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.

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Some references:

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.

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