Week 1: Introduction to set theory: some axioms of the Zermelo-Fraenkel set theory, Russell's paradox, proper classes and sets; Glossary of set theory: union, intersection, power set, ordered pairs and binary Cartesian products; Functions: injective, surjective and bijective functions, composition of functions, direct and inverse images, sets of functions; Cantor's theorem on power sets
Week 2:Equivalence relations, partitions and quotients; Choice functions, Cartesian products of arbitrary families and the Axiom of Choice (AC), Equinumerosity; Cantor-Schroeder-Bernstein (CSB) theorem: proof 1 by Julius Konig, proof 2 via Knaster-Tarski fixed point theorem
Week 3:Standard number systems: Natural numbers, arithmetic of natural numbers using recursion, Integers, Rational numbers, Real numbers; Applications of the CSB theorem to sets constructed using standard number systems (tools include Cantor's middle third set and continued fractions); Equivalence of strong and weak induction principles
Week 4:Linearly ordered sets; Ordinal numbers: well-ordered sets, transitive sets, transfinite induction, ordinal arithmetic, Well-Ordering Theorem (WOT); Cardinal numbers: cardinal arithmetic assuming WOT
Week 5:Partially ordered sets (posets): strict and weak, Glossary of order theory: maximum, minimum, maximal and minimal elements, up and down subsets, Hasse diagram, chains and antichains; Order-preserving (monotone) and order-reflecting maps, order isomorphism; Order-theoretic and algebraic lattices, lattice homomorphisms; Zorn's lemma (ZL), Equivalence between AC, ZL and WOT (without proof), Application of ZL to construct a basis of a vector space (non-examinable)
Week 6:Boolean algebras as complemented distributive lattices; Glossary of boolean algebras: atoms and coatoms, filters, equivalence between different types of filters: maximal, prime and ultrafilters; Homomorphism and isomorphism between boolean algebras; Boolean prime filter theorem; Stone's representation theorems for boolean algebras: finite and infinite versions
Week 7:Introduction to logic; Propositional logic syntax: language and meta-language, formulas, unique readability of formulas; Propositional logic semantics: valuations, logical equivalence of formulas, Lindenbaum-Tarski algebra
Week 8:Conjunctive and disjunctive normal forms; Adequate sets of connectives; Satisfiable sets of formulas, logical/semantic consequence relation; Hilbert-style deductive calculus: sequents and formal proofs, deductive/syntactic consequence relation
Week 9:Finite character of proof; Deduction theorem; Consistent sets of formulas; Soundness and completeness theorem for Hilbert-style deductive calculus; Compactness theorem; Konig's lemma as an application of compactness
Week 10:Predicate logic syntax: language and meta-language, terms, formulas; Predicate language semantics: structures, interpretation/value of a term, truth of a formula, logical/semantic consequence relation, substructures and structure homomorphisms
Week 11:Theories, models, elementary equivalence; Ultraproducts, Los' theorem (proof non-examinable), construction of the ordered field of hyperreals as an application of the ultraproduct construction; Compactness via Los' theorem
Week 12:Upward and downward Lowenheim-Skolem theorems (without proof); Introduction to categoricity of theories; Quantifier elimination; Godel's incompleteness theorems (without proof)