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Axiomatic Set Theory
2003/2004; 2nd Trimester
Institute for Logic, Language & Computation
Universiteit van Amsterdam

Instructor: Dr Benedikt Löwe
Grader: Brian Semmes
Vakcode:
Time: Wednesday 3-5, Thursday 11-1
Place: P.227 (Wednesday during January), P.016 (Wednesday during February and March), P.015A (Thursday)
Course language: English
Intended Audience: Mathematics students in their third or fourth year, MoL students

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 in the Department of Geometry 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.

Topics covered will include:

  • Axioms of Set Theory
  • Foundations of Mathematics
  • The Axiom of Choice
  • Basic Descriptive Set Theory
  • Ordinals and Cardinals
  • Cardinal Arithmetic
  • Combinatorial Set Theory
  • Measurable Cardinals

We will start to follow the textbook Yiannis N. Moschovakis, Notes on Set Theory, Springer-Verlag 1994 which covers the first five topics. After that, we shall continue with Chapters 5, 8, 9 and 10 of Thomas Jech, Set Theory, The Third Millenium Edition, revised and expanded, Springer-Verlag 2003.

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 2004/05 continuing the material of this course. It is possible to write a Master's thesis in set theory (either for an M.Sc. in Mathematics or an M.Sc. in Logic) based on the material of these two courses (Axiomatic Set Theory and Advanced Topics in Set Theory).

Lectures.

  • 1st Lecture (Jan 7). Set Theory as subfield of mathematics. Set Theory as foundations of mathematics. History of the axiomatic method. The birth of set theory. Relations. Functions. Equinumerosity. Countability. (p.1-8 in Moschovakis' book)
  • 2nd Lecture (Jan 8). More about functions. Informal discussion of natural and real numbers. Countable sets: the integers, the rational numbers. Diagonal counting method. Countability of countable unions of countable sets. Cantor's Theorem: uncountability of the set of real numbers. Power set version of Cantor's Theorem. (p.8-12 and p.15 in Moschovakis' book)
  • 3rd Lecture (Jan 14). Finite sequences. Cardinality of the set of real numbers. The Cantor-Schröder-Bernstein Theorem. Axiom of Full Comprehension. Russell's Paradox. Axiom of Extensionality. Axiom of Pairing. Empty Set Axiom. Axiom of Separation (Aussonderungsaxiom). (p.13-18, p.21-22, p.24-25 in Moschovakis' book)
  • (Jan 15). Lecture Cancelled: Provability Logic: New Frontiers
  • 4th Lecture (Jan 21). The Axiomatic Method. Peano Axioms. Systems of Arithmetic. Connections between structures for set theory and directed graphs. Power Set Axiom. Union Axiom. Axiom of Infinity. Zermelo set theory Z-. The non-existence of a set of all sets. (p.25-26, p.53 in Moschovakis' book)
  • 5th Lecture (Jan 22). The ordered pair. Kuratowski's definition. Cartesian products. Relations, functions, injections, surjections revisited. Equivalence Relations. Equivalence Classes. Quotients. Topological Spaces. Groups. (p.34-41, p.45-46 in Moschovakis' book)
  • 6th Lecture (Jan 28). Disjoint unions. Existence of the natural numbers. Zermelo numbers. Von Neumann numbers. The Recursion Theorem. Recursion with Parameters. (p.53-59 in Moschovakis' book)
  • 7th Lecture (Jan 29). Addition and Multiplication of natural numbers. Ordering of the natural numbers. Well-orders. Integers. Rational Numbers. Dedekind Cuts. Cauchy Sequences. The real numbers. Examples of transfinite recursions. (p.59-63 and parts of Appendix A (p.209-237) in Moschovakis' book)
  • 8th Lecture (Feb 4). Well-orders. Wellorderable sets. Least elements in well-orders. Successors and limits. Initial segments of wellorders. Order preserving functions. Order isomorphisms. (p.93-98 in Moschovakis' book)
  • 9th Lecture (Feb 5). Transfinite Induction. Transfinite Recursion. Operations on well-orders: successor, addition, multiplication. Initial segment order of well-orders. Total ordering of well-orders (1st half of the proof). (p.98-104 in Moschovakis' book)
  • 10th Lecture (Feb 11). Total ordering of well-orders (2nd half of the proof). Wellfoundedness of the class of wellorders. Ordinals. Operations on ordinals: successor, union. (p.104-105, p.174 and p.195 in Moschovakis' book)
  • 11th Lecture (Feb 12). The Replacement Axiom. Zermelo-Fraenkel set theory ZF-. The General Recursion Theorem for function-like formulas. The transitive closure of a set. The von Neumann isomorphism. The ordinal of a wellorder. Properties of the von Neumann isomorphism. Wellfoundedness and transitivity of the class of ordinals. The Burali-Forti paradox. (p.169-175 and p.189-196 in Moschovakis' book)
  • (Feb 18). Exam Week.
  • (Feb 19). Exam Week.
  • 12th Lecture (Feb 25). Recursion on the ordinals. Ordinal Arithmetic: Addition and Multiplication. Examples for ordinals. Ordinal Exponentiation. Hartogs' Theorem. The Aleph Sequence. (p.197-200 and p.106-107 in Moschovakis' book)
  • 13th Lecture (Feb 26). Wellorderings on N and aleph1. Limit points in the initial ordinals. Subtraction and division of ordinals. The Cantor Normal Form Theorem. More examples for ordinals. Examples for ordinal computations.
  • 14th Lecture (Mar 3). Even and odd ordinals. Choice Functions. The Axiom of Choice. Zermelo-Fraenkel set theory with Choice ZFC-. The product version of the Axiom of Choice. Zermelo's Wellordering Theorem. Cardinal numbers. Cardinal addition and multiplication. Hessenberg's Theorem. (p.43-44, p.117-121 and p.136-137 in Moschovakis' book)
  • 15th Lecture (Mar 4). Infinite sums and products of cardinals. Cardinal exponentiation. The Beth Sequence. The Continuum Hypothesis. The Generalized Continuum Hypothesis. Cofinality. Regular Cardinals. Singular Cardinals. Some examples for cofinalities. (p.19, p.43-44, p.69, p.141, p.205 in Moschovakis' book)
  • 16th Lecture (Mar 10). König's Theorem (on infinite sums and products). The Gimel Function. The Cofinality of the Continuum. Consequences of the Axiom of Choice (without proofs). Baire Space. The Metric and Topology of Baire Space. (p.137-139, p. 145-147 in Moschovakis' book)
  • 17th Lecture (Mar 11). Trees. Branches of Trees. Tree Representations. Trees and closed sets. Perfect sets. The Cardinality of perfect sets. The Cantor-Bendixson sequence of a closed set. (p. 148-149 in Moschovakis' book)
  • 18th Lecture (Mar 17). The Cantor-Bendixson Theorem. Extensions of the Cantor-Bendixson Theorem: Hausdorff's Theorem (without proof). Bernstein's Theorem. The Axiom of Foundation. Infinite Descending Sequences. (p. 149-150, p. 160-161, p.178-179 in Moschovakis' book)
  • 19th Lecture (Mar 18). Z, ZF, ZFC. The von Neumann hierarchy. Consistency of the Axiom of Foundation. Inaccessible Cardinals. Inaccessibility of inaccessible cardinals. General remarks about large cardinal notions. (Chapter 12 in Jech's book)
Homework:
Additional Exercises (not part of the official homework):
Last update : March 26th, 2004