Mathematical Logic

Syntax of First-Order Logic: First Order Languages, Terms and Formulas of a First Order language, First Order Theories. Semantics of First-Order Languages: Structures of First-Order Languages, Truth in a Structure, Model of a Theory. Propositional Logic: Tautologies and Theorems of propositional Logic, Tautology Theorem. Proof in First Order Logic, Metatheorems of a first order theory, e.g. , theorems on constants, equivalence theorem, deduction and variant theorems etc., Consistency and Completeness, Lindenbaum Theorem. Henkin Extension, Completeness theorem, Extensions by definition of first order theories, Interpretation theorem. Model Theory: Embeddings and Isomorphisms, L¨owenheim-Skolem Theorem, Compactness theorem, Categoricity, Complete Theories. Recursive functions, Arithmatization of first order theories, Decidable Theory, Representability, Godel’s first Incompleteness theorem.