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From: olcott <polcott333@gmail.com>
Newsgroups: sci.logic
Subject: Re: Formal systems that cannot possibly be incomplete except for
 unknowns and unknowable
Date: Mon, 5 May 2025 10:39:56 -0500
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On 5/5/2025 4:50 AM, Mikko wrote:
> On 2025-05-05 02:23:56 +0000, olcott said:
> 
>> When we define formal systems as a finite list of basic facts and 
>> allow semantic logical entailment as the only rule of inference we 
>> have systems that can express any truth that can be expressed in 
>> language.
>>
>> Also with such systems Undecidability is impossible. The only 
>> incompleteness are things that are unknown or unknowable.
> 
> A formal system has a formal language. Unless the language is too
> restricted for most interesting purposes the negation of every
> sentence is another sentence. In a consistent system some sentence
> is unprovable. If the negation of that system is also unprovable
> then the system is incomplete.
> 

My system skips merely "provable" and goes directly to "provably true".
Expressions such as "This sentence is not true" and its negation
are not provably true, thus rejected as semantically unsound.

> None of the features of specified above prevents an expressible unprovable
> sentence that has an unprovable negation. While adding one of the pair
> to the basic facts there is nothing to prevent an infinite set of such
> pairs.
> 

Basic facts are stipulated to be a finite set
of general knowledge of the world.


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