All meetings online via MS Teams at 2:30 pm.
Session 1, 11/04: Real numbers. The Continuum Hypothesis, Suslin's Problem, the Perfect Set Theorem for closed sets, Borel sets.
Session 2, 11/25: Review of Model Theory. Gödel's completeness and incompleteness theorems, definability, directed systems and direct limits, models of set theory; filters, ultrafilters, ultraproducts and ultrapowers.
Session 3, 11/30: Constructibility, Part I. Definability; definitions by transfinite recursion and definability of the rank function; the Levy hierarchy of formulas; history, motivation behind, and definition of L; absoluteness; proof that L satisfies the axiom of constructibility; proof that L is a model of ZF containing all ordinals.
Session 4, 12/2: Constructibility, Part II. Consistency of V = HOD; proof that L is a model of the Axiom of Choice; the condensation lemma; proof that L is a model of the Generalized Continuum Hypothesis.
Session 5, 12/7: Forcing, Part I. Intuition; partial orders, generic filters; names.
Session 6, 12/9: Forcing, Part II. Generic extensions; canonical names; the forcing language and the forcing relation; proof that forcing preserves pairing, union, foundation; proof that M[G] extends M, contains G, is contained in every model of ZFC which extends M and contains G, and has the same ordinals as M.
Session 7, 12/16: Forcing, Part III. The Forcing Theorem; proof that forcing preserves ZFC; Cohen forcing; the countable chain condition; a model of ZFC where the Continuum Hypothesis fails.
Exam date: 01/12.
Session 1: Jech, Set Theory, chapter 4; Wikipedia.
Session 2: Jech, Set Theory, chapters 7 and 12; Wikipedia.
Sessions 3, 4: Jech, Set Theory, chapter 13; Wikipedia. Further reading: Devlin, Constructibility, chapters 1,2; Gödel, "The consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis," Proceedings of the National Academy of Sciences 24(12).
Sessions 5, 6, 7: Kunen, Set Theory: An Introduction to Independence Proofs, chapter 7.