For many mathematicians, Kurt Gödel’s work is the end of a story, or of several stories. The typical introductory class in logic will culminate with the Incompleteness Theorems. That the Axiom of Choice and the Continuum Hypotheses are independent of Zermelo-Fraenkel set theory is about as far as most mathematicians get; I suspect very few have really worked through Gödel’s consistency proof, much less Cohen’s theorem showing the independence. In philosophy of mathematics, some of us may have heard that Gödel was (or became) a full-blown Platonist, but we don’t know more than that.
Of course, for those who are actively engaged in those disciplines, things look very different. Gödel’s last contributions happened fifty years ago, after all, and mathematics has not stood still, and neither has philosophy. This collection of essays in honor of Gödel’s centennial in 2006 offers some glimpses of the more recent development and some reassessment of Gödel’s ideas and contributions.
The book opens with two essays on the massive project, recently completed, of publishing Gödel’s collected works. The hard part of the project was to decipher the unpublished material, some of which was written in an obscure form of shorthand. Choices also had to be made: what to include, how much to annotate, etc. The second article includes a list of materials that were not included; it is aptly named “Future tasks for Gödel scholars.”
Once that is taken care of, the book becomes quite technical. There are three main sections: on proof theory, set theory, and philosophy of mathematics. In each case, we get quasi-historical essays examining Gödel’s work and ideas, essays on the current state of some of the problems he worked on, and a few proposals for future work.
In all of these fields, I am a complete outsider, and some of the essays, particularly in proof theory, were over my head. I found “reverse mathematics,” which attempts to see just how much is needed to deal with specific bits of classical mathematics, an interesting idea. This movement (for lack of a better word) seems to be an attempt to see how much of Hilbert’s Finitism still makes sense after the Incompleteness Theorems. The essays on set theory are also very interesting. In particular, they gave me at least a glimmer of the role of large cardinal axioms, something that I have always found rather puzzling. I was also interested in Gödel’s attempt to decide the Continuum Hypothesis by coming up with some sort of “intuitively clear” axiom that could be added to ZFC.
The philosophical essays were the reason I decided to read the book, but in the end I found them disappointing. Too many of them focused on Gödel’s intellectual development instead of his ideas. I wanted to read some more serious engagement of his Platonism; instead, too many of the essays took it as given that Gödel’s position cannot be sustained, and went on in other directions.
Mathematicians who are interested in finding out what has happened since Gödel might well start here, at least if they are willing to plow through the more technical patches. Sometimes jumping into the deep part of the pool is just the thing.
Fernando Q. Gouvêa wishes there was enough time to read and study everything that is interesting.
Part I. General: 1. The Gödel editorial project: a synopsis Solomon Feferman; 2. Future tasks for Gödel scholars John W. Dawson, Jr., and Cheryl A. Dawson; Part II. Proof Theory: 3. Kurt Gödel and the metamathematical tradition Jeremy Avigad; 4. Only two letters: the correspondence between Herbrand and Gödel Wilfried Sieg; 5. Gödel's reformulation of Gentzen's first consistency proof for arithmetic: the no-counter-example interpretation W. W. Tait; 6. Gödel on intuition and on Hilbert's finitism W. W. Tait; 7. The Gödel hierarchy and reverse mathematics Stephen G. Simpson; 8. On the outside looking in: a caution about conservativeness John P. Burgess; Part III. Set Theory: 9. Gödel and set theory Akihiro Kanamori; 10. Generalizations of Gödel's universe of constructible sets Sy-David Friedman; 11. On the question of absolute undecidability Peter Koellner; Part IV. Philosophy of Mathematics: 12. What did Gödel believe and when did he believe it? Martin Davis; 13. On Gödel's way in: the influence of Rudolf Carnap Warren Goldfarb; 14. Gödel and Carnap Steve Awodey and A. W. Carus; 15. On the philosophical development of Kurt Gödel Mark van Atten and Juliette Kennedy; 16. Platonism and mathematical intuition in Kurt Gödel's thought Charles Parsons; 17. Gödel's conceptual realism Donald A. Martin.