# 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.

Motivating pictures:

### Exam

There will be an oral exam on (all of) the topics covered in class. (Dates are in the SIS.) If there will be interest, there will be also one exam in September, probably in the last week, possibly on September 26.
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 Konèický (in Czech).

### Log from classes

1. class 22.2.2017
Motivation: contradictions in the naive set theory. Starting with the axioms of Zermelo-Fraenkl: Axioms of Existence, Extensionality, Axiom Scheme of Comprehension. $x \in \emptyset$

Somewhat detailed notes from the class.

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

Exercises

• Prove that the "set of all sets" does not exists. [Hint: if it does, use the Axiom Scheme of Comprehension and deduce a contradiction, as we did in class.]
• Assume a weaker version of the Axiom of Existence: A set exists. Use the Comprehension Schema to deduce the version we started with. [Hint: If $A$ is a set, consider $\{x \in A \mid x \not= x \}$.]
2. class 1.3.2017
More axioms: The Axiom of Pair, The Axiom of Union, and The Axiom of Powerset. Ordered pair. Relations, existence of cartesian product, etc.

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

3. class 8.3.2017
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 15.3.2017
Recursive definitions, arithmetics.

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

5. class 22.3.2017
Cardinality of sets. Cantor-Bernstein. Finite sets.

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

6. class 29.3.2017
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.

7. class 5.4.2017
Dedeking cuts. Cardinal numbers.

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

8. class 12.4.2017
Well-ordered sets. Ordinals (beginning).

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

9. class 19.4.2017
More ordinals.

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

10. class 26.4.2017
Plan: 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.

11. class 3.5.2017
How to use ordinals 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.

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

no class on 17.5.2017
All classes canceled -- sporting day.
no class on 24.5.2017
I cannot attend this day, unfortunately. (Sorry for any complications this may cause.)
13. class on 25.5.2017 -- usual time (17:20-18:50) in S9
More infinite combinatorics: uncountable Ramsey, trees, Konig's lemma with applications.

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