Metmathematics I
Math 502
Fall 2007
Course Meeting: 2:00pm MWF 208 in Stevenson Hall
(north of SEO, past the library)
Call Number: 26522
Instructor: Alice Medvedev
Office: 716 SEO
Office Hours: M 3-5, T 3-4
phone: 312-413-9578
e-mail: alice@math.uic.edu
course webpage:
http://www.math.uic.edu/~alice/math502F07
Grading
There are 12 problem sets. As you may have noticed, I accept late work. I will also accept, grade, and somewhat count new and improved solutions to old homework problems. Should you decide to take advantage of this, write them up well, and hand in both the new and the original solutions, no later than Wednesday, December 5th. You may work together on homework problems (and I encourage you to do so), but you should write your solutions yourself and acknowledge that you have worked together.
There will be a take-home final exam due Wednesday, December 12th, at noon. I will leave Chicago, so you will not be able to hand it in late. Copies will be handed out in class on Wednesday, December 5th and also posted here: Final Exam - Don't look past first page until you're ready to take it!
Homework Assignments
- Problem Set 0
- one more problem now due Wednesday, September 5th
- Problem Set 1, due Wednesday, September 12th
- Problem Set 2, due Wednesday, September 19th
- Problem Set 3, due Wednesday, September 26th
- Problem Set 4, due Wednesday, October 3rd
- Problem Set 5, due Wednesday, October 10th
- Problem Set 6, due Wednesday, October 17th
- Problem Set 7, due Wednesday, October 24th
- Problem Set 8, due Monday, November 4th
- Problem Set 9, due Monday, November 12th
- Problem Set 10, due Monday, November 19th
- Problem Set 11, due Wednesday, November 28th
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.
Prerequisites
Graduate standing. No previous background in logic is assumed. As many examples will come from Algebra, Math 516 is a useful corequisite.
Text
I intend to follow
Mathematical Logic by H.-D. Ebbinghaus, J. Flum, and W. Thomas fairly closely.
We are also using Models of Peano Arithmetic by R. Kaye.
Textbook and Content:
The textbook, Mathematical Logic by H.-D. Ebbinghaus, J. Flum, and W. Thomas, is best acquired from amazon.com, cheaper used, more expensive with complete electronic access. This should not be construed as endorsement of either amazon's electronic access arrangement, with which i've been mildly pleased so far, or the book, which i've been finding a bit more tedious and belaborious than necessary. Nevertheless, it should be an adequate reference (so that you don't have to rely on lecture notes), and it offers much more historical and philosophical motivation than i can. In particular, Chapter 1 is a great introduction, and Chapter 7 should be read through every couple weeks during this course, to keep track of what we're doing and why.
I will use the book's notation, and the book's general organization of material, with one exception: partial isomorphisms from chapter 12 will appear early and throughout the course.
The aim of this course will be Godel's Incompleteness Theorem, which needs some model theory, some proof theory, and some recursion theory. An approximate list of topics:
- Motivation: (Now is a good time to read Chapter 1.)
- Serious foundational worries: the historical origin of mathematical logic.
- What do we know about integers?
- Symbol sets and structures: logic as a useful way of thinking about mathematics.
- Formal syntax of first-order logic. (Chapter 2)
- Formal semantics of first-order logic. (Chapter 3)
- Proof theory. (Chapter 4)
- Completeness of first-order logic, through Henkin constructions.
(Chapter 5)
- Compactness of first-order logic. (Chapter 6)
- Recursion theory (Chapter 10 and Chapter 0 of Kaye)
- Peano arithmetic: coding (Chapter 10 and Chapter 3 of Kaye)
Peano arithmetic: non-standard models (Chapter 1 of Kaye)
- Godel's incompleteness theorems (Chapter 10 and Chapter 3 of Kaye)
- Model Theory of Algebraically closed fields, not on homework or exams.
Last updated 11/15/2007