Course.
Math 116c Spring 2018
10:30-11:55 TR, 187 Linde Hall (Zoom most likely)

Instructor.
Aristotelis Panagiotopoulos
102 Linde Hall
panagio at caltech.edu
Office hours. Mondays 4-5 PM

In this course we will study the interactions between mathematical logic, set theory, and real analysis. During the first part of the quarter we will cover topics in Descriptive Set Theory and in the second part of the quarter we will develop the theory of Forcing and use it to establish various independence results in set-theory (and time permiting, in analysis, and algebra). Here is a rough sketch of the topics we are going to cover:
(1)Topics in Descriptive Set Theory. Theory of Borel and (co)analytic sets, Baire category methods, regularity properties of definable sets, determinacy of infinite games and its relationship with large cardinal assumptions.
(2) Forcing and applications. The Continuum Hypothesis is independent from ZFC, Axiom of Choice is independent from ZF, symmetric models and ergodic theory, Whitehead's problem for abelian free groups (time permitting), and automorphism problem for the Calkin Algebra (time permitting).

Books. This class will be based on a complilation of book chapters from Kechris' "Classical descriptive set theory" for the first half of the quarter and Kunnen's "Set Theory, An Introduction To Independence Proofs" for the second half of the quarter. Some additional resources which are going to be used include: Srivastavas' "A course on Borel sets", Anushes notes , and Rosendal's notes , Jech's "The axiom of Choice", Weaver's "Forcing for Mathematicians."

Slides.
Week 1, Day 1. Historic context, Polish spaces, examples.
Week 1, Day 2. Closure properties, Gδ-sets, filters, Baire category theorem, generic properties.
Week 2, Day 1. The generic linear order, the generic continuous function, tube lemma, trees, topology on the Baire space, closed subsets of the Baire space.
Week 2, Day 2. Ranks on Well-founded trees, transfer theorems, Suslin-Luzin-Cantor schemes.
Week 3, Day 1. Perfect Set Property, transfer theorems, operation A, analytic sets.
Week 3, Day 2. More characterizations of analytic sets, examples, Borel sets are strictly fewer than analytic, sets, universal closed and analytic sets, complete analytic sets.
Week 4, Day 1. Methods for showing non-Borelness, examples of (co)analytic non-Borel sets (WF,DIFF,DIV), Classical regularity properties.
Week 4, Day 2. Sets without perfect set property, projective hierarchy, analytic sets have perfect set property, introduction to independence results, the universe V of all sets.
Week 5, Day 1. Gödel's constructible universe L, a Δ¹_2 set that is not Lebesgue measurable, games, strategies, axiom of determinacy.
Week 5, Day 2. Games with rules, cut and choose game for PSP, open and closed games are determined, large cardinals
Week 6, Day 1. Measurable cardinals, Ramsey properties, Rowbottom's theorem, Kleene-Brower ordering.
Week 6, Day 2.Measurable cardinals imply analytic determinacy, historic remarks, sketch of Borel determinacy.
Week 7, Day 1. Review of basic set theory.
Week 7, Day 2. Transitive structures, Relativization, Mostowski collapse, Reflection
Week 8, Day 1. Intro to forcing, Forcing notions, existence of generics.
Week 8, Day 2. Names, basic properties of the generic extension M[G], the fundamental theorem of forcing
Week 9, Day 1. Sketch of proof of the fundamental theorem of forcing, M[G] satisfies ZFC, Forcing CH
Week 9, Day 2. Δ-system lemma, c.c.c. for posets, Forcing negation of CH, further directions: symmetric models and iterated forcing

Homework.
List of suggested exercises.

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