Chapter 12 A bibliography of recursive analysis and recursive topology

Abstract

Publisher Summary This chapter provides a bibliography for recursive analysis and recursive topology. Recursive mathematics investigates the “constructive” nature of mathematical results when “constructive” is interpreted via recursive function theory or Turing computability, whereas the classical laws of logic are adopted intact. Recursive analysis focuses this approach of study to classical analysis. The chapter describes some of the more recent approaches to recursive analysis, including sequence approach, filter approach, and representation approach. The sequence approach to computability has been inspired by the observation that a real function (defined on the unit interval) is computable in the sense of Grzegorczyk/Lacombe if and only if it maps computable sequences to computable sequences and admits a computable modulus of continuity. In the filter approach, the objects of study are not points, they are neighborhoods. The approach explains the existing phenomena on pathology of domains of recursive real functions while presenting new results and machinery to apply to recursive analysis. In the representation approach, classical theorems can be expressed in highly effective versions. It provides a direct way to investigate complexity in analysis.