# Metmathematics I

# Math 502/Phil 596

# Fall 2003

Call Number: 67590/84970 MWF 11-12 302AH

Instructor: David Marker

Office: 411 SEO

Office Hours: M 1:30-3:00, W 9:00-11:00, Th 10:30-12:00

phone: (312) 996-3069

e-mail: marker@math.uic.edu

course webpage:
http://www.math.uic.edu/~marker/math502

### Description

A first graduate course in mathematical logic.
We will introduce the fundamental themes of mathematical logic
(truth, provability, and computability), discuss their
interconnections and examine the power and limits of formal methods.
The main results will be Godel's Completeness and Incompleteness
Theorems and Tarski's decidability results for the real and complex
fields. Specific topics covered will include.
- Mathematical structures
- Formal proofs
- The Completeness Theorem
- The Compactness Theorem and elementary model theory
- Introduction to computability theory
- The Incompleteness Theorem
- Decidability results for real closed fields and algebraically closed fields

### Texts

- H.-D. Ebbinghaus, J. Flum and W. Thomas,
*Mathematical Logic* Second
Edition, Springer-Verlag, 1994.
- N. Cutland,
*Computability: An introduction to recursive function
theory*, Cambridge University Press, 1986.
- J. Shoenfield,
* Mathematical Logic*, A.K. Peters, 2001

I will ** not** be following any of these texts completely.
The treatment of material at the begining of the course on structures, truth
and formal proofs will be similar to the treatment in Ebbinghaus-Flum-Thomas.
The treatment of computability will follow Cutland. Shoenfield's *
Mathematical Logic* is the classic text in the subject. It is an excellent
reference for some of the more advanced subjects we will cover and contains
a wealth of interesting material.
### Prerequisites

Graduate standing. No previous background in logic is assumed. As many examples will come from Algebra, Math 516 is a useful.
### Grading

I will give out about 8 problem sets. You may work together on homework problems (and I encourage you to do so), but when you turn in the problem you should acknowledge that you have worked together.
There will probably be a one hour final exam testing basic concepts,
definitions, and statements of theorems.
### Lecture Notes

- Survey of Basic Set Theory
- Sections 1--5: Languages and Structures,
Theories, Formal Proofs, Goedel's Completeness Theorem, Basic Model
Theory
- Sections 6--9: Models of Computation,
Universal Machines and Undecidability, Recursively Enumerable and Arithmetic
Sets, Further Topics in Computability Theory
- VERY PRELIMINARY VERSION Section 10
Godel's Incompleteness Theorem

### Homework Assignments

Dave Marker's Home Page

Last updated 11/21/03