# How is mathematics formulated - with models of formal systems?

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?

• I am trying hard to understand: why do you start speculating instead of just grabbing a book and see how it is done? If you don't know any logic but want to understand it, the natural thing to do would be studying it? – Jishin Noben Feb 7 at 10:44
• So you don't know any logic, but you state with some confidence that there is no unique model of ZFC. You need to give us more context about what you have actually studied and about why you are interested in an answer to your question. – Rob Arthan Feb 7 at 20:19
• @RobArthan I don't really know any logic; I know that you can prove certain things are independent of ZFC e.g. continuum hypothesis by forcing? But I do not know what forcing entails; beyond a vague notion that it is something like choosing a target statement, and adding (in an inductive way?) axioms to a base system to reach a target statement. I'm interested because I'm taking a class in Set Theory and I don't really understand how things are formalised – Joshua Lin Feb 8 at 1:46