Math 574 - Topics in Logic and Foundation

Spring 2011

Overview

Lectures: MWF 12:20-1:10pm, Room 106 McAllister
Instructor: Jan Reimann
Office: 318B McAllister
Office hours: Mo 2-4, We 1:30-2:30
Email:
Personal Website: http://math.psu.edu/reimann/

Content

We will cover the basics of Descriptive Set Theory. Among the topics are: Polish Spaces, Borel and projective sets, regularity properties of sets of reals, effective descriptive set theory, Borel equivalence relations.

Material

There will be lecture notes for the course available on this website.

Titlepage and Notation
Bibliography
Lecture: Perfect subsets of the real line
Lecture: Polish spaces
Lecture: Excursion – The Urysohn space
Lecture: Trees
Lecture: Borel sets
Lecture: Borel sets as clopen sets
Lecture: Measure and category
Lecture: The Axiom of Choice
Lecture: Effective Borel sets
Lecture: The Structure of Borel sets
Lecture: Continuous Images of Borel Sets
Lecture: Analytic Sets
Lecture: Regularity Properties of Analytic Sets
Lecture: The Projective Hierarchy
Lecture: The Constructible Universe
Lecture: Constructible Reals
Lecture: Co-Analytic Sets
Lecture: Σ¹₂ Sets
Lecture: Recursive Ordinals and Ordinal Notations
Lecture: Π¹₁ Sets of Natural Numbers
Lecture: Co-analytic Ranks
Lecture: Hyperarithmetical Sets

Recommended reading:

S. Gao, Invariant Descriptive Set Theory, CRC Press, 2009
A. S. Kechris, Classical Descriptive Set Theory, Springer, 1995
R. Mansfield and G. Weitkamp, Recursive Aspects of Descriptive Set Theory, Oxford University Press, 1985
Y. N. Moschovakis, Descriptive Set Theory, 2nd edition, American Mathematical Society, 2009
G. Sacks, Higher Recursion Theory, Springer, 2010

Prerequisites

Basic knowledge in Mathematical Logic (Math 557) and in Topology (in particular metric spaces).

Exams

There will be a take home final at the end of the semester.

Homework

Homework will be assigned each Wednesday and will be due in class the following Wednesday in class. Homework will be graded and the two lowest scores will be dropped. Late homework will not be accepted. There will be no exception to this rule. Of course it may happen that you cannot turn in homework because you were ill or for some other valid reason. This is why the two lowest scores will be dropped.

Homework 1, due January 26, 2011
Homework 2, due February 9, 2011
Homework 3, due February 16, 2011
Homework 4, due February 23, 2011

Final Homework, April 18, 2011

Course Grade

The final grade will be determined as follows: 70% homework, 30% final exam.

Academic Integrity

All Penn State Policies regarding ethics and honorable behavior apply to this course.

Collaboration: Collaboration among students to solve homework assignments is welcome. This is a good way to learn mathematics. So is the consultation of other sources such as other textbooks. However, every student has to hand in an own set of solutions, and if you use other people's work or ideas you have to indicate the source in your solutions.
(In any case, complete and correct homework receives full credit.)