Sujet : Re: Formal systems that cannot possibly be incomplete except for unknowns and unknowable
De : mikko.levanto (at) *nospam* iki.fi (Mikko)
Groupes : sci.logicDate : 06. May 2025, 11:02:38
Autres entêtes
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References : 1 2 3
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On 2025-05-05 15:39:56 +0000, olcott said:
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.
The references to truth and semantics make the system informal.
-- Mikko
Formal systems that cannot possibly be incomplete except for unknowns and unknowable
Re: Formal systems that cannot possibly be incomplete except for unknowns and unknowable