Mathematical Logic

This is a Dover reprint of Stephen Cole Kleene’s classic book, published originally by John Wiley and Sons in 1967. A pdf scan of the original 1967 version is available on line at the Internet Archive. This presumably means the book is in the public domain.

Kleene (1909–1994), intended this book to be an introduction to mathematical logic at the undergraduate level, in contrast to his earlier and more celebrated Introduction to Metamathematics (1952), which was intended as a graduate text. The older book is often considered to be the better of the two. Indeed, this book contains many references to the older book and some of its parts are actually abridged versions of the older book.

The book is about 400 pages and divided into two parts. The first part introduces elementary mathematical logic, that is, propositional and predicate calculus. The second part delves into logic and the foundations of mathematics (computability, decidablity, formal number theory, Turing machines, incompleteness, completeness etc.). There are many exercises and an extensive bibliography.

Since its appearance fifty years ago there have been myriad other introduction to logic texts published, many of which might be considered more accessible to undergraduates, such as Herbert Enderton’s A Mathematical Introduction to Logic. Nevertheless I find Kleene’s book to still be quite lucid introduction that might work well as a text for honors students.

When a book enters the pantheon of historical classics is not always clear, but certainly the author’s stature as one of the great logicians of the twentieth century makes the book’s exposition interesting in and of itself.

Steven Deckelman is a professor of mathematics at the University of Wisconsin-Stout, where he has been since 1997. He received his Ph.D from the University of Wisconsin-Madison in 1994 for a thesis in several complex variables written under Patrick Ahern. Some of his interests include complex analysis, mathematical biology and the history of mathematics.