Maybe I'm lost again with definitions and wrong assumptions.
Completeness maybe is a litle more general in the sense that you don't need the formal concept of being computable?
But it is hard to imagine a formal system that is not computable.
Can someone talk about provability without defining first computability?
To talk about unprovability, it is required first to define computability?
All this depends of the bootstrap procedure one use to bring logic and mathematics alive?
Edit: To talk about some concrete sentence: "Logic is complete in the sense that we have a consistent formal system that proves all sentences that are valid, that is, true under all interpretations."
In this sentence, the part "a consistent formal system that proves (all) sentences" what is the name of this property? I guess I wrongly called it provability...
Ups... "A consistent formal system that proves sentences" is what the syntatic side of the formal system does by itself. Nothing special in proving sentences. It has no name.
To prove all valid sentences is special. It has a name: completeness.
To prove any invalid sentence is special too. It is called unsoundness.
To not prove any invalid sentence is called soundness.
Valid is not an easy thing to be. Being true under all interpretations.
An interpretation is a mapping between the syntactic side and a semantic interpretation given by a "model" that not only maps elements from one side to the other but it maps the provability over syntactic elements to a "relationship" on that particular "model".
If we talk about propositional logic, to interpret a sentence P is deciding if it is TRUE of if it is FALSE. Any "model" has two elements: TRUE and FALSE. The interpretation must explain too, how to interpret semantically any provable statements (ie whatever new statement I could create following syntactic rules with P,Q,R,... and symbols like and, or, not).
Regarding other logics: there are two ways of changing propositional logic without changing the dictionary. One would be to accept a different set of axioms and inference rules (pure syntax) and another is to change the semantics, for example requiring more than two elements in any interpretation.
If I keep all propositional syntactic rules and I only change the interpretation, allowing for example three values, what does it mean to be valid now?
Returning to the syntactic side, I would try to make sense of decidability.
Proving a sentence only means to start writing new long and complicate sentences just following syntactic rules.
Apparently the only way we know how to produce this new sentences is using Turing machines (computing new sentences). Showing that a given sentence cannot be generated by a Turing machine given some syntactical rules, is undecidability of the sentence under these syntactic rules.