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The Notre Dame Lectures
Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. In the fall of 2000, the logic community at the University of Notre Dame, Indiana hosted Greg Hjorth, Rodney G. Downey, Zoé Chatzidakis and Paola D'Aquino as visiting lecturers. Each of them presented a month-long series of expository lectures at the graduate level. This volume, the eighteenth publication in the Lecture Notes in Logic series, contains refined and expanded versions of those lectures. The four articles are entitled 'Countable models and the theory of Borel equivalence relations', 'Model theory of difference fields', 'Some computability-theoretic aspects of reals and randomness' and 'Weak fragments of Peano arithmetic'.
The Notre Dame Lectures
In fall 2000, the Notre Dame logic community hosted Greg Hjorth, Rodney G. Downey, Zoe Chatzidakis, and Paola D'Aquino as visiting lecturers. Each of them presented a month long series of expository lectures at the graduate level. The articles in this volume are refinements of these excellent lectures.
Reverse Mathematics
Reverse mathematics studies the complexity of proving mathematical theorems and solving mathematical problems. Typical questions include: Can we prove this result without first proving that one? Can a computer solve this problem? A highly active part of mathematical logic and computability theory, the subject offers beautiful results as well as significant foundational insights. This text provides a modern treatment of reverse mathematics that combines computability theoretic reductions and proofs in formal arithmetic to measure the complexity of theorems and problems from all areas of mathematics. It includes detailed introductions to techniques from computable mathematics, Weihrauch style ...
Computability Theory and Its Applications
This collection of articles presents a snapshot of the status of computability theory at the end of the millennium and a list of fruitful directions for future research. The papers represent the works of experts in the field who were invited speakers at the AMS-IMS-SIAM 1999 Summer Conference on Computability Theory and Applications, which focused on open problems in computability theory and on some related areas in which the ideas, methods, and/or results of computability theory play a role. Some presentations are narrowly focused; others cover a wider area. Topics included from "pure" computability theory are the computably enumerable degrees (M. Lerman), the computably enumerable sets (P....
Automorphisms of the Lattice of Recursively Enumerable Sets
A version of Harrington's [capital Greek]Delta3-automorphism technique for the lattice of recursively enumerable sets is introduced and developed by reproving Soare's Extension Theorem. Then this automorphism technique is used to show two technical theorems: the High Extension Theorem I and the High Extension Theorem II. This is a degree-theoretic technique for constructing both automorphisms of the lattice of r.e. sets and isomorphisms between various substructures of the lattice.
The Incomputable
This book questions the relevance of computation to the physical universe. Our theories deliver computational descriptions, but the gaps and discontinuities in our grasp suggest a need for continued discourse between researchers from different disciplines, and this book is unique in its focus on the mathematical theory of incomputability and its relevance for the real world. The core of the book consists of thirteen chapters in five parts on extended models of computation; the search for natural examples of incomputable objects; mind, matter, and computation; the nature of information, complexity, and randomness; and the mathematics of emergence and morphogenesis. This book will be of interest to researchers in the areas of theoretical computer science, mathematical logic, and philosophy.
A Hierarchy of Turing Degrees
[Alpha]-c.a. functions -- The hierarchy of totally [alpha]-c.a. degrees -- Maximal totally [alpha]-c.a. degrees -- Presentations of left-c.e. reals -- m-topped degrees -- Embeddings of the 1-3-1 lattice -- Prompt permissions.
Logical Methods
The twenty-six papers in this volume reflect the wide and still expanding range of Anil Nerode's work. A conference on Logical Methods was held in honor of Nerode's sixtieth birthday (4 June 1992) at the Mathematical Sciences Institute, Cornell University, 1-3 June 1992. Some of the conference papers are here, but others are from students, co-workers and other colleagues. The intention of the conference was to look forward, and to see the directions currently being pursued, in the development of work by, or with, Nerode. Here is a brief summary of the contents of this book. We give a retrospective view of Nerode's work. A number of specific areas are readily discerned: recursive equivalence ...
