Set Theory of the Real Line
Wintersemester 2014/2015

Mathematisches Institut Abteilung f├╝r Math. Logik

Die ist die Homepage der Vorlesung "Set Theory of the Real Line" im Wintersemester 2014/2015.


Giorgio Laguzzi

(Please feel free to contact me at for asking any question about the course.)

Ort und Zeit

Di 14 - 16, SR 125, Eckerstr. 1.
Do 10 - 12, SR 404, Eckerstr. 1.

Klausur, Ort und Zeit: Eckerstrasse 1, Room 311, 19.03.2015, 14h


The aim of this course is to give an introduction to the study of the real line from the set-theoretical viewpoint. When dealing with the real numbers, it is a common practice in set theory to work with the Baire space, i.e., the set of infinite sequences of natural numbers endowed with the Baire topology. As a consequence, one can investigate questions concerning measure and category in terms of combinatorial properties of infinite sequences and trees. We will develop a careful study of the ideals of null and meager sets, as well as the regularity properties, such as the Baire property, the Lebesgue measurability and the perfect set property, and we will further see the connections with infinite games.


  1. T. Bartoszynski, H. Judah, - Set Theory - On the structure of the real line, AK Peters, 1995.
  2. K. Kunen, Set Theory - An introduction to independence proofs, North Holland, 1980.
  3. A. Levy, Basic Set Theory, Springer, 1979.
  4. T. Jech, Set Theory, Springer, 3rd Milleniuum edition, 2003.


WEEK 1: relationship between the real line, the Cantor space and the Baire space.
(Exercise sheet 1)
WEEK 2: projective hierarchy, definable sets in second order arithmetic and Borel codes.
(Exercise sheet 2)
WEEK 3: cardinal invariants, combinatorial characterization of meager sets and cov(M).
(Exercise sheet 3)
WEEK 4: Inequalities between cardinal invariants.
(Exercise sheet 4)
WEEK 5: Generic extension and forcing method.
(Exercise sheet 5)
WEEK 6: embeddings and iterated forcing.
(Exercise sheet 6)
WEEK 7: examples of tree-forcing and their basic properties.
(Exercise sheet 7)
WEEK 8: examples of tree-forcing and their basic properties, part II.
(Exercise sheet 8)
WEEK 9: proper forcing, quotient and preservation of properness.
(Exercise sheet 9)
WEEK 10: Second preservation theorem: preserving ``no dominating reals are added''.
(Exercise sheet 10)
WEEK 11: Consistency of cov(M)=cov(N)> b and other strict inequalities.
(Exercise sheet 11)
WEEK 12: Levy collapse and Solovay's model.
(Exercise sheet 12)
WEEK 13: Generalized regularity properties of the reals.
(Exercise sheet 13)
WEEK 14: Regularity of Delta^1_2 and Sigma^1_2 sets.
(Exercise sheet 14)