Effective Mathematics of the Uncountable

Noam Greenberg is a Rutherford Discovery Fellow of the Royal Society of New Zealand where his research interests include computability theory, algorithmic randomness, reverse mathematics, higher recursion theory, computable model theory and set theory. Denis Hirschfeldt is a Professor of Mathematics at the University of Chicago and has previously held visiting positions at the University of Wisconsin, Madison and the University of Notre Dame. He was a recipient of the 2010 Sacks Prize of the Association for Symbolic Logic for expository writing. Joel David Hamkins is a professor at the City University of New York and has held visiting positions at the University of California, Berkeley, Kobe University, Carnegie Mellon University, the University of Muenster, the University of Amsterdam and New York University. Russell Miller holds an appointment as Professor of Mathematics jointly between Queens College and the CUNY Graduate Center. His research applies computability to other areas of mathematics, including model theory, set theory, commutative algebra, differential algebra, graph theory and topology.

List of contributors; Preface; 1. Introduction; 2. Borel structures: a brief survey Antonio Montalbán and André Nies; 3. Infinite time Turing machines and an application to the hierarchy of equivalence relations on the reals Samuel Coskey and Joel David Hamkins; 4. Some results on R-computable structures W. Calvert and J. E. Porter; 5. Effective model theory via the Σ-definability approach Alexey Stukachev; 6. Computable structure theory using admissible recursion theory on ω1 Noam Greenberg and Julia F. Knight; 7. E-recursive intuitions Gerald E. Sacks; 8. Local computability and uncountable structures Russell Miller; 9. Reverse mathematics, countable and uncountable: a computational approach Richard A. Shore.