The Annotated Gödel: A Reader's Guide to his Classic Paper on Logic and Incompleteness

by Hal Prince

T he Annotated Gödel offers a guided tour of Kurt Gödel's 1931 article on incompleteness, which demonstrated unexpected limits to the power of many logical systems. Today we call these results Gödel's First and Second Incompleteness Theorems. The book includes the complete article in a new English translation, interleaved with commentary that guides the reader through Gödel's work, step by step. The commentary concentrates on Gödel’s exposition. It describes what he is doing at each point, and how it relates to other parts of the article. It elaborates on his proofs by outlining them, for example, or by making a table of his variables and their uses, or by filling in gaps in his arguments.

The translation uses modern mathematical notation and terminology. It replaces Gödel's function and relation names, based on German word fragments, with English equivalents. Its language is less formal than that of the existing translations, which date from the 1960s. The book assumes some familiarity with mathematical definitions and proofs, at the level of an undergraduate abstract math course. It also assumes some knowledge of formal logic, from an introductory course or the equivalent.

Genres: Mathematics, Philosophy, Nonfiction, Science

183 pages, Hardcover

Published August 18, 2022

Reviews

[Harris Bolus · Rating 5 out of 5]

Prince brings a great deal of thought and background reading into his exposition of Godel’s paper, and his pace reminds me of an incredibly patient professor. This is a wonderful book.

[Dante Criscio · Rating 4 out of 5]

Gödel is clearly a genius, and I appreciate Hal Prince for all his hard work of explaining and translating. I'm so glad that this book exists, as I loved "The Annotated Turing" and it made frequent references to this material. Now I feel like I have a much clearer understanding of what Gödel was proving.

I found the idea of primitive-recursive functions amazing but also very natural, as it's basically the way that children learn math: every number can be incremented, addition is repeated incrementation, multiplication is repeated addition, etc. The idea that complex propositions like "Is Prime" and "Is Provable" can be built from this is incredible.

I did find this book much harder to understand than "The Annotated Turing". It could be because the content is legitimately trickier, but sometimes I wish that the author dumbed it down for me just a little more.

[William Schram · Rating 5 out of 5]

Kurt Gödel was a mathematician. His work changed our understanding of mathematics. The Annotated Gödel is a book by Hal Prince. In the book, Prince explains Gödel's Incompleteness Theorems section by section. Prince also provides a new translation of the book that modernizes the text.

The Incompleteness Theorems are statements that proclaim their incompleteness. Gödel invented a method called Gödel Numbering and used arithmetic to provide proof. It's akin to saying this sentence is false.

Prince's book is not for the layman, but it does make Gödel's statements and methods more straightforward.

I enjoyed the book. It still wasn't easy to understand, but I did enjoy it. Thanks for reading my review, and see you next time.

[Lucille Nguyen · Rating 4 out of 5]

Good overview and English translation of Godel's landmark paper. Some of the notation choices for the mathematical logic are closer to what one might see in theoretical computer science than modern mathematical research, but exceptionally readable if one is a programmer as Prince was. Good if one wants to get deeper into the guts of Godel's completeness and incompleteness results.

[TheEoJMan · Rating 5 out of 5]

Gave me a deeper understanding of Gödel’s famous paper. Before I only had a vague popular understanding of it—can’t say I truly understand the whole work, but I’ve got a clearer idea of it now. Read so I could better grasp Turing’s “On Computable Numbers…”.