The logician Kurt Goedel (1906-1978) published a paper in 1931 formulating what have come to be known as his 'incompleteness theorems', which prove, among other things, that within any formal system with resources sufficient to code arithmetic, questions exist which are neither provable nor disprovable on the basis of the axioms which define the system. These are among the most celebrated results in logic today. In this volume, leading philosophers and mathematicians assess important aspects of Goedel's work on the foundations and philosophy of mathematics. Their essays explore almost every aspect of Godel's intellectual legacy including his concepts of intuition and analyticity, the Completeness Theorem, the set-theoretic multiverse, and the state of mathematical logic today. This groundbreaking volume will be invaluable to students, historians, logicians and philosophers of mathematics who wish to understand the current thinking on these issues.
Juliette Kennedy is an Associate Professor in the Department of Mathematics and Statistics at the University of Helsinki.
1. Introduction: Goedel and analytic philosophy: how did we get here? Juliette Kennedy; Part I. Goedel on Intuition: 2. Intuitions of three kinds in Goedel's views on the continuum John Burgess; 3. Goedel on how to have your mathematics and know it too Janet Folina; Part II. The Completeness Theorem: 4. Completeness and the ends of axiomatization Michael Detlefsen; 5. Logical completeness, form, and content: an archaeology Curtis Franks; Part III. Computability and Analyticity: 6. Goedel's 1946 Princeton bicentennial lecture: an appreciation Juliette Kennedy; 7. Analyticity for realists Charles Parsons; Part IV. The Set-Theoretic Multiverse: 8. Goedel's program John Steel; 9. Multiverse set theory and absolutely undecidable propositions Jouko Vaananen; Part V. The Legacy: 10. Undecidable problems: a sampler Bjorn Poonen; 11. Reflecting on logical dreams Saharon Shelah.