Gödel, Escher, Bach: an Eternal Golden Braid
By Douglas R. Hofstadter
This is a fine book, and has helped me very much in my understanding of Gödel’s Theorem and formal systems in general. Hofstadter develops the necessary mathematics from the ground up through his own formal system called Typographical Number Theory (TNT). Not all chapters in the book, nor all the contents of each chapter need to be read in order to see Gödel’s proof developed to its conclusion. I think the book is brillant, and highly recommend it.
Metalogic – An Introduction to the Metatheory of Standard First Order Logic
By Geoffrey Hunter
This work makes available to readers without specialized training in mathematics complete proofs of the fundamental metatheorems of standard (i.e., basically truth-functional) first order logic. This is my favorite and perhaps the easiest introduction of the topic that I own. If I could only own one introduction to this field it would be this book. (By the way, Peter Suber of Earlham College taught a course on Logical Systems in the 2002-03 school year where he used this text and the Hofstadter text. All of the homework assignments and handouts are available at his site. Very valuable!)
An Introduction to Gödel’s Theorems
By Peter Smith
This is one of the newer books that have come out regarding Gödel’s Incompleteness Theorems. Peter Smith has a page dedicated to the book (Logic Matters). Hopefully, this will be the second book that I work through. When asked about books with good explanations of Gödel’s Incompleteness Theorems, here is what Peter Smith had to say about Smullyan’s book below…
My favourite book is perhaps Smullyan’s Gödel’s Incompleteness Theorems, but in some ways it is very terse.
Smith’s book appears much less terse, and is being hailed by some as “extraordinary, and a cause for celebration because Smith combines in a logic text a level of coverage, rigor, accessibility, and funness of read that may be unprecedented.”
Gödel’s Incompleteness Theorems (GIT)
By Raymond M. Smullyan
This will be the book that I initially work through. Raymond Smullyan is my favorite mathematical logic author. He is brilliant, and has a way of explaining rather difficult topics in very accessible ways. This introductory volume has been said to be “the most direct and serious account of Gödel’s incompleteness theorems, and is accessible to anyone with a basic acquaintance of mathematical logic.”
