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\textbf{Syllabus for Math.\ 609}
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Math.\ 609-610 is divided into three parts, model theory, computability
theory, and set theory. For Fall, 2002, model theory came first, followed
by the first part of computability. In model theory, the usual material
includes some propositional logic, and then focuses on predicate
logic. There are the basic Completeness, Compactness, and
L\"{o}wenheim-Skolem-Tarski Theorems. There are examples of theories with
different numbers of countable models, including the Ehrenfeucht examples
and something involving quantifier elimination. We do the Omitting Types
Theorem, and the Ryll-Nardjewski Theorem. There is a discussion of
prime, atomic, and saturated models. The final goal is the theorem of
Vaught saying that there is no countable complete theory with exactly two
countable models, up to isomorphism.
The first part of computability includes Turing machines, primitive
recursive functions, partial computable functions, Kleene normal form,
and a sketch of the proof of equivalence of Turing computable and
partial computable functions. We discuss computable and computably
enumerable sets. There are some interesting examples such as Ackermann's
function, the busy-beaver function, and the halting set. There is a
discussion of many-one reducibility and one-one reducibility, the Myhill
Isomorphism Theorem, and productive and creative sets. Adding the
existence of a simple set, we get the fact that among c.e.\
sets, there is one which is neither computable nor $1$-complete. We
describe oracle machines and relative computability, Turing reducibility,
Turing degrees, the jump function on sets, and on degrees, and the
Kleene-Post Theorem.
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For Fall, 2002, there was only one student in Math.\ 609. Because
this student was already familiar with some of the material, it was
possible to cover the basic material quickly, and add supplementary
material. Thus, the course included Morley's Categoricity Theorem, plus
some notes of Marker on model theory of fields. Instead of a mid-term,
the student (by his own choice) gave several lectures on Marker's notes.
There was regular homework. The final will be of the usual
kind---written, covering just the basic material---but with
problems that are more challenging than usual.
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