Philo. 106b Math Logic
Professor Alan Berger
e-mail: berger@brandeis.edu
Office Hours: TBA (On Wednesday late afternoon
and/or and by appt.)
Attention: If you are a student with a documented disability on record
at Brandeis University and wish to have a reasonable accommodation made
for you in this class, please see me immediately. Any case of dishonesty
is a serious academic infraction and is subject to disciplinary action.
This includes cheating on test, using other materials (includes
Internet) without citing the source.
Philosophy 106b (designed primarily for students
who have some degree of mathematical
sophistication)
Grading
You will be graded on the basis of one in class
midterm exam, and one final exam during final exam
week.
Texts
There will be two texts, the one that we will use
up until the midterm is Mendelsohn's Introduction
to Mathematical Logic. Since the first part of the
book does not vary from one version of the book to
any other version of the book, you may purchase any
used version of any volume. For the second half of
the course, you will have to purchase my notes and
bring them to class regularly.
The aim and Purpose of this Course
The central aim of this course is to develop a firm
grip on what a formal system is, and to study and
prove some of the most famous and important
theorems in mathematical logic, i.e., theorems
about any arbitrary and rich enough system. In
particular, some typical theorems that we will
prove are Godel's two incompleteness Theorems,
Tarski's Theory of Truth, a few paradoxes,
especially concerning the Lowenheim-Skolem Theorem.
We will also discuss some of the philosophical
implications of Godel's Theorems, especially with
respect to the Hilbert Program. Along the way we
will learn notions such as recursive and primitive
recursive functions and other machinery that Godel
had to invent in order to prove his two famous
incompleteness theorems. We will also have to
study Robinson Arithmetic, and we will prove
Church's Theorem as well as discuss Church's
Thesis.
Specific Topics and Approximate Schedule
The first half of the course will consist in
setting up a very elementary formal system,
Propositional Logic, and prove such theorems such
as that all truth functional expressions are
expressible in this system, that there can be one
truth functioal connective from which all other
truth functional connectives can be defined, and
that Propositional logic is complete. We will then
introduce the semantics for our next formal system,
Predicate logic. This will bring us up to the
first half of the course. In addition to having to
do exercises, we will be be using any version of
Elliot Mendelsohn's classic textbook, Introduction
to Mathematical Logic.
For the second half of the course, we will learn
all the important material that you will be
responsible for knowing in detail on the final. We
will learn Robinson Arithmetic, Arithmetization,
Godel numbering, diagonalization, recursive, and
primitive recursive functions, and the two Godel
Incompleteness Theorems. We will be using my
personal notes to cover this material. YOu will be
required to bring my notes to class.
Only For Graduate Students, Advanced or Interested
Students wanting special Course Credit
In addition to attending the regular Mathematical
Logic class, graduate students and advanced or very
interested honors students will meet with me one
hour a week to discuss further details of Godel's
two incompleteness theorems. We will also go over
in more detail, Tarski's Theory of Truth and the
"paradox" concerning the Lowenheim-Skolem Theorem.
We will also look at the philosophical implications
of the Godel Theorems on the Hilbert Program. All
of these topics depend upon whether time permits.