How does one start doing 'Mathematics' from the ground up? My naive intuition (I don't know any logic) was that it's something like this:
1) Write down some symbols; so that we can use them for doing Math with. They don't have any meaning yet.
2) Formalise a logical system; e.g. propositional logic, or first order logic. This formalises what strings of symbols can be meaningful, and which just don't make any sense. We can already start 'doing math'; but none of the objects have any meaning yet - we have not defined our domain of discourse?
3) Formalise our domain of discourse; e.g. Sets (ZFC) or something similar. Now we have additional axioms telling us how the objects behave; how to make new objects with given objects, etc.
4) Do math.
This seems to make sense to me; what I get confused about is that when we write down say the axioms for ZFC; there is not a unique model of ZFC - and this can be important depending on what we are studying. How does model theory then fit into the picture? And does my outline seem fairly legitimate?