With this book, Raymond Smullyan once again demonstrates his command of both logic and the English language: he understands and can explain it. In general, each chapter begins with a brief explanation of the primary topic, followed by a set of problems (exercises) where the reader is expected to think their way through to the solution. Solutions to those problems are given at the end of each chapter.
Smullyan also gives a powerful existence proof of the fundamental principle that mathematics is the manipulation of symbols together with an operator, the latter being the replacing of one set of characters with another set of characters. While doing this he illustrates some of the deepest mathematical concepts.
Smullyan opens with some examples of paradoxes; in this case they are undecidable expressions. The island of knights (who always tell the truth) and knaves (who never tell the truth) serves to set the stage for what is to come. Beginning with self reference in expressions and operators, Smullyan moves on to explain and illustrate the conclusions of Gödel regarding what can be proved or disproved.
The concept of omega-inconsistency is covered. It is defined as the situation where for the number 1 there is a proof that 1 does not have the property, there is a separate proof that the number 2 does not have the property, and so on, continuing through all the natural numbers. But at the same time there is a separate proof that there is in fact some number n such that n has the property. Smullyan then goes on to state that there are indeed consistent systems that are omega inconsistent.
The foundations of mathematics can seem esoteric and horrendously complicated; the common joke is that it took Russell and Whitehead 300 pages to prove that 1 + 1 = 2 in “Principia Mathematica.” Smullyan uses much less time and much clearer notation and prose to prove some of the most foundational of mathematics concepts.
Charles Ashbacher splits his time between consulting with industry in projects involving math and computers, teaching college classes and co-editing The Journal of Recreational Mathematics. In his spare time, he reads about these things and helps his daughter in her lawn care business.