Axiom System and facts/rules Im studying First Order Logic and im trying to make order of the various concepts.
I think i have understood what is an axiom system (some people call it "Formal System"). But i have two questions:
1) I haven't understood the difference beetween a theory and a formal system. (however i understood the definition of theory)
2) i havent understood how facts and rules are connected to the concepts of formal system and theory.
Can you help me ?
 A: Theory in mathematical logic is used with different (but related) meanings.
In a broad sense, we understood as a theory in e.g. FOL a set of non-logical axioms on top of the deductive calculus, like Peano's axioms or Tarski's axioms.
In the first case, we have the so-called first-order theory of arithmetic.
It is made of: a (first-order) language, logical axioms (one of many equivalent versions of axioms for predicate calculus), a set of rules (usually, at least modus ponens) and the set of specific arithmetical axioms (the first-order version of Peano original axioms).

In model theory we call (complete) theory associated with a structure $A$, denoted: $\text{Th}(A)$:

the set of all first-order sentences over the signature [i.e. language] of $A$ which are satisfied by $A.$ 

I.e.

$\text{Th}(A) = \{ \sigma \mid A \vDash \sigma \}$.

A third related meaning is:

a theory is a set of sentences closed under logical implication.

That is, a set of sentences $T$ is a theory iff for any
sentence $σ$ of the language: $T \vDash σ \text { iff } σ ∈ T$.
If we define $\text{Mod}(T)$ to be the class of all models of the set $T$ of sentences, we have that $\text {Th} (\text { Mod}(T))$ is the set of all sentences which are true in all models of $T$. But this is just the set of all sentences logically implied by $T$, or set of consequences of $T$.
For example, first-order theory of arithmetic is the set of consequences of the axioms for arithmetic.
