Re: Formal systems that cannot possibly be incomplete except for unknowns and unknowable

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