\def\RCAo{\hbox{\rm RCA}_0}
\def\WKLo{\hbox{\rm WKL}_0}
\def\ACAo{\hbox{\rm ACA}_0}
\def\ATRo{\hbox{\rm ATR}_0}
\def\CAo{\hbox{\rm CA}_0}
\magnification=\magstep2
\nopagenumbers
\noindent
COURSE ANNOUNCEMENT
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COURSE: Math 565, Foundations of Mathematics II (Fall 1999).
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INSTRUCTOR: Stephen G. Simpson ({\tt simpson@math.psu.edu}).
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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.
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TEXTBOOK: Stephen G. Simpson, {\it Subsystems of
Second Order Arithmetic}, Springer-Verlag, 1998. Copies of the book
are available for students to borrow.
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\noindent
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.
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TOPICS:
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1. Subsystems of Second Order Arithmetic.
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2. Recursive Comprehension, Arithmetical
Comprehension, Weak K\"onig's Lemma.
Development of analysis, algebra, and
combinatorics within the formal systems $\RCAo$, $\ACAo$, and $\WKLo$.
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3. Reversals for $\ACAo$ and $\WKLo$.
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4. Stronger systems: $\ATRo$, $\Pi^1_1$-$\CAo$.
Combinatorics and descriptive set theory in these systems. Reversals.
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5. Models of subsystems of Second Order
Arithmetic: $\beta$-models, $\omega$-models, non-$\omega$-models.
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6. Conservation theorems. Philosophical
significance of these results.
\bye