I'm a really beginner in Mathematical Logic.I'm currently reading Shoenfield Mathematical's Logic and i'm having a hard time trying to relate the concept of Formal Systems with the concept of Axiom System ( or Axiomatic System ).
I know any Formal System has 3 constituents : a language, a set of axioms ( certain expressions in that language ) , and rules of inference.
What about an Axiom System ?
Shoenfield describes it more or less as :
"An axiom system is the entire eddifice which a mathematician constructs, consisting of basic concepts and axioms ( describing them ) and derived concepts and derived theorems ( describing them )."
I'm not well to sure how he defined it.
Did he mean an Axiom System is just a set ( of axioms and derived theorems ) ?
Is the reason he didn't introduce the need of a language in an Axiom System, the fact that an axiom is mainly idea ( which can or can not be formalized through a syntax ) ?
What is the reason he didn't introduce in the definition of Axiom Systems, the need of rules of inference ( to derive the Theorems ) ?
How does the concept of Axiom Systems differ from the concept of Formal Systems ?
P.S : I made a search and discovered Mathematical Logic is mainly divided into Proof Theory, Model Theory and Recursion Theory.Where would my question fit ? I also can't seem to solve my doubt by other Mathematical Logic books ( Stephen Klenee, Van Dallen, Machover ) because i couldn't find the description of Formal Systems and Axiom Systems, they usually start right away with some kind of Calculus and First-order. A book recommendation which would cover this ( formal system, language, axiom system, etc ) would also be really helpful.