Axiomatic Set Theory 2005/2006; 2nd Semester Institute for Logic, Language & Computation Universiteit van Amsterdam

Set Theory is both an area of mathematics (the study of sets as a kind of mathematical object) and an area of mathematical logic (the study of axiom systems of set theory as special axiomatic frameworks). As an area of mathematics, Set Theory has applications in all areas of pure mathematics, most notably set-theoretic topology. (Students planning to specialize in this research area, for example at the Vrije Universiteit will greatly benefit from having a firm understanding of the basics of Set Theory.)

This course will cover the basics of axiomatic set theory presented in a mathematical fashion. Knowledge of logic is not a prerequisite, though familiarity with the axiomatic method is.

We will follow the textbook Keith Devlin, "The Joy of Sets" (amazon.de link). If you are a member of the NSA or VIA student associations, you can buy the book in the basement of the Euclides building for a discount of 10%.

Grading will be based on weekly exercises. There will be no exam. There will be a Master level course Advanced Topics in Set Theory in the first semester of 2006/07 continuing the material of this course. It is possible to write a Master's thesis in set theory (either for an MSc in Mathematics or an MSc in Logic) based on the material of these two courses (Axiomatic Set Theory and Advanced Topics in Set Theory).

Lectures.

• First Lecture: Feb 8, 2006. The protagonists: cardinals and ordinals. Cantor, his background, Cantor's theorem, the Cantor-Bendixson derivative. Set-theoretic operations. First axioms: empty set, extensionality, union, pairing, power set.
  Georg Cantor (1845-1918): Biography.
• Second Lecture: Feb 9, 2006. Frege's Comprehension Axiom and its inconsistency (Russell's paradox). Ordered pairs as sets. Naive Set Theory: relations, functions, partial orders, wellfoundedness.
  Homework. PDF file (Deadline: Feb 16, 2006).
• Third Lecture: Feb 15, 2006. Wellfoundedness: examples. Functions: injectivity, surjectivity, bijectivity. Wellfoundedness and induction. Basic theory of wellorders.
• Discussion (B Semmes): Feb 16, 2006.
  Homework. PDF file (Deadline: Feb 23, 2006).
• Fourth Lecture: Feb 22, 2006. More on wellorders. The representation theorem for wellorders.
• Fifth Lecture: Feb 23, 2006. Proof of the representation theorem for wellorders.
  Homework. PDF file (Deadline: Mar 2, 2006).
• Sixth Lecture: Mar 1, 2006. Axiomatic Set Theory. The language of set theory LAST. LAST-describability: abbreviations in LAST, formalization of the notion of a group. Repetition of basic axioms: Union, Pairing, Powerset, Emptyset. LAST-Comprehension: Russell's paradox. The von Neumann-hierarchy. The infinity axiom. Separation axiom. No largest set.
• Seventh Lecture: Mar 2, 2006. Axiom systems of set theory and their names. The Axiom Scheme of Replacement. Definition of V_ω with replacement.
  Homework. PDF file (Deadline: Mar 9, 2006).
• Eighth Lecture: Mar 8, 2006. The Recursion Theorem for ordinals. The Recursion Theorem for the class of ordinals. Definition of V_α using recursion. The axiom of foundation. Connection to the von Neumann hierarchy. Mirimanoff rank.
• Discussion (B Semmes): Mar 9, 2006.
  Homework. PDF file (Deadline: Mar 16, 2006).
• Ninth Lecture: Mar 15, 2006. More about the von Neumann hierarchy. Fragments of set theory in levels of the von Neumann hierarchy. The Axiom of Choice.
• Discussion (B Semmes): Mar 16, 2006.
  Homework. PDF file (Deadline: Mar 23, 2006).
• Tenth Lecture: Mar 22, 2006. The Axiom of Choice. Zermelo's Well-ordering theorem. Zorn's lemma and one application. The equivalence of Zorn's lemma and the Axiom of Choice. Hartogs alephs.
• Eleventh Lecture: Mar 23, 2006. Existence of Hartogs alephs. Coding of wellorderings of a set as subsets.
  Homework. PDF file (Deadline: Apr 6, 2006; note: two weeks!).
• Mar 29, 2006. Exam Week. No Class.
• Mar 30, 2006. Exam Week. No Class.
• Twelfth Lecture: Apr 5, 2006. Construction of the algebraic structures N, Z, Q, and R in set theory. Grassmann Identities. First glance at ordinal operations.
• Discussion (B Semmes): Apr 6, 2006.
  Homework. PDF file (Deadline: Apr 13, 2006).
• Thirteenth Lecture: Apr 12, 2006. Ordinal operations: addition and multiplication.
• Discussion (B Semmes): Apr 13, 2006.
  Homework. PDF file (Deadline: Apr 20, 2006).
• Fourteenth Lecture: Apr 19, 2006. Ordinal operations: exponentiation. Cantor Normal Form. γ, δ, and ε numbers.
• Fifteenth Lecture: Apr 20, 2006. Normal operations, existence of fixed points for normal operations. Countability of ordinals and uncountable ordinals: &omega₁.
  Homework. PDF file (Deadline: Apr 27, 2006).
• Sixteenth Lecture: Apr 26, 2006. Cardinalities: the class relation of equipollence. Cardinalities are proper classes. Scott's Trick. Injections and bijections. The Schröder-Bernstein Theorem. Cardinals.
• Seventeenth Lecture: Apr 27, 2006. Unions of sets of cardinals are cardinals. Definition of the ω operation. Every cardinal is some ω_α.
  Homework. PDF file (Deadline: May 4, 2006).
• Eighteenth Lecture: May 3, 2006. Cardinal addition and multiplication. Cardinal operations differ from the ordinal operations. The aleph notation. The Gödel β function. Hessenberg's Theorem. Cardinal addition and multiplication trivialize.
• Discussion (B Semmes): May 4, 2006.
  Homework. PDF file (Deadline: May 11, 2006).
• Nineteenth Lecture: May 10, 2006. Cardinal exponentiation. Cantor's theorem revisited. The Continuum Hypothesis. The Generalized Continuum Hypothesis. Infinite sums and products. König's Lemma (in cardinal arithmetic).
• Twentieth Lecture May 11, 2006. Applications of König's Lemma. Cofinality. Singular and regular cardinals. Easton's Theorem (without proof).
  Homework. PDF file (Deadline: May 18, 2006).
• Twenty-first lecture: May 17, 2006. The function κ^λ. Hausdorff's formula. Some computations of cardinals. Behaviour of the continuum function at singular cardinals. Silver's Theorem (without proof). The Singular Cardinals Hypothesis SCH. Strong limit cardinals. Weakly inaccessible cardinals. Inaccessible cardinals. If κ is inaccessible, then V_κ is a model of ZFC. The existence of inaccessible cardinals can't be proved in ZFC.
• May 18, 2006. Class cancelled.
  Homework. PDF file (Deadline: May 25, 2006).