Math 39: Introduction to Mathematical Logic

Course Description

Mathematics depends on the notion of "mathematical proof" but what does that mean? In this class we will give a rigorous definition of what a "proof" is and we will learn what it means when we say that a theorem is "true". We will learn exactly when and why things we prove are true and exactly when and why we can find proofs for true statements. We will learn what a "paradox" is and how the curious phenomenon of self-referentiality led to Godel's famous Incompleteness Theorems.
This is not an honors level class although some of the material we will be treating is sophisticated. The prerequisites for this course (which will be useful) are calculus, familiarity with vector spaces, and light experience with writing mathematical proofs. Exceptions can be made but only after talking with the instructor. Students interested in taking the honors version of this course (being offered next school year) should also consult the instructor.

Textbook

Friendly Introduction to Mathematical Logic by Christopher C. Leary Published by Prentice Hall ISBN: 0-13-010705-0

The textbook will be available at Wheelock Books. I suggest that you search online for the best price.
The publisher gives the following description of the book:
"With the idea that mathematical logic is absolutely central to mathematics, this tightly focused, elementary text discusses concepts that are used by mathematicians in every branch of the subjectÑa subject with increasing applications and intrinsic interest. It features an inviting writing style and a mathematical approach with precise statements of theorems and correct proofs. Students are introduced to the main results of mathematical logicÑresults that are central to the understanding of mathematics as a whole."

Homework will be assigned daily and collected every Monday at the beginning of class. Assignments can be found on the "homework" page of the course website. Late assignments will receive no credit although I will still grade your work if you desire. Assignments will awarded a grade between 1 and 5 points each. The rubric I will use for grading the assignments is roughly as follows:

5 points: The assignment is written up clearly, thoroughly and correctly. Every problem has a correct and complete solution, showing all necessary details and addressing all subtleties.

4 points: Every problem has been attempted. Most of the solutions are correct, with at most very minor errors or omissions. The explanations are all reasonably clear and complete.

3 points: Every problem has been attempted. Most of the solutions are correct although the student may be missing some vital parts of certain arguments or the arguments may not be clearly explained.

2 points: One or two problems have not been attempted or the work for these problems is cursory. The solutions may be only partially correct and the explanations are poorly presented, or the solution may be a strong attempt at a problem which is entirely wrong.

1 point: There appears to be some reasonable attempt at doing the assignment. Several problems have not been attempted or the work for these problems is cursory. The solutions may be only partially correct and the explanations are poorly presented, or the solution includes nonsensical ideas.


Since this is (possibly your first) proof-based course, intense effort will be made to improve your exposition and presentation of proofs. For every assignment you turn in, I expect that:
You use full sentences with punctuation;
Your handwriting is legible or the assignment is typed up;
Your name appears at the top right corner of the front page;
The assignment number is written on the top of the front page; and
The pages are stapled together (not dog-eared) and the paper is not frayed or wrinkled.

In other words, I want you to take pride in the work you hand in. Points may be deducted for any violations of these standards. I encourage you to consult the departmental writing editor, Jane Whittington, for guidance on writing-up assignments.

Group Work

Some days of lecture will include some time for group work. I will pass out a worksheet or post problems on the board to be worked on during class. The solutions will occasionally (but not always) be collected. You will get points for every group assignment based not on whether you finished or got all the right answers, but rather that everyone in your group worked together and understood every argument that was written down. I will post solutions to every group assignment on the group work webpage for those who are interested in seeing the solutions.

Examinations

There will be one midterm exam and a final in this course.

Midterm exam:
    Friday, October 28, in class.

Final Exam:
    Saturday, December 4, location TBA.

I will be giving a pop quiz one day in class. It will be on a day when you least expect it. The quiz will be simple but if you are not in lecture that day, you will not be allowed to make it up unless you have given me 24 HOUR NOTICE VIA BLITZ that you will not be in class that day. The pop-quiz will be worth 5 bonus points!!! I promise that you won't be able to predict which day of lecture it will occur.

Grades

The grades in this course will be calculated as follows:

I am fond of bonus points. There might be bonus homework problems, bonus questions on exams, perhaps a bonus writing assignment, and class participation may serve as a bonus in the case of borderline final grades. One day in lecture there will be a pop-quiz worth 5 bonus points. I promise that you won't be able to predict which day of lecture it will occur. See the pop quiz section for more details.

Collaboration on homework is highly encouraged; that is, it's a great idea to talk about the problems with each other and try to solve them together. However, you must write up homework solutions independently and in your own words. If you consult any person or source other than the course textbook, your class notes, or myself, you must acknowledge the source in your homework write-up. Failure to do so is an act of plagiarism.

For graded work other than homework, I will give specific details about what types of assistance (notes, books, friends, etc.) are acceptable. Please talk to me immediately if you have any questions about whether or not a particular form of assistance is acceptable.

Students with disabilities who will be taking this course and may need disability-related classroom accommodations are encouraged to make an appointment to see me as soon as possible. I will do my best to accommodate any reasonable requests. I also recommend stopping by the Academic Skills Center in Collis Center to register for support services.