I have a question that I hope can clarify the scopes of model theory and proof theory. I have the following naïve understanding of the two areas (please correct me if I'm wrong):
- Model theory is about the relationship between languages and interpretations, or theories and their models. It gives a semantic account of logical consequence ($\vDash$).
- Proof theory is about logical calculi and provability. It studies syntactic accounts of logical consequence ($\vdash$). As there are many calculi, there are many different consequence relations.
This explanation suggests that the study of semantic consequence belongs to model theory and the study of syntactic consequence belongs to proof theory. This seems to be confirmed in Wikipedia's article about logical consequence:
The two prevailing techniques for providing accounts of logical consequence involve expressing the concept in terms of proofs and via models. The study of the syntactic consequence (of a logic) is called (its) proof theory whereas the study of (its) semantic consequence is called (its) model theory.
My question is as follows: Soundness and completeness theorems of first-order logic relate the semantic consequence relation $\vDash$ and syntactic consequence relations $\vdash$ for logical calculi. Are these theorems a part of proof theory, model theory or something else? These notions are not meaningful unless one talks about a certain logical calculus, which I take to be proof theoretic. But then proof theory seems to depend on model theory. I'm hoping this question can be used to clarify the relationships and differences between metalogic, model theory and proof theory.
