Math 39
Logic
Last updated May 31, 2008 12:24:23 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.

We will see how the Completeness Theorem can be applied to develop a logically valid way of doing calculus with infinitesimals, or infinitely small numbers. (So you really can compute a derivative by dividing an infinitely small dy by an infinitely small dx.) We will also briefly discuss Godel's Incopleteness Theorem, which tells us that it is impossible (mathematically impossible) to write down a complete system of axioms for the natural numbers: Any valid system of axioms we come up with is necessarily incomplete in that there are facts about the natural numbers that are true but cannot be proved from those axioms.

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 8:45 - 9:50 (x-hour) Thu 9:00 - 9:50 103 Bradley Hall

Instructor

 Professor Marcia Groszek Office: 104 Choate House Office Hours: Monday and Thursday 10:30-12:30, and by appointment.

Exams

 Exam 1 In class, Wednesday, October 6. Exam 2 In class, Friday, October 29. Exam 3 In class, Monday, November 15. Final Exam Sunday, December 5. 11:30am-2:30pm, room to be announced.

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 particpation (10%), written homework (30%), in-class exams (30% total) and a final exam (30%).

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.

Written homework 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.

In-class exams will not require lengthy proofs and will be graded on content.

The final exam will have a take-home portion, which, like written homework, will be graded both on content and on clarity of writing.

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 the take-home portion of the final exam, 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 12:24:23 EDT