[Math] Clarifying the definition of an axiomatic system

logicmodel-theoryproof-theory

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:

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

  2. What is the reason he didn't introduce in the definition of Axiom Systems, the need of rules of inference ( to derive the Theorems ) ?

  3. 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.

Best Answer

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.

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