In 2004, Torkel Franzén wrote Inexhaustibility: A Non-Exhaustive Treatment, a study of Gödel’s incompleteness theorems written for logicians. In this book, he gives a self-contained introduction to Gödel’s incompleteness theorems for non-mathematicians.
The first three chapters give a brief introduction to the life and work of Gödel, a not-at-all brief introduction (which takes up almost one third of the book) to the incompleteness theorems (along with the philosophical issues that they raise for mathematicians), and the notion of computability and undecidable sets. The remainder of the book then deals with the limitations of the incompleteness theorems: what they say and imply and — more importantly — what they do not say or imply.
Just as physicists have seen the Heisenberg Uncertainty Principle abused in countless non-technical situations, so mathematicians have found Gödel’s theorems misused in contexts (such as theological or legal) which have absolutely no formal systems in play. The main purpose of this book is “to set out the content, scope, and limits of the incompleteness theorem in such a way as to allow a reader with no knowledge of formal logic to form a sober and soundly based opinion of … various arguments and reflections invoking the theorem.” Franzén gives many examples of misconceptions found in Internet discussions or post-modern criticisms.
There are two audiences which may find this book of great use: the beginning mathematics student who is new to Gödel’s work, and the non-mathematician who wants to know what Gödel’s work really means. There are many who fall into the second category, especially people who abuse the incompleteness theorems. Unfortunately, while these people would benefit greatly from this book, one wonders if they would actually take the time to read it. (A modest proposal: anyone who incorrectly invokes the incompleteness theorem must read this text!) In any case, the text is accessible to the non-mathematician, although the reader will have to contend with statements such as
“Specifically, [Peano Arithmetic] proves R if and only if for every n, if n is the Gödel number of a proof of R, then there is an m < n such that m is the Gödel number of a proof of not-R.”
But, if the reader is serious about understanding the scope and limitations of Gödel’s theorems, this book will serve them well.
Donald L. Vestal is Associate Professor of Mathematics at Missouri Western State University. His interests include number theory, combinatorics, and a deep admiration for the crime-fighting efforts of the Aqua Teen Hunger Force. He can be reached at firstname.lastname@example.org.