Math 69
Honors Mathematical Logic
Last updated August 02, 2011 13:48:53 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.

Prerequisite for this course: Experience with mathematical structures and proofs, as offered by such courses as Mathematics 71, 54, or 24. If you are unsure about your preparation, please talk to Professor Groszek.

Math 69 satisfies the culminating experience requirement for mathematics majors.

Math 69 is appropriate for any graduate student who wants to take a course in logic.

Textbook

A Mathematical Introduction to Logic by Herbert Enderton, 2nd edition
(Available at Wheelock Books or the Dartmouth Bookstore).

Scheduled Lectures

 Groszek MWF 11:15 - 12:20 (x-hour) Tu 12:00 - 12:50 028 Haldeman Center.

Instructor

 Professor Marcia Groszek Geoff Goehle Office: 330 Kemeny Hall Office: 212 Kemeny Hall Office Hours: Tu 1:00-2:30, Th 10:00-11:30, and by appointment. Office Hours: TBA, and by appointment. Phone: 646 - 2313 or BlitzMail (preferred) Phone: 646 - 9808 or BlitzMail (preferred)

Exams

There will be two midterm exams, both take-home. Most likely one will be during the fourth week of class and one during the seventh. There will be a final project or paper rather than a final exam.

Here are some resources that may be helpful for writing mathematics papers (and, in some cases, for writing proofs for homework): 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 from the previous class and/or to present their solutions.

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

Your course grade will be based on class participation, written homework, exams and final project. 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.

Homework, midterm exams (counted as a single unit), and the final project will be weighted approximately equally. Class participation will be decisive in borderline cases, and may raise or lower your grade by a single step (B to B+, not B to A).

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 August 02, 2011 13:48:53 EDT