Set Theory (NAIL063) -- Robert ©ámal

Basic course about set theory: starting with axioms of ZFC, covering ordinals, cardinals, axiom of choice, and ending with infinitary combinatorics.

This is the page for the English class. The Czech version of the class is taught by Jan Kynčl.

Motivating pictures:

Exam

There will be an oral exam on (all of) the topics covered in class. The dates of the exams are already in the system, let me know if they don't work for you.
With the notes from each class, there are some recommended exercises. While I will NOT check them, grade them, etc., it is still recommended to try to solve them, to make sure you understand what the class is about.

Literature

Many recommended books are mentioned in the SIS description of the class. We loosely follow the book by Hrbacek and Jech. You may want to consult notes of Václav Končický (in Czech).

Log from classes

1. class 20.2.2017
Motivation: contradictions in the naive set theory. Starting with the axioms of Zermelo-Fraenkl: Axioms of Existence, Extensionality, Axiom Scheme of Comprehension. Def. of empty set and of set intersection.

Somewhat detailed notes from the class.

(Notes are written using a nice app called Workflowy.)

Exercises

2. class 27.2.2018
More axioms: The Axiom of Pair, The Axiom of Union, and The Axiom of Powerset. Ordered pair.

Somewhat detailed notes from the class. Do not miss the Exercises at the end of the notes.

3. class 6.3.2018
Building the natural numbers using inductive sets and the Axiom of Infinity. Proving induction, etc.

Somewhat detailed notes from the class. Do not miss the Exercises at the end of the notes.

4. class 13.3.2018
Cartesian product, relations, functions. Recursive definitions (start). Next week we will finish the proof of the Recursion theorem.

Somewhat detailed notes from the class. Do not miss the Exercises at the end of the notes.

5. class 22.3.2017
Proof of the Recursion theorem, arithmetics of natural numbers. Cardinality of sets.

Somewhat detailed notes from the class. Do not miss the Exercises at the end of the notes.

6. class 29.3.2018
Cantor-Bernstein. Finite sets.

Somewhat detailed notes from the class. Do not miss the Exercises at the end of the notes.

7. class 3.4.2018
Countable sets. Cantor theorem and uncountability of reals.

Somewhat detailed notes from the class. Do not miss the Exercises at the end of the notes.

8. class 10.4.2018
Dedeking cuts. Cardinal numbers.

Somewhat detailed notes from the class. Do not miss the Exercises at the end of the notes.

9. class 17.4.2018
Well-ordered sets (main result: every two well-ordered sets are "comparable"). Ordinals (just a start).

Somewhat detailed notes from the class. Do not miss the Exercises at the end of the notes.

10. class 24.4.2018
Properties of ordinals, relation to well-ordered sets. Transfinite induction and recursion. Sketch: arithmetic of ordinals.

Somewhat detailed notes from the class. Do not miss the Exercises at the end of the notes. The notes include a detailed proof of the theorem about addition of ordinals, that perhaps seemed too complicated in class.

no class on 1.5. and 8.5.2018 -- State holidays
To alleviate the loss of two classes, here is some material for you to go over, to learn a bit more of the material and get to some of the more interesting stuff. Won't be needed to pass the exam, though. You may start with the Wikipedia page on ordinal arithmetic. More details are provided in the following lecture notes. Finally, an application of all of this to questions in combinatorics of finite graphs is the hydra game, or Goodstein sequences.
11. class 15.5.2018
How to use ordinals to define cardinals. Axiom of choice - various forms and applications.

Somewhat detailed notes from the class. Do not miss the Exercises at the end of the notes.

12. class 10.5.2017
Applications of AC. Infinite Ramsey theorem -- $R(\aleph_0) = \aleph_0$.

Somewhat detailed notes from the class. Do not miss the Exercises at the end of the notes.