Mathematical Logic

Mathematical logic is a subfield of mathematics and theoretical computer science, which bears close connections to metamathematics, the foundations of mathematics. This field explores the applications of formal logic to mathematics.

Objectives

The unifying themes in mathematical logic include the study of the expressive power of formal systems and the deductive power of formal proof systems.

Prerequisites

None

Syllabus

Propositional Calculus : Language, Unique Readability, Logical connectives, truth assignments, Semantic Implication, adequacy of connectives, disjunctive and conjunctive canonical form, Compactness Theorem for Propositional Logic, applications. First-Order Predicate Calculus: Language, Variables, Notions of Free and Bound Variables, substitution, programming analogy, the concept of structure, language's interpretation, Tarski definition of truth. Axiomatization of First-Order Logic: the concept of axiomatic system, analogies with algorithmic concepts, the concept of , Theorems of Correctness and Gödel's Theorem of Completeness. Gödel's incompleteness theorems and Gödel-Church's Theorem. Proof Theory of Propositional and Predicate Calculus: Gentzen system, propositional approach, cut elimination, Tableau systems and completeness.