Archive for the ‘Syllabus’ Category

116b- Syllabus

January 15, 2008

Math 116 provides an introduction to the basic concepts and results of mathematical logic and set theory. Math 116B will be devoted to computability theory and the incompleteness theorems.

(Additional topics may include the behavior of countable models and Hilbert’s 10th problem.)

Our approach to incompleteness will be somewhat non-standard and will allow us to discuss subsystems of second-order arithmetic.

Grading Policy: The grade for this course will be based on homework assignments. There will be no exams.

Solutions to homework problems should be written individually, although collaboration is allowed. All references used to solve a problem should be explicitly mentioned, including those students you collaborated with. You cannot look up solutions from any source.

No late submissions of solutions are allowed, except for medical problems (note needed from the health center) or serious personal difficulties (note needed from the Dean’s office).

Please try to solve as many problems as it seems reasonable from each set. Let me know if you find some problems to be too hard or too easy or to contain mistakes. Feedback is greatly appreciated.

Textbook: There is no required textbook. The following suggested references may be useful:

R. Cori and D. Lascar. Mathematical logic. A course with Exercises (Part II), OUP, 2001, ISBN 0198500505

T. Franzén. Inexhaustibility, ASL (A K Peters), ISBN 0198500505 J. Shoenfield. Mathematical logic, ASL (A K Peters), ISBN 1568811352S. Simpson. Subsystems of second order arithmetic, Springer, ISBN 3540648828

117b – References

December 31, 2006

There won’t be a required textbook. In lecture, I will mention references according to the topics being discussed. General references that may be useful are:

The incompleteness phenomenon, by M. Goldstern and H. Judah. AK Peters (1998), 1-56881-093-8

Hilbert’s tenth problem, by Y. Matiyasevich. MIT Press (1993), 0-262-13295-8 or 978-0-262-13295-4

Mathematical logic, by J. Shoenfield. ASL (2001), 1-56881-149-7

117b – Grading Policy

December 31, 2006

Grading will be based on Homework assignments. There will be no exams.

Solutions to homework problems should be written individually, although collaboration is allowed. All references used to solve a problem should be explicitly mentioned, including those students you collaborated with. However, you cannot look up solutions from any source (including other students, earlier years’ solution sets, books, etc).

No late submissions of solutions are allowed, except for medical problems (note needed from the health center) or serious personal difficulties (note needed from the Dean’s office).

Please, and this is very important, make sure that what you turn in is your final solution, as opposed to a draft or your scratch work. Be neat and professional about the appearance of your work. However, if you cannot find a complete solution to a problem, turn in a partial solution, indicating what is missing. Again, this does not mean that you can turn in your scratch work for problems you only have partially completed, but make clear at the beginning of your solution that you did not finish the problem.  

117b – About this course

December 31, 2006

We continue the introduction to the basic mathematical theory of computation. The term will roughly cover three topics.

First, we will introduce the notion of relative computability, use this to define the structure D of Turing degrees, and proceed to prove a few structural results about D.

Second, we will study the connection between undecidability, mathematical logic, and number theory;  in particular, we will provide proofs of Gödel’s incompleteness theorems and of the undecidability of Hilbert’s tenth problem.

Finally, we will discuss examples of undecidability arising in diverse areas of mathematics and computer science.