcott333@gmail.com>
Newsgroups	comp.theory, sci.logic
Subject	Tarski / Gödel and redefining the Foundation of Logic
Date	2024-07-21 08:20 -0500
Organization	A noiseless patient Spider
Message-ID	<v7j1u4$3o7r$2@dont-email.me> (permalink)
References	(23 earlier) <v7d4mr$2svvi$1@dont-email.me> <v7dqs3$30pvh$1@dont-email.me> <v7ft98$3fbg8$1@dont-email.me> <v7gdmn$3hlc2$3@dont-email.me> <v7ikah$1hri$1@dont-email.me>

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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.

>>> 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. 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).

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

-- 
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

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Re: DDD correctly emulated by HHH is correctly rejected as non-halting. Mikko <mikko.levanto@iki.fi> - 2024-07-14 11:49 +0300
  Re: DDD correctly emulated by HHH is correctly rejected as non-halting. olcott <polcott333@gmail.com> - 2024-07-14 09:48 -0500
    Re: DDD correctly emulated by HHH is INcorrectly rejected as non-halting. Richard Damon <richard@damon-family.org> - 2024-07-14 14:22 -0400
    Re: DDD correctly emulated by HHH is correctly rejected as non-halting. Mikko <mikko.levanto@iki.fi> - 2024-07-15 10:57 +0300
      Re: DDD correctly emulated by HHH is correctly rejected as non-halting. --- You are not paying attention olcott <polcott333@gmail.com> - 2024-07-15 08:32 -0500
        Re: DDD incorrectly emulated by HHH is incorrectly rejected as non-halting. --- You are not paying attention Richard Damon <richard@damon-family.org> - 2024-07-15 22:19 -0400
        Re: DDD correctly emulated by HHH is correctly rejected as non-halting. --- You are not paying attention Mikko <mikko.levanto@iki.fi> - 2024-07-16 10:18 +0300
          Re: DDD correctly emulated by HHH is correctly rejected as non-halting. --- You are not paying attention olcott <polcott333@gmail.com> - 2024-07-16 09:46 -0500
            Re: DDD INcorrectly emulated by HHH is INcorrectly rejected as non-halting. --- You are not paying attention Richard Damon <richard@damon-family.org> - 2024-07-16 21:12 -0400
            Re: DDD correctly emulated by HHH is correctly rejected as non-halting. --- You are not paying attention Mikko <mikko.levanto@iki.fi> - 2024-07-17 10:08 +0300
              Re: DDD correctly emulated by HHH is correctly rejected as non-halting. --- You are not paying attention olcott <polcott333@gmail.com> - 2024-07-17 08:14 -0500
                Re: DDD emulated by HHH is incorrectly rejected as non-halting. joes <noreply@example.org> - 2024-07-17 17:08 +0000
                Re: DDD correctly emulated by HHH is correctly rejected as non-halting. --- You are not paying attention Mikko <mikko.levanto@iki.fi> - 2024-07-18 10:55 +0300
                Re: DDD correctly emulated by HHH is correctly rejected as non-halting. --- You are not paying attention olcott <polcott333@gmail.com> - 2024-07-18 08:36 -0500
                Re: DDD correctly emulated by HHH is correctly rejected as non-halting. --- You are not paying attention Mikko <mikko.levanto@iki.fi> - 2024-07-19 10:30 +0300
                Re: DDD correctly emulated by HHH is correctly rejected as non-halting. --- You are not paying attention olcott <polcott333@gmail.com> - 2024-07-19 08:48 -0500
                Re: DDD incorrectly emulated by HHH is incorrectly rejected as non-halting. --- You are not paying attention Richard Damon <richard@damon-family.org> - 2024-07-19 11:28 -0400
                Re: DDD correctly emulated by HHH is correctly rejected as non-halting. --- You are not paying attention Mikko <mikko.levanto@iki.fi> - 2024-07-20 11:42 +0300
                Re: DDD correctly emulated by HHH is correctly rejected as non-halting. --- You are not paying attention olcott <polcott333@gmail.com> - 2024-07-20 08:22 -0500
                Re: DDD correctly emulated by HHH is correctly rejected as non-halting. --- You are not paying attention Mikko <mikko.levanto@iki.fi> - 2024-07-21 12:27 +0300
                Tarski / Gödel and redefining the Foundation of Logic olcott <polcott333@gmail.com> - 2024-07-21 08:20 -0500
                Re: Tarski / Gödel and redefining the Foundation of Logic Richard Damon <richard@damon-family.org> - 2024-07-21 13:53 -0400
                Re: Tarski / Gödel and redefining the Foundation of Logic Mikko <mikko.levanto@iki.fi> - 2024-07-22 11:14 +0300
                Re: Tarski / Gödel and redefining the Foundation of Logic olcott <polcott333@gmail.com> - 2024-07-22 09:40 -0500
                Re: Tarski / Gödel and redefining the Foundation of Logic Richard Damon <richard@damon-family.org> - 2024-07-22 20:12 -0400
                Re: Tarski / Gödel and redefining the Foundation of Logic Mikko <mikko.levanto@iki.fi> - 2024-07-23 11:07 +0300
                Re: Tarski / Gödel and redefining the Foundation of Logic olcott <polcott333@gmail.com> - 2024-07-23 09:53 -0500
                Re: Tarski / Gödel and redefining the Foundation of Logic Richard Damon <richard@damon-family.org> - 2024-07-23 22:16 -0400
                Re: Tarski / Gödel and redefining the Foundation of Logic Mikko <mikko.levanto@iki.fi> - 2024-07-25 11:55 +0300
                Re: Tarski / Gödel and redefining the Foundation of Logic olcott <polcott333@gmail.com> - 2024-07-25 10:51 -0500

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