Basic course about set theory: starting with axioms of ZFC, covering ordinals, cardinals, axiom of choice, and ending with
infinitary combinatorics.
Motivating pictures:
- 1. class 22.2.2017
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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
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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
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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
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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
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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
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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
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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
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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
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More ordinals.
Somewhat detailed notes from the class. Do not miss the Exercises at the end of the notes.
- 10. class 26.4.2017
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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
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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
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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
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All classes canceled -- sporting day.
- no class on 24.5.2017
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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
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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.
