Class:  Math 571  Spring 2002
        Introduction to Mathematical Logic
        MWF 9:55-10:45  B105 Van Vleck

Instructor:  A. Miller
 403 Van Vleck Hall 262-2925
 office hours:   MWF  1:00-2:00
 miller@math.wisc.edu
 www.math.wisc.edu/~miller/m571     All class handouts will be here.

Prerequisite: M340 or consent of instructor.
However my experience is that students who have not taken 441 or 541
(abstract algebra - groups, rings, fields) never have a clue what this
course is about.  See Q & A below.

Text: Enderton, Herbert B.,  A MATHEMATICAL INTRODUCTION TO LOGIC,  2nd ed.
Harcourt/Academic Press.

Topics: The syntax and semantics of propositional and predicate logic.
Formalization of the notions of first-order formula, proof, and model. The
main goal of this course is to completely understand the following theorems
and their proofs.

 Compactness Theorem: Given any set T of sentences, if every finite subset
 of T has a model, then T has a model.

 Completeness Theorem: Given T and a sentence A, then if every model of T is
 a model of A, then T proves A.

 Incompleteness Theorem: For any true theory T which is strong enough to do
 some of elementary arithmetic and whose axioms can be effectively listed,
 there is a true sentence A of arithmetic such that T does not prove A.

Grading:  There will be one midterm in the evening and the Final exam at the
regular time.  There will be a daily assignment of 3 or 4 problems. Homework
solutions are due in lecture exactly one week from when they are assigned. 
Please put them on the desk in front of the room at the beginning or end of
class.

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Q. I'm presently taking M441 and have found that I really need to focus on
proof skills.  Proof has become a vital part of school mathematics and as
you may know is often neglected. As someone who taught secondary mathematics
for six years, I recognize this need and appreciate the need for teachers
such as myself to understand proof.  Is 571 the right course for me?

A. I think you are mistaken when you think that M571 will teach you how to do
proofs in mathematics.  Mathematical logic is concerned with the abstract
theory of proofs - e.g. we prove theorems about the formal notion of
"proof". I recommend that you take M541, M551, or M521 before you take M571,
since these courses are less abstract.  Another course, M371 is specifically
designed to teach students how to do proofs in mathematics.