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.