A Problem Course in Mathematical Logic

Author : Stefan Bilaniuk, Department of Mathematics, Trent University
Publication Date : September 2003
Free License : GNU Free Documentation License, Version 1.2

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Book Excerpts:

This book is a free text intended to be the basis for a problem-oriented course(s) in mathematical logic and computability for students with some degree of mathematical sophistication. Parts I and II cover the basics of propositional and first-order logic respectively, Part III covers the basics of computability using Turing machines and recursive functions, and Part IV covers Godel's Incompleteness Theorems. They can be used in various ways for courses of various lengths and mixes of material. The author typically uses Parts I and II for a one-term course on mathematical logic, Part III for a one-term course on computability, and/or much of Part III together with Part IV for a one-term course on computability and incompleteness.

In keeping with the modified Moore-method, this book supplies definitions, problems, and statements of results, along with some explanations, examples, and hints. The intent is for the students, individually or in groups, to learn the material by solving the problems and proving the results for themselves. Besides constructive criticism, it will probably be necessary for the instructor to supply further hints or direct the students to other sources from time to time. Just how this text is used will, of course, depend on the instructor and students in question. However, it is probably not appropriate for a conventional lecture-based course nor for a really large class.

The material presented in this text is somewhat stripped-down. Various concepts and topics that are often covered in introductory mathematical logic and computability courses are given very short shrift or omitted entirely. Instructors might consider having students do projects on additional material if they wish to to cover it.

Prerequisites:

The material in this text is largely self-contained, though some knowledge of (very basic) set theory and elementary number theory is assumed at several points. A few problems and examples draw on concepts from other parts of mathematics; students who are not already familiar with these should consult texts in the appropriate subjects for the necessary definitions. What is really needed to get anywhere with all of the material developed here is competence in handling abstraction and proofs, including proofs by induction. The experience provided by a rigorous introductory course in abstract algebra, analysis, or discrete mathematics ought to be sufficient.

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