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Mathematical Logic

Mathematical logic is a subfield of mathematics, which is related to the formal systems in association with the way they encode intuitive concepts of mathematical objects such as sets and numbers, proofs, and computation.

Mathematical logic was earlier known as “symbolic logic” (as opposed to philosophical logic) and “metamathematics.” The former term is still in use as in the Association for Symbolic Logic, but the latter term is in use for certain aspects of proof theory. Mathematical logic was the name given to symbolic logic by Giuseppe Peano.

Mathematical logic is not so much the logic of mathematics as it is the mathematics of logic. It includes those parts of logic, which can be modeled and studied mathematically. It also includes areas of pure mathematics, such as model theory and recursion theory, in which definability is central to the subject matter.

The research in mathematical logic has played an important role in the study of foundations of mathematics.
Mathematical logic is often divided into four subfields. The "Handbook of Mathematical Logic" (1977) divides mathematical logic into four parts; model theory, proof theory, set theory, and recursion theory.

Model theory: The theory studies the models of various formal theories. The set of all models of a particular theory is called an elementary class; classical model theory seeks to determine the properties of models in a particular elementary class, or determine whether certain classes of structures form elementary classes. The method of quantifier elimination is used to show that models of particular theories cannot be too complicated.

Proof theory: It is the study of formal proofs in various logical deduction systems. These proofs are represented as formal mathematical objects, facilitating their analysis by mathematical techniques.

Set theory: It is the study of sets that are abstract collections of objects. The basic concepts of set theory such as subset and relative complement are often called naive set theory. Modern research is in the area of axiomatic set theory that uses logical methods to study which propositions are provable in various formal theories such as ZFC or NF.

Recursion theory: The theory is also called computability theory. It studies the properties of computable functions and the Turing degrees, which divide the uncomputable functions into sets which have the same level of uncomputability. Recursion theory also includes the study of generalized computability and definability.