Foundations Of Mathematics: Logic At Harvard Essays In Honor Of W. Hugh Woodin's 60th Birthday March 27-29, 2015 Harvard University, Cambridge, Ma (contemporary Mathematics)

This volume contains the proceedings of the Logic at Harvard conference in honor of W. Hugh Woodin's 60th birthday, held March 27–29, 2015, at Harvard University. It presents a collection of papers related to the work of Woodin, who has been one of the leading figures in set theory since the early 1980s. The topics cover many of the areas central to Woodin's work, including large cardinals, determinacy, descriptive set theory and the continuum problem, as well as connections between set theory and Banach spaces, recursion theory, and philosophy, each reflecting a period of Woodin's career. Other topics covered are forcing axioms, inner model theory, the partition calculus, and the theory of ultrafilters. This volume should make a suitable introduction to Woodin's work and the concerns which motivate it. The papers should be of interest to graduate students and researchers in both mathematics and philosophy of mathematics, particularly in set theory, foundations and related areas. 6. Applications to extreme amenability and calculation of the universal minimal flow7. Reformulation of the Ramsey Property for homogeneous measure algebras References Topological Ramsey numbers and countable ordinals 1. Introduction 2. Preliminaries 3. The closed pigeonhole principle for ordinals 4. The ordinal omega plus one 5. Stepping up by one 6. Ordinals less than omega squared 7. The ordinal omega squared 8. The anti-tree partial ordering on ordinals 9. The ordinal omega squared plus one 10. The weak topological Erdős-Milner theorem 11. Questions References Open determinacy for class games1. Introduction 2. A broader context 3. The truth-telling game 4. Clopen determinacy is equivalent over \GBC to \ETR 5. Proving open determinacy in strong theories 6. Questions References Open problems on ultrafilters and some connections to the continuum 1. The cut spectrum 2. A uniqueness theorem 3. On internal maps 4. Bases for goodness Appendix References Obtaining Woodin's cardinals 1. Introduction 2. The Cantorian versus the Zermelian realms 3. Reflection 4. The Ackermannian realm and reflection 5. Global Reflection Norming Infinitesimals Of Large Fields / H.g. Dales -- The Enumeration Degrees: Local And Global Structural Interactions / Theodore A. Slaman And Mariya I. Soskova -- Ramsey Properties Of Finite Measure Algebras And Topolotical Dynamics Of The Group Of Measure Preserving Automorphisms: Some Results And An Open Problem / A.s. Kechris, M. Sokic, And S. Todorcevic -- Topological Ramsey Numbers And Countable Ordinals / Andres Eduardo Caicedo And Jacob Hilton -- Open Determinacy For Class Games / Victoria Gitman And Joel David Hamkins -- Open Problems On Ultrafilters And Some Connections To The Continuum / M. Malliaris And S. Shelah -- Obtaining Woodin's Cardinals / P.d. Welch -- Woodin's Axiom (*), Or Martin's Maximum, Or Both? / Ralf Schindler -- Translation Procedures In Descriptive Inner Model Theory / Grigor Sargsyan -- Implications Of Very Large Cardinals / Scott Cramer -- What Makes The Continuum / Justin Tatch Moore -- Set-theoretic Foundations / Penelope Maddy. Andrés Eduardo Caicedo [and Three Others], Editors. Includes Bibliographical References. Contains the proceedings of the Logic at Harvard conference in honour of W. Hugh Woodin's 60th birthday. The topics cover many of the areas central to Woodin's work, including large cardinals, determinacy, descriptive set theory and the continuum problem, as well as connections between set theory and Banach spaces, recursion theory, and philosophy, each reflecting a period of Woodin's career.