Math 69
Logic
Last updated June 25, 2009 14:49:00 EDT

General Information

Mathematical logic, in the sense of this course, applies mathematical tools to model and study the activity of working mathematicians.

Working mathematicians prove theorems (among other things.) We will study formal deductions in a formal language, which are proofs formulated in a precise framework, so precise that they can be studied mathematically: We can prove things about them. Our results about formal deductions can illuminate the proofs found in mathematicians' research papers and in mathematics students' homework papers.

We will begin the course with the study of the language of sentential logic, a formal language that is not powerful enough to model the language used by working mathematicians. Building on this, we will study the more powerful language of first order logic, and the notion of formal deduction in first order logic. We will prove Godel's Completeness Theorem, a major result, which states that this notion of formal deduction completely captures the power of mathematical proof in the following sense: Given any axiom system (for example, the axioms for a real vector space) and any statement in our language, either that statement can be proved from the axioms by means of a formal deduction, or else we cannot hope to prove the statement because there is a context (in our example, a real vector space) in which the axioms are true but the statement is false.

In this, the honors version of the course, we will also see the proof of Godel's Incompleteness Theorem. The Incompleteness Theorem is one of the most important mathematical results of the twentieth century, with important practical and philosophical consequences for mathematics.

An important philosophical consideration through our study will be the notion of "effective" or "algorithmic" process. For example, Godel's Incompleteness Theorem will tell us that there is no algorithm to list all the true statements about the natural numbers. In other words, it is impossible to program a computer to answer all possible questions (even in our restricted formal language) about the natural numbers.

Textbook

A Mathematical Introduction to Logic by Herbert B. Enderton, second edition
(Available at Wheelock Books)

Scheduled Lectures

 Winkler MWF 11:15 - 12:20 (x-hour) Tue 12:00 - 12:50 028 Haldeman Center

Instructor

 Professor Pete Winkler Office: 231 Kemeny Hall Office Hours: Monday 3-4, Wednesday 10-11, Friday 2-3, and by appointment.

Exams

There will be one in-class hour exam (Friday Feb 2), and a final exam in the scheduled slot for MWF 11:15 courses (8:00 am Saturday, March 10, venue to be announced). If for some reason you anticipate having to miss an exam, please let me know now.

Homework Policy

Homework will be assigned each class day, with the written part due at the beginning of the next meeting of the class. Part or even all of a given assignment might be questions to just think about prior to the next class. Written assignments will be graded or checked off; late assignments will be checked off only.

Your course grade will be based on class participation, written homework, and exams. The three major grade components are written homework, the midterm exam, and the final exam. The final will be about twice as long as the midterm, will cover all the material, and will count twice as much as the midterm; the midterm will count somewhat more than the written assignments. Class participation will decide borderline cases, and may raise or lower your grade by a single step (for example, from a B to a B+.)

Class participation grades will be based on the following: Are you present in class? Are you prepared? When the class divides into small groups, do you fully participate in your group's work? Do you contribute to class discussions, or present homework solutions to the class, when you are asked to? Class particpation grades are not based on whether you talk a lot or say brilliant things, or even on whether your answers to questions are correct.

Written homework and exams will be graded on clarity of writing as well as content. Writing clear and understandable proofs is important in all areas of mathematics, but particularly critical in the study of mathematical logic.

The Honor Principle

Academic integrity and intellectual honesty are an integral part of academic practice. This does not mean that you can't work on homework together or get ideas and help from other people. It does mean that you can't present somebody else's work or ideas without giving them due credit.

Feel free to discuss homework problems with other people and to work together to answer them. You must write up the answers yourself without copying from anybody. (This means you cannot copy down a joint solution arrived at by a group working together, even if you were part of the group. You must write up the solution in your own words.) You must also acknowledge any sources your consulted or people you worked with; working with other people or consulting other sources will not lower your grade.

Of course, no help may be given or received on exams.

Disabilities

Students with disabilities who will be taking this course and may need disability-related classroom or exam accommodations are encouraged to make an appointment to see the instructor as soon as possible. Also, they should stop by the Academic Skills Center in Collis Center to register for support services.

Pete Winkler
Last updated June 25, 2009 14:49:00 EDT