Overview

Lectures: MWF 1:25 PM - 2:15 PM, Room 104 Osmond Lab
Instructor: Jan Reimann
Office: 318B McAllister
Office hours: Tu 11-12, We 11-12
Email:
Personal Website: http://www.math.psu.edu/reimann/

Content

This course explores foundational issues of mathematics. At the center will be two notions: undecidability and incompleteness. Topics include:

  • Undecidability of first order number theory (arithmetic)
  • The Gödel Incompleteness Theorems
  • Hilbert's 10th Problem
  • Undecidability in mathematics: Word problems, homeomorphism of manifolds, etc.
  • Decidability of the real number system: the Tarski-Seidenberg Theorem and quantifier elimination
  • Informal and formal set theory
  • The Continuum Hypothesis
  • The Axiom of Choice

Material

The core material is covered in Lecture Notes by Stephen Simpson, available online.

Supplementary reading:

A classic introduction to mathematical logic is Mathematical Logic, by Joseph Shoenfield, Association of Symbolic Logic, A K Peters/CRC Press, 2001.

If you would like to read more about quantifier elimination in algebraic structures, Model Theory - An Introduction by David Marker, Springer, 2002 is an excellent source (Chapter 3).
(An online version of the book is available for Penn State members (use a campus computer or the VPN server) at dx.doi.org/10.1007/b98860)

For set theory, T. Jech, Set Theory (dx.doi.org/10.1007/3-540-44761-X) is a comprehensive reference.

Exams

There will be no final or midterm exam in the traditional form. However, at the end of the semester, each participant has to give a short presentation (~ 30 min) on a topic that extends or elaborates one of the topics covered in class. Each student is also expected to complete a short (5-10 pages) survey paper about the same topic. The topics will be assigned during the first half of the semester.

Homework

Homework will be assigned each week and will be due the following week in class. Homework will be graded and the lowest score will be dropped.

Course Grade

The final grade will be determined as follows: 75% homework, 25% project.

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.)