Thinking about algebra (e.g. group theory) makes a lot of this make more sense. The definition of a group is a “theory”; any particular group is a “model”. This isn’t a huge revelation or anything, but it’s easier to think about these ideas in the context of algebra (where different structures that behave similarly are commonplace) rather than arithmetic (where we like thinking about one “true” picture).
Thinking about algebra (e.g. group theory) makes a lot of this make more sense. The definition of a group is a “theory”; any particular group is a “model”. This isn’t a huge revelation or anything, but it’s easier to think about these ideas in the context of algebra (where different structures that behave similarly are commonplace) rather than arithmetic (where we like thinking about one “true” picture).