Mathematical logic is a branch of mathematics derived from symbolic logic and includes the subfields of model theory, proof theory, recursion theory and set theory. It is closely related to the formal logic in philosophy originated by Aristotle, but mathematical logic is a more complete method of checking arguments. Mathematical logic uses formal proof systems that are used to prove certain theorems. Here's how to understand mathematical logic.
Study sentential logic as the first encounter with mathematical logic. This includes truth tables and the use of "and," "or" and "not" in symbolic logic. This level of study should also include first order logic, which adds quantifiers such as "for all" and "there exists" to the language.
Continue with proof theory, which is the study of symbolic manipulation. This will require a formal language consisting of a set of symbols and a syntax. These elements comprise formulas that are used to build axioms for the theories of that language.
Advance to first order model theory, which describes the structures that will satisfy a set of axioms. Logical formulas are used to determine the sets that may be defined in a given structure.
Begin a study of set theory. This should include very large infinite sets to show that a "set" is an ambiguous concept.
Take up recursion theory next. This field is the study of membership of a given set by determining what can be calculated about that set in a finite number of steps. Recursion theory involves concepts such as degree structures, ideas about reducibility and relative computability.