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 aleph_{1}. 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:
- Homework Assignment #1. (Deadline. January 14th, 2004)
Exercises x2.2 and x2.3 (p.18 of Moschovakis' book).
- Homework Assignment #2. (Deadline. January 22nd,
2004) Portable Document Format File.
PostScript File.
- Homework Assignment #3. (Deadline. January 29th,
2004) Portable Document Format File.
PostScript File.
Solution
to 3.1 (due to Brian Semmes).
- Homework Assignment #4. (Deadline. February 5th,
2004) Exercises x5.1, x5.2 and x5.19 (p.69 and p.71
of Moschovakis' book).
(Note. For x5.1 and x5.2, you may use Theorems 5.12
and 5.15 without proof.)
- Homework Assignment #5. (Deadline. February 12th,
2004)
Exercises
7.17 and 7.18 (p.97 of Moschovakis' book) and
x7.9 (p.110 of Moschovakis' book).
- Homework Assignment #6. (Deadline. February 26th,
2004)
Exercises
11.11 (p.174), 11.14 (p.175),
x11.1 (p.185), and x7.14 (p.112).
- Homework Assignment #7. (Deadline. March 4th,
2004) Portable Document Format File.
PostScript File.
- Homework Assignment #8. (Deadline. March 11th,
2004) Portable Document Format File.
PostScript File.
- Homework Assignment #9. (Deadline. March 18th,
2004) Portable Document Format File.
PostScript File.
Additional Exercises (not part of the official homework):
Last update : March 26th, 2004