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From: Mikko <mikko.levanto@iki.fi>
Newsgroups: comp.theory
Subject: Re: DDD correctly emulated by HHH is correctly rejected as non-halting. --- You are not paying attention
Date: Sun, 21 Jul 2024 12:27:45 +0300
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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.

>> 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
> https://liarparadox.org/Tarski_275_276.pdf

-- 
Mikko