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

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.
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.
--
Mikko