Completeness

Definition. A Proof Theory is complete if whenever $P_1,P_2,\dots ,P_n\models Q$ (truth/valid) then $P_1,P_2,\dots ,P_n⊢Q$ (proof).

Reminder from Logic: $|-$ → Proof Theory, pronounced “proves” $|=$ → Semantics, pronounced “entails” / “valid” / “semantic entailment”

Related

• Soundness

I am still confused at how these two are related. https://math.stackexchange.com/questions/3536805/what-is-true-about-a-proof-system-that-is-complete-but-not-sound

If a proof system is unsound and complete, then all the semantically valid inferences can be proven, but in addition it proves some sequents that are not actually valid.