P506 | 25674 | McCarty

Our primary task will be a close and thorough study of the First Incompleteness Theorem of Gödel - not only as metamathematical result but also as contribution to philosophy. Besides getting a clear understanding of the theorem itself and exploring a number of different ways to prove it, we will have an opportunity to look briefly into a variety of logical areas adjacent to Gödel's Theorem, including simple computability theory, definability in formal arithmetic, arguments by diagonalization, and nonstandard models of arithmetic. In an attempt to appreciate the historical and philosophical significance of Gödel's ideas, we will also read and discuss original articles and essays by Gödel, Turing, Hilbert and Wittgenstein, as well as contemporary philosophical discussions of the incompleteness phenomena 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 is supplemented by a coursepack, and readings from source materials in the relevant areas of logic, history, and philosophy. Written exercises will be assigned on a regular basis, and quizzes may be administered. Besides exercises and quizzes, students will be required to complete a midterm and a final examination, and to give a classroom presentation. Students will be expected to have attained, prior to attempting P506, a firm grasp of the techniques of basic mathematical logic equivalent to that covered by a strict and thorough graduate-level introduction to metamathematics. Among other things, it will be assumed that students are well acquainted with proofs of the completeness and compactness theorems for propositional and first- order predicate logics, and with proofs of the Löwenheim-Skolem Theorem.