Instructor: Dr Benedikt Löwe
Time: Wednesday, 17-19
Course language: English
Teaching Assistant: Brian
Intended Audience: M.Sc. students of Logic and Mathematics
Prerequisites: This course assumes knowledge comparable to the course
"Axiomatic Set Theory".
Literature: Kenneth Kunen, Set Theory, An Introduction to Independence Proofs,
North-Holland 1980 [Studies in Logic and the Foundations of Mathematics 102]
In this class, we shall cover the important technique of forcing. Our motivation is
Hilbert's First Problem: the question whether there is a bijection between the first uncoutable
ordinal omega1 and the set of real numbers (this statement is called the
continuum hypothesis CH). In the early 1960s, Paul Cohen
developed the technique of forcing that allows to construct models of set theory in which CH
is true and those in which it is false.
This class will be much more metamathematical than other set theory classes and will require
some preparatory work on models of set theory before we can start developing the theory of forcing.
Grading is based on homework assignments.
- September 7, 2005.
Class cancelled because of the
Annual ILLC Boat Trip.
All students enrolled in the class are cordially invited to join us for the boat trip.
- September 14, 2005. (First Lecture.) General
The continuum problem.
A very informal introduction to the methodology of forcing. Relativization
Homework: Exercises IV.(6) and IV.(8).
- September 21, 2005. (Second Lecture.)
Literature on higher set theory. Absoluteness.
Homework: PDF file.
- September 28, 2005.
(Third Lecture.) More on absoluteness. Absoluteness of recursive
definitions. The Hkappa and inaccessible
cardinals. The consistency strength hierarchy.
Homework: Exercises IV.(10) and IV.(30).
- October 5, 2005.
(Fourth Lecture.) Lévy Reflection Theorems. Mostowski
Collapse (without proof).
Defining Definability (without any details).
Definition of OD and HOD.
Homework: Exercises III.(10), III.(14) and V.(5) (pages 108 and
163 in Kunen).
- October 12, 2005.
(Fifth Lecture.) The constructible hierarchy. The Axiom of
Constructibility V=L. The Condensation Lemma. Notation for
partial orders in forcing: stronger than, chain conditions, etc..
Examples. Definition of MA(kappa).
Homework: Exercises VI.(2), VI.(3), II.(8), and II.(16) (pages 180,
87 and 88 in Kunen).
- October 19, 2005.
(Sixth Lecture.) More examples of partial orders.
MA(aleph0). P-names. The generic
extension. Minimality of the generic extension.
Homework: Exercises VII.A1 (p.237), VII.A5, VII.A6, VII.A7, VII.A10
(p.238), VII.B1 (p.239). Due on November 2, 2005!
- October 26, 2005.
No Class (Exam week)
- November 2, 2005. (Seventh Lecture.) Proof of some
ZFC axioms in M[G]. The (semantic) forcing relation. The syntactic
forcing relation. The Forcing Lemma.
Homework: Exercises VII.A12 (p.239), VII.B5 (p.240).
- November 9, 2005. (Eighth Lecture.) The Forcing Lemma
(ctd.). ZFC in M[G].
Homework: PDF file.
- November 16, 2005. (Ninth Lecture.) Consistency proofs
with forcing. Collapsing cardinals. Adding reals. Cardinal preservation.
Every c.c.c. forcing preserves cardinals. Delta System lemma. Adding
kappa many Cohen
reals is c.c.c.
Homework: Exercises II.1 (p.86), VII.A2 (p.237).
- November 23, 2005. (Ninth Lecture.) Preserving
cofinalities and preserving regularity. Computing the exact cardinality of
the continuum in the Cohen model. Adding subsets of larger cardinals.
Homework: Read p.234-236 of Kanamori's book on Prikry forcing. Find
out why Prikry forcing is an example of a forcing that preserves cardinals
but not cofinality (Exercise 18.5). Also do Kunen's exercise VII.(G1) on
- November 30, 2005. (Tenth Lecture.) Simultaneously
2aleph-0 and 2aleph-1.
lambda-closed forcing and preservation of cardinals.
Collapses. Preservation of aleph1 and reals.
Homework: Exercises VII.C1, VII.C2, VII.C3, VII.C4 (p.242),
VII.G5 (p.246). Due on December 14, 2005!
- December 7, 2005.
- December 14, 2005.
- December 21, 2005.
No Class (Exam week)
Last update : November 30th, 2005