Math 69
Logic
Last updated May 31, 2008 12:24:35 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

 Groszek MWF 1:45 - 2:50 (x-hour) Thu 1:00 - 1:50 202 Moore Hall

Instructor

 Professor Marcia Groszek Office: 104 Choate House Office Hours: Monday and Thursday 10-12, and by appointment.

Exams

Exams in this class will all be take-home.

Tentatively, there will be two midterms, one due on January 31 and one due on February 21, and a final. Midterms will be assigned about 5 days before they are due, and will include only material covered up to the point when they are assigned.

For the final exam, you will apply some of the theorems and methods of the course to a specific mathematical structure, and write up your conclusions in the form of a short paper. More details, including the specific topic of your paper and specific questions it should answer, will be available later. You will have approximately the last two weeks of class, during which no other written homework will be assigned, to work on your paper.

Here are some resources that may be helpful for writing your final paper (and, in some cases, for writing proofs for homework): The mathematics department writing specialist, is Ms. Jane Whittington, and her web page is here . The booklet Sources sets out guildelines for citing and acknowledging sources. The Student Center for Research, Writing and Information Technology in Baker Library provides a number of resources, including peer tutors. Here is a guide to writing in mathematics courses; the intended audience is first term calculus students who are writing rather short papers describing the solutions to calculus problems. Here is a paper about writing mathematics; the intended audience is undergraduate mathematics majors writing serious mathematics papers.

Homework Policy

Homework will be assigned each class day, and wil include both discussion questions and a written homework assignment.

Be prepared at each class to discuss the discussion questions assigned the previous class, or to present solutions to the class.

Written homework will generally be due on Mondays. Late (written) homework will receive partiol credit; missing homework will count as zero.

Your course grade will be based on class participation, written homework, and exams. The three major grade components are written homework, the midterm exams, and the final paper. They will be weighted approximately equally. 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+.)

If your grades on the three major grade components are significantly different from each other, the following policies may be relevant: You are guaranteed at least a grade of one full letter grade below the average of your highest two of the three major components, suitably modified to take class participation into account. Nobody who gets a grade of at least C- on the final paper will fail the course; nobody who gets a grade of at least B- on the final paper will get below a C- in the course.

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 particpate 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 both on content and on clarity of writing. Writing clear and understandable proofs is critical in all areas of mathematics, but particularly called for 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. For take-home exams, unless otherwise specified, the only sources you may consult are your textbook, your class notes and the instructor.

Disabilities

Students with disabilities who will be taking this course and may need disability-related classroom 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.

Marcia J. Groszek
Last updated May 31, 2008 12:24:35 EDT