COURSE ANNOUNCEMENT COURSE: Math 565, Foundations of Mathematics II (Fall 1999). INSTRUCTOR: Stephen G. Simpson (simpson@math.psu.edu). TIME: Hours will be set for the convenience of the participants. To register for this course, it may be helpful to use the schedule number: 561048. TEXTBOOK: Stephen G. Simpson, Subsystems of Second Order Arithmetic, Springer-Verlag, 1998. Copies of the book are available for students to borrow. DESCRIPTION: This is a course on Reverse Mathematics. The goal of Reverse Mathematics is to classify specific mathematical theorems according to which set existence axioms are needed to prove them. The theorems that we consider are from areas such as elementary real analysis, countable algebra, countable combinatorics, and separable Banach spaces. It turns out that the power of the Zermelo-Fraenkel axioms is excessive. Instead, we use set existence axioms formulated in the language of Second Order Arithmetic. Very often it turns out that, if a given theorem is proved from the right set existence axiom, then the axiom is actually equivalent to the theorem. For example, the Arithmetical Comprehension Axiom is equivalent to the Bolzano-Weierstrass Theorem. TOPICS: 1. Subsystems of Second Order Arithmetic. 2. Recursive Comprehension, Arithmetical Comprehension, Weak Koenig's Lemma. Development of analysis, algebra, and combinatorics within the formal systems RCA_0, ACA_0, and WKL_0. 3. Reversals for ACA_0 and WKL_0. 4. Stronger systems: ATR_0, Pi^1_1-CA_0. Combinatorics and descriptive set theory in these systems. Reversals. 5. Models of subsystems of Second Order Arithmetic: beta-models, omega-models, non-omega-models. 6. Conservation theorems. Philosophical significance of these results.