Math 90-10; CRN 5467
3 Credits
TR 9:20-10:35 
Professor Lee Stanley

Knights, Knaves and Gödel

Gödel’s Theorems about the Incompleteness and Undecidability of various formal axiomatic systems for certain branches of mathematics are widely viewed as one of the milestones of intellectual history : an illustration of their impact is that Einstein, Gödel and Alan Turing (whose work in mathematics was based on that of Gödel and helped to lay the foundations for digital computing), alone among mathematical scientists appeared in Time™ Magazine’s “Men of the Century in Science” issue in 1999.  The theorems have their roots in the ancient Greek “liar paradoxes” (what can be said about the statement “this statement is false”?),  and, along with the 19th century discovery of non-Euclidean  geometries, inalterably changed  the understanding of mathematical truth.  Controversial interpretations of the theorems have even been offered in support of intellectual relativism, post-modernism and against the possibility of so-called strong versions of artificial intelligence.

The main goal of the course is to develop an understanding of what the theorems do (and do not) say.  Following a sequence of logic puzzles, ingeniously crafted by the renowned magician,  juggler, Taoist, chess-problemist and mathematical logician, Raymond Smullyan, we will progress rapidly and imperceptibly from recreational problems about liars (knaves) and truth-tellers (knights) to the point of coming to grips with the deepest issues about the very nature of mathematics which are embodied in Gödel’s theorems.

In order to succeed in this course, a student should have some mathematical aptitude and a strong interest in mathematics for its own sake, but intellectual curiosity and the willingness and capacity to deal with abstraction are more important than familiarity with any particular body of material.

Course materials

The course will be built around two books by Raymond Smullyan:  Forever Undecided  and The Lady or the Tiger.  Depending on the progress made, late in the semester, these materials may also be supplemented by some more technical handouts about Gödel’s theorems.

Required assignments

Attendance at lectures is required.  In addition to lecture, the coursework will consist primarily of the following components: Weekly Homework Assignments (starting with Week 2): In the early weeks of the course, the assignments will consist mainly of writing up solutions to the logic puzzles appearing in The Lady or the Tiger.  As the semester progresses, the logic puzzles will be supplemented by more technical material related to Gödel’s theorems and other related topics in mathematical logic.  The homework assignments will count for 40% of the final grade. One Midterm, given in class, sometime between weeks 7 and 9, depending on progress.  This will count for 10% of the final grade. The final Paper/Project will be a group effort, involving a group of 3 – 4 students, and will be devoted to a more indepth development of a topic related to Gödel’s Theorem.  The topics will chosen by Week 7 in consultation with Professor Stanley. Presentations of the final papers/projects will be given in class in the final two weeks of the course.  The final paper/project will count for 30% of the final grade.