Clarifying the definition of an axiomatic system 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 ).
What I know:
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.
Questions:

*

*Did he mean an Axiom System is just a set ( of axioms and derived theorems ) ?


*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 ?
Attempt at self answering
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.
 A: A useful suggestion is Richard Kaye, The Mathematics of Logic, (Cambridge U.P., 2007).
In Chapter 3 : Formal systems, he describes formal systems as :

kinds of mathematical games with strings of symbols and precise rules.

Rules are of two basic kind :


*

*rules of formation : how to generate well formed (i.e.admissible) strings

*rules of transformation : how to produce new (well formed) strings from existing ones. 
The following chapters deal with the typical formal systems of Math Log : Propositional Logic and First-Order Logic.
A: Euclid's Elements satisfies the criteria for being an axiomatic system. It does not, however, satisfy the criteria for being a formal system; the reason being that, from the point of view of formalism, certain steps in Euclid's proofs are left implied or tacit. In other words: all formal systems are axiomatic, but not all axiomatic systems are formal.
In nearly all practicality - except when studying formal systems - one would rarely do work in a bona fide formal system, but always in an axiomatic, or one is not doing math.
