Kurt Goedel (1906-1978) did groundbreaking work that transformed logic and other important aspects of our understanding of mathematics, especially his proof of the incompleteness of formalized arithmetic. This book on different aspects of his work and on subjects in which his ideas have contemporary resonance includes papers from a May 2006 symposium celebrating Goedel's centennial as well as papers from a 2004 symposium. Proof theory, set theory, philosophy of mathematics, and the editing of Goedel's writings are among the topics covered. Several chapters discuss his intellectual development and his relation to predecessors and contemporaries such as Hilbert, Carnap, and Herbrand. Others consider his views on justification in set theory in light of more recent work and contemporary echoes of his incompleteness theorems and the concept of constructible sets.
Solomon Feferman has been a Professor of Mathematics and Philosophy at Stanford University since 1956, from which he retired in 2004. He is a Fellow of the American Academy of Arts and Sciences, was President of the Association for Symbolic Logic in 1980-2, and was the recipient of the Rolf Schock Prize for Logic and Philosophy in 2003. Feferman was editor-in-chief of the Collected Works of Kurt Goedel (1986-2003). Charles Parsons holds an AB (mathematics) and PhD (philosophy) from Harvard University and studied for a year at King's College, Cambridge. He was on the faculty at Harvard University from 1962-65 and 1989-2005 and at Columbia University from 1965-89. His publications are mainly in logic, philosophy of mathematics, and Kant. He was an editor of the posthumous works of Kurt Goedel (Collected Works, Volumes III-V). Stephen G. Simpson is a mathematics professor at the Pennsylvania State University. He has lectured and published widely in mathematical logic and the foundations of mathematics. Simpson is the developer of the foundational program known as Reverse Mathematics and the author of Subsystems of Second Order Arithmetic, 2nd Edition.
Part I. General: 1. The Goedel editorial project: a synopsis Solomon Feferman; 2. Future tasks for Goedel scholars John W. Dawson, Jr, and Cheryl A. Dawson; Part II. Proof Theory: 3. Kurt Goedel and the metamathematical tradition Jeremy Avigad; 4. Only two letters: the correspondence between Herbrand and Goedel Wilfried Sieg; 5. Goedel's reformulation of Gentzen's first consistency proof for arithmetic: the no-counter-example interpretation W. W. Tait; 6. Goedel on intuition and on Hilbert's finitism W. W. Tait; 7. The Goedel 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. Goedel and set theory Akihiro Kanamori; 10. Generalizations of Goedel'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 Goedel believe and when did he believe it? Martin Davis; 13. On Goedel's way in: the influence of Rudolf Carnap Warren Goldfarb; 14. Goedel and Carnap Steve Awodey and A. W. Carus; 15. On the philosophical development of Kurt Goedel Mark van Atten and Juliette Kennedy; 16. Platonism and mathematical intuition in Kurt Goedel's thought Charles Parsons; 17. Goedel's conceptual realism Donald A. Martin.