...
Show More
This book serves as an excellent introduction to Gödel's theorem, clearly distinguishing what it is and what it isn't. It presents numerous examples of how the theorem has been utilized by people and delves into the discussion of why only certain of those uses are valid. The book is of moderate technicality. It provides sufficient information to enable an understanding of the mathematics underlying the theorem, yet only sketches out various proofs. However, the organization of the book is somewhat cumbersome. For instance, computability and Gödel numbering are extensively employed in chapter 2 but are not fully explained until chapter 3. The author justifies this choice by stating that only a superficial understanding is initially required, and deeper explanations can follow. Nevertheless, I found this to be confusing whenever it occurred. After perusing this book, I have a significantly better grasp of the concepts of completeness and consistency of formal systems. I also comprehend how computability factors into the incompleteness theorem. This is undoubtedly an informative book. The second half of the book examines various applications that people have made of Gödel's incompleteness theorems and how those applications often have little to do with the actual theorem. It also explores some ways in which Gödel's theorem is related to Kolmogorov Complexity and Turing machines. The author is mainly responding to statements made by others, and his responses are sometimes interesting and at other times rather tedious. Overall, despite its organizational flaws, this book offers valuable insights into Gödel's theorem and its related concepts.
