21-600 Mathematical Logic I
Instructor: Peter Andrews
MWF 11:30am-12:20pm
HH B103
12 Units
Description: The study of formal logical systems which model the
reasoning of mathematics, scientific disciplines, and everyday
discourse. Propositional calculus and first-order logic. Syntax,
axiomatic treatment, derived rules of inference, proof techniques,
computer-assisted formal proofs, normal forms,
consistency, independence, semantics, soundness, completeness, the
Lowenheim-Skolem and Downward Lowenheim-Skolem Theorems, compactness.
TEXTBOOK: Peter B. Andrews, An Introduction to Mathematical Logic and
Type Theory: To Truth Through Proof, Second Edition, Kluwer Academic
Publishers, now published by Springer, 2002.

21-602 Set Theory I
Instructor: Ernest Schimmerling
MW 3:30-4:50
Porter Hall A19D
12 Units
Description: First semester graduate level set theory. Godel proved that
if ZF is consistent, then so is ZFC + GCH; this is the most important
theorem that will be covered in the course. The main topics are ZFC,
cardinal arithmetic, the rank hierarchy, the H(kappa) hierarchy,
the reflection theorem, absoluteness theorems, relative consistency, 
infinitary combinatorics (diamond principles, Aronszajn and Suslin trees, 
etc.), the model HOD, the constructible universe, and descriptive set theory.
TEXTBOOK OR REFERENCES: Kenneth Kunen, "Set theory : An introduction to
independence proofs"

21-603 Model Theory I
Instructor: Rami Grossberg (rami@cmu.edu)
MWF 1:30
Room TBA
12 Units
DESCRIPTION: Model theory is one of the four major branches of
mathematical logic. There are many applications of model theory to
algebra (e.g. field theory, algebraic geometry, number theory, and group
theory), analysis (non-standard analysis, complex manifolds and the
geometry of Banach spaces) and theoretical computer science (via finite
model theory) as well as set theoretic topology and set theory. This
course is the first in a sequence. The purpose of this course to
introduce the student to some of the most important ideas of the field
with a special attention to "classification theory" which is the most
important subfield of modern model theory.  The main focus of the course
is Morely's categoricity theorem, the related conceptual infrastructure
and extensions.

Topics will include: compactness theorem, model completeness, elementary
decideability results,
Henkin's omitting types theorem, prime models. Elementary chains of
models, some basic two-cardinal theorems, saturated models
(characterization and existence), basic results on countable models
including Ryll-Nardzewski's theorem. Indiscernible sequences, and
connections with Ramsey theory, Ehrenfeucht-Mostowski models.
Introduction to stability (including the equivalence of the
order-property to instability), chain conditions in group theory
corresponding to stability/superstablity/omega-stability, strongly
minimal sets, various rank functions, primary models, and a proof of
Morley's categoricity theorem. Basic facts about infinitary languages
and abstract elementary
classes, computation of Hanf-Morley numbers.

PREREQUISITE: This is a graduate level course, while at the beginning
the pace will be slow, the pace will speed up in the second half. In the
past many students where undergraduates, so the prerequisites are kept
to the minimum of "an undergraduate-level" course in logic. Motivated
undergraduate students are encouraged to contact the instructor.
TEXT: Rami Grossberg, A course in model theory, a book in preparation.
Most of the material (and more) appears in the following books:
1. C. C. Chang and H. J. Keisler, Model Theory, North-Holland 1990.
2. Bruno Poizat, A course in Model Theory, Springer-Verlag 2000.
3. S. Shelah, Classification Theory, North-Holland 1991.
I will not use a text, but will provide access to my notes on a weekly
basis.
EVALUATION: Will be based on weekly homework assignments (20%), a 50
minutes midterm (20%) and a 3 hours in-class comprehensive final written
examination (60%).
COURSE WEBPAGE: www.math.cmu.edu/~rami/mt1.10.desc.html

80-413/713  Category Theory
Instructor: Steve Awodey
TR 1:30 - 2:50PM
PH A22
12 units
Description:
Category theory, a branch of abstract algebra, has found many
applications in mathematics, logic, and computer science. Like such
fields as elementary logic and set theory, category theory provides a
basic conceptual apparatus and a collection of formal methods useful for
addressing certain kinds of commonly occurring formal and informal
problems, particularly those involving structural and functional
considerations. This course is intended to acquaint students with these
methods, and also to encourage them to reflect on the interrelations
between category theory and the other basic formal disciplines.A text
will be provided.
Prerequisites: one course in logic or algebra.

80-414/714 COMPUTABILITY: Machines & Minds
Instructor: Wilfried Sieg
W 2:00am-4:20pm
BH 150
10-12 Units
Description:  This deeply interdisciplinary seminar is divided into
three parts. Part 1 reviews the emergence of the central concept and
arguments for Churchıs or Turingıs Thesis; it presents then, in sharp
contrast, an axiomatic characterization of serial and parallel
computability. Part 2 takes up the considerations of Gödel and Turing to
relate machines and the human mind, in particular, with respect to
mathematics. Finally, Part 3 discusses the use of computations in
cognitive psychology to model aspects
of human minds; the focus is on (Post-) production systems and
parallel-distributed processes.

21-800 Advanced topics in Logic: large cardinals and inner models I
Instructor: Ernest Schimmerling
MWF 2:30
Porter Hall 226B
12 Units
Topics: The hierarchy of large cardinal axioms; the fine structure of
the constructible universe; the beginnings of core model theory;
applications.
This is the first part of a two semester series that will eventually
reach the level of inner model theory where iteration trees play a key role.
Prerequisites: Students who have not taken 21-602 or the equivalent
would need to take both simultaneously and stay ahead in 21-602 with the
help of the instructor.  Such plans would need to be discussed before
the semester begins at the earliest possible date.

80-813 Seminar in the Philosophy of Mathematics
Instructors: Jeremy Avigad (Carnegie Mellon) and Kenneth Manders (University of
Pittsburgh)
F 10-12:30
BH 150 (at CMU) and 1001-B Cathedral of Learning (at Pitt)
12 units

Algebra and number theory in the nineteenth century

A number of questions regarding the types of equations that can be solved in the
integers and in the reals have their origins in antiquity, when mathematics was
held to be the science of quantity, both continuous (magnitude) and discrete
(number). The beginning of  the nineteenth century brought striking advances
along these lines.  For example, Gauss gave a detailed analysis of the integers
that can  be represented by a given quadratic form, and Abel and Galois showed
that the general quintic equation has no solution by radicals.

A good deal of effort in the nineteenth century was devoted to making sense of
these results, and by the end of the century the ideas had been recast in
algebraic structural terms. Galois theory and the study of quadratic forms are
now invariably presented in terms of field extensions and their properties. This
shift is prototypical of the transition to the "modern" view of mathematics.

In this seminar, we will trace the development of these ideas. We will focus, as
much as possible, on the original sources, with an eye towards obtaining a
better understanding of the methodological considerations that drove these
developments.