P506 | 3249 | McCarty

Our primary task will be a thorough study of the First Incompleteness Theorem of Godel – not only as metamathematical result but also as a contribution to mathematics and a problem for philosophy. Besides getting a clear understanding of the theorem itself and exploring different ways to prove it, we will have an opportunity to look briefly into a variety of logical areas adjacent to Godel's theorem, especially simple computability theory, definability in formal arithmetic, arguments by diagonalization and (time permitting) nonstandard models. In an attempt to gauge the historical and philosophical significance of Godel's ideas, students will read and lead discussions of original articles and essays by Godel, Turing, Hilbert, and Wittgenstein and contemporary philosophical articles by such scholars as Dummett and Feferman. Our main textbook will be the paperback volume Computability and Logic by George Boolos and Richard Jeffrey. This will be supplemented by a set of photo-copied course notes and by readings from source materials in logic. Written homework will be assigned on a regular basis. Besides homework, students will be required to complete one or more midterms and a final examination and to give a classroom presentation on an article of historical or philosophical significance to the topics of the course. It is recommended that students attain a grasp of the techniques of mathematical logic equivalent to that represented by P505 prior to taking P506. In particular, it will be assumed that students are familiar with proofs of the completeness and compactness theorems for first-order predicate logic.