As a beginner, I'm overwhelmed by the usage of terminology , such as theory, model, interpretation, structure et al, which are omnipresent in Mathematical logic.
Here's my understanding about them:
Creating a theory is synonymous to axiomatization, which is selecting a finite number of sentences, which is expected to be satisfiable,which means these sentences are true in at least one model.
The motivation of distinguishing between theory and model is that a theory always suffers from the indecidability of some meaningful sentences e.g. continuum hypothesis, which can't be avoided once and for all by adding more axioms in the theory. However, as expected in a model, there is no such thing indecidable as continuum hypothesis, or we don't talk about them in the context of a model. Usually, a model is defined as consisting of an underlying set, a set of relations, and a set of functiosn, which are defined intentionally to address certain unsolved problems in the theory. It seems to me it only make sense to talk about a model in the context of a particular sentence of interest, because no model can be fully axiomatized.
I'm also confused with the usage of interpretation, structure, and model. Are they interchangeable?