This course will provide an introduction to set theory and independence results. The following is an overview of the material:

1. Introduction to the axioms of Zermelo-Frankel set theory with Choice (ZFC); how models of set theory serve as models of mathematics; the relationship between classes and sets.
2. Basic tools in set theory: ordinals, cardinals, transfinite induction and recursion; the Borel heirarchy; stationary sets, closed unbounded sets and the Pressing Down Lemma; the \(\Delta\)-System Lemma; set-theoretic trees and sequences.
3. Topics in combinatorial set theory: Jensen's \(\diamondsuit\); Martin's Axiom \(\mathrm{MA}_{\kappa}\); Souslin trees; gaps, towers, and coherent sequences; measure and category; Devlin-Shelah and Solovay coding.
4. Forcing: basic theory and definitions; the countable chain condition and the consistent failure of CH; countably closed forcings and the consistency of CH; Cohen and random reals.
5. Iterated forcing: two stage iterations and factorization; finite support iterations; the consistency of \(\mathrm{MA}_{\kappa}\); the Solovay model.
6. The \(L(\mathbb{R})\) Absoluteness Theorem

Text: Most of the course will follow these lecture notes (the version of these notes from the start of the semester is here). These lecture notes are a work in progress and will be updated and corrected as the course progresses. If you find corrections which should be made, please let me know. Kunen's Set Theory: an Introduction to Independence Results will also serve as a supplemental text. This text is available electronically through the math library. Additional reference material will be posted later.

Background: Students are expected to have a solid foundation in undergraduate mathematics. Good proofwriting skills are essential. Ideally students will have had some exposure to basic measure theory and in particular be familiar with the notion of a Borel set. This can be found in, e.g., Rudin's Real and complex analysis or in my notes on descriptive set theory. Ideally students should have some familiarity with first order logic, although this will mostly be important toward the end of the course. MATH 4810 or 6810 are more than adequate. Those who would like some additional background reading on first order logic should consult Lou van den Dries's lecture notes.

Grading: This course can be taken for either a letter grade or with the S/U option. Students who complete (nearly) all of the assignments either correctly or with occasional minor errors will receive an A; minor errors on most homework sets, frequent serious errors, or multiple missing assigments may result in an A- or some form of B. C and below indicates a lack of comprehension of the basic concept in the course. The homework will be set up so that more advanced students can challenge themselves while students who need to master the basics can do this as well. A grade of A+ is reserved for (near) perfect work which includes the completion of the more challenging homework problems.

Homework should be submitted via Gradescope. If you are unable to submit homework by Gradescope, please either email the assignment to the instructor or submit a hardcopy.

Detailed Schedule:

Week 16

• Monday (12/5) Solovay's theorem, part 2

Week 13

• Monday (11/14): Names, interpretation, and the fundamental theorem of forcing.
• Wednesday (11/16): Prikry forcing.
• Friday (11/18): Iterated forcing.

Week 12

• Monday (11/7): Forcings to add \(\theta\) Random and Cohen reals.
• Wednesday (11/9): the \(\kappa\)-c.c. and CON(not CH)
• Friday (11/11): \(\sigma\)-closed posets and CON(CH)

Week 11

• Monday (10/31): Introduction to forcing, forcing syntax and its properties.
• Wednesday (11/2): Syntactic analysis of the forcing relation, part 1.
• Friday (11/4): Synatactic analysis of the forcing relation, part 2.

Week 10 Optional additional homework.

• Monday (10/24): Speciality of trees with no uncountable chains.
• Wednesday (10/26): Speciality of trees with no uncountable chains, continued.
• Friday (10/28): The Hechler poset.

Week 9 .

• Monday (10/17): The \(\Delta\)-System Lemma.
• Wednesday (10/19): (no class - will be made up at future date)
• Friday (10/21): (no class - will be made up at future date)

Week 8 Homework assignment 5 (due 10/26).

• Monday (10/10): Fall Break (no class)
• Wednesday (10/12): Knaster's condition, and the productivity of the c.c.c..
• Friday (10/14): Additivity of measure; Solovay's almost disjoint coding; \(\sigma\)-centered posets, \(\mathfrak{p}\), and Bell's Theorem.

Week 7

• Monday (10/3): Trees and linear orders.
• Wednesday (10/5): Jensen's \(\diamondsuit\) and his construction of a Souslin tree.
• Friday (10/7): Martin's Axiom.

Week 6 Homework assignment 4 (due 10/12).

• Monday (9/26): ZFC and the Generalized Continuum Hypothesis in \(\mathbf{L}\); condensation.
• Wednesday (9/28): Souslin's problem.
• Friday (9/30): Analysis of c.c.c. linear orders and Souslin's problem.

Week 5 Homework assignment 3 (due 9/26).

• Monday (9/19): Definability and its (un)definability; elementarity, clubs and stationary sets.
• Wednesday (9/21): The Pressing Down Lemma.
• Friday (9/23): Ulam matrices; Gödel's constructible universe \(\mathbf{L}\).

Week 4

• Monday (9/12): Zorn's Lemma.
• Wednesday (9/14): Inaccessible cardinals.
• Friday (9/16): The Reflection Theorem

Week 3 Homework assignment 2 (due 9/14).

• Monday (9/5): Labor Day (no class).
• Wednesday (9/7): The Mostowski collapse and other uses of transfinite recursion.
• Friday (9/9): Cardinals and the Axiom of Choice.

Week 2

• Monday (8/29): Transfinite recursion on well-founded relations.
• Wednesday (8/31): Ordinal arithmetic.
• Friday (9/2): Cantor Normal Form; Mostowski Collapse

Week 1 Homework assignment 1 (due 9/2 via Gradescope).

• Monday (8/22): Course overview.
• Wednesday (8/24): Axioms of ZFC, part 1: Extensionality, Emptyset, Pairing, Union, Powerset, Separation; ordered pairs, Cartesian product, classes.
• Friday (8/26): Regularity, ordinals, transfinite induction.