Math 69
Mathematical Logic (honors)
Last updated January 27, 2017 09:27:59 EST

## 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 (Second edition) by Herbert Enderton
(Available at Wheelock Books and elsewhere)

Scheduled Lectures

 Groszek MWF 12:50 - 1:55 (x-hour) Tu 1:20 - 2:10 Haldeman 028

We will use the x-hour only when necessary to make up missed classes, or for optional extras. We will definitely use the x-hour on January 17, to make up for the class missed on January 16, MLK day.

Instructor

 Professor Marcia Groszek Office: 330 Kemeny Hall Office Hours: Mon 2:30-4:00, Thu 1:30-3:00, and by appointment Contact via email.

Office hours are always drop in; you need not make an appointment. If you have a conflict with regularly scheduled office hours, you can make an appointment for another time.

Exams

There will be two take-home exams and a final paper.

The first exam will be distributed around January 23 and due on January 30. The second midterm will be distributed around February 13 and due on February 20. The final paper will be assigned during the last two weeks of class and due on the day reserved for our final exam, March 13. (We will not have a final exam on that day.)

A preliminary version of the final paper assignment is here. An example of an expository paper is here.

Here are some resources that may be helpful for writing mathematics papers (and, in some cases, for writing proofs for homework): Here you can find some guildelines for citing and acknowledging sources. The Institute for Writing and Rhetoric 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

Written homework will be assigned daily and due weekly. Homework assigned on a given day is due on Wednesday of the following week. Homework is due at 12:00 noon unless you bring it to class with you, in which case it is due at 12:50 PM. Homework handed in any later than this does not get full credit. If you are not going to be in class, you may hand in your homework at the instructor's office or by email, in which case it is due at noon.

Late homework receives partial credit, depending on how late it is. Generous partial credit for homework only a few minutes or hours late is intended to cover missing class due to minor illness, malfunctioning alarm clock, or other such things.

Late homework will be excused only in case of serious, unpredictable events such as documented illness or family emergency.

If you must miss a class, it is your responsibility to submit all homework on time, and to arrange to get notes from a classmate.

The course grade will be based upon the scores on the midterm exams, homework, and final paper. These three components will be weighted equally. In borderline cases, factors such as class participation, demonstration of the ability to work independently and collaboratively, or a steady record of improvement will be considered.

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 is at the core of our mission as mathematicians and educators, and we take it very seriously. We also believe in working and learning together.

Collaboration on homework is strongly encouraged.

On written homework, you are encouraged to work together, and you may get help from others, but you must write up the answers yourself. If you are part of a group of students that produces an answer to a problem, you cannot then copy that group answer. You must write up the answer individually, in your own words.

On exams, you may not give or receive help from anyone. You should discuss the exams only with the instructor, for clarification of problems. You may use your textbook, notes, homework, and any materials distributed in class. You may not use outside sources, including but not limited to other textbooks and online sources.

Special Considerations

Students with disabilities enrolled in this course and who may need disability-related academic adjustments and services are encouraged to see their professor privately as early as possible in the term. Students requiring disability-related academic adjustments and services must consult the Student Accessibility Services office (301 Collis Student Center, 646-9900, Student.Accessibility.Services at Dartmouth.edu). Once SAS has authorized services, students must show the originally signed SAS Services and Consent Form and/or a letter on SAS letterhead to their professor. As a first step, if students have questions about whether they qualify to receive academic adjustments and services, they should contact the SAS office. All inquiries and discussions will remain confidential.