Course description: Some basic questions of logic are "What is a valid argument?", "How to distinguish valid arguments from invalid ones?", "What is a proof?", "What does it mean that a statement is true?", and "How are these notions related?". During the course, we will focus mainly on classical first-order logic, which is a standard model for mathematical proofs. We are going to define mathematical structures, proofs, truth and investigate their connection, as well as, the axiomatic method and its limitations. And we will peek into some logics beyond first-order one.

Topics covered:

• Sentential logic, truth assignments, truth tables, tautological implication.
• Boolean functions, disjunctive normal form, compactness theorem.
• First-order languages, terms, formulas, free variables, sentences.
• Truth and models, logical implication, definability.
• Homomorphism, isomorphism, automorphism, elementary equivalence.
• Formal deduction, modus ponens, deduction theorem, contraposition, reductio ad absurdum.
• Soundness and completeness theorems.
• Finite models, Löwenheim–Skolem theorems, Skolem's paradox.
• Theory of a class of structures, axiomatizable and finitely axiomatizable theories.
• Complete, ℵ0-categorical and κ-categorical theories, Łoś–Vaught test for completeness.
• Interpretations between theories, syntactical translation. Basic ideas of nonstandard analysis.
• Elimination of quantifiers, theory of natural numbers, undecidability, Gödel's incompleteness theorem.
• Gödel's second incompleteness theorem for set theory.
• Second-order logic, many-sorted logic, general structures.