Gödel’s Theorems

The big book

An Introduction to Gödel’s Theorems was first published by CUP in 2007 with the second edition appearing in 2013.

A corrected version of the second edition is now available as a freely downloadable PDF.

Many people, however, prefer if possible to work from a physical book. So you can also get a print-on-demand copy of this version as a very inexpensive large format book from Amazon: US link; UK link, or find on your local Amazon by using the ASIN identifier B08GB4BDPG in their search field.

The book previously appeared in the series ‘Cambridge Introductions to Philosophy’ from CUP. But don’t let that mislead you! IGT is actually a fairly techie logic book, originally intended for advanced philosophy undergraduates and postgraduates. It is quite long (388 pp.) and is full of theorems — so many mathematics students should find it useful too. Still, it does aim to give a relatively relaxed and approachable exposition of the technicalities around and about the incompleteness theorems and related results, and it does also provide a modest amount of more philosophical commentary on the interpretation and significance of the theorems.

A shorter book

Gödel Without (Too Many) Tears is a much shorter book (146 pp.) based on my notes for the lectures I used to give to undergraduate philosophers taking the Mathematical Logic paper in Cambridge. You can think of this as a cut-down version of the longer book, aiming to give some of the key technical facts about the incompleteness theorems without too many digressions. It is still full of technical exposition and quite short on philosophical asides — the main aim is to put a reader in a position where they can begin to understand what’s going on in discussions of supposed philosophical implications of the incompleteness theorems.

This book is now in a second edition and is available as a freely downloadable PDF.

But again, many will find it easier to work from the decently produced but extremely inexpensive Amazon print-on-demand paperback, with the ISBN 1916906341.

Corrections to current versions

  • IGT: For just a handful of minor known corrections for the current downloadable PDF/print-on-demand Amazon version, see here.
  • GWT: Again for just a handful of minor known corrections for the current versions, both PDF and printed, see here.

Since the second editions of both books are freely available to download, I deem the earlier editions to be obsolete, and will no longer be maintaining lists of corrections for them

Exercises

There are exercises with solutions for early chapters in IGT2. I may add further exercises one day, but this isn’t a top priority:

More on Gödel’s Theorems

  • Lectures on the First Incompleteness Theorem — just four introductory lectures given in Easter term 2011 as a supplement to Thomas Forster’s earlier Part III Maths course on Computable Function Theory. (The first three don’t require any background in the theory of computation over and above a grip on the idea of a primitive recursive function and the idea of coding: only the fourth appeals to results like the unsolvability of the halting problem.)
  • Back to Basics: Revisiting the Incompleteness Theorems. The notes for a three-lecture series given to mathematicians at a Cambridge weekend workshop for graduates in 2009. They complement the book by approaching things in a rather different order.
  • Expounding the First Theorem — extensive (though far from completed) notes on the expository tradition. Version 2: From 1931 to 1953.

Other relevant notes