Lecture Notes in Logic, 41 

Effective Mathematics of the Uncountable

Noam Greenberg, Joel David Hamkins, Denis Hirschfeldt, Russell Miller, Editors

Year: 2013

204 pages. Hardcover.
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Classical computable model theory is most naturally concerned with countable domains. There are, however, several methods – some old, some new – that have extended its basic concepts to uncountable structures. Unlike in the classical case, however, no single dominant approach has emerged, and different methods reveal different aspects of the computable content of uncountable mathematics. This book contains introductions to eight major approaches to computable uncountable mathematics: descriptive set theory; infinite time Turing machines; Blum-Shub-Smale computability; Sigma-definability; computability theory on admissible ordinals; E-recursion theory; local computability; and uncountable reverse mathematics. This book provides an authoritative and multifaceted introduction to this exciting new area of research that is still in its early stages. It is ideal as both an introductory text for graduate and advanced undergraduate students, and a source of interesting new approaches for researchers in computability theory and related areas.

Table of Contents

  1. Introduction
  2. Borel structures: a brief survey
  3. Infinite time Turing machines and an applicaiton to the hierarchy of equivalence relations on the reals
  4. Some results on R-computable structures
  5. Effective model theory via the Σ-definability approach
  6. Computable structure theory using admissible recursion theory on ω1
  7. E-recursive intuitions
  8. Local computability and uncountable structures
  9. Reverse mathematics, countable and uncountable: a computational approach