Philosophy | Logical Theory II
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.