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Effective descriptive set theory

Effective descriptive set theory is the branch of descriptive set theory dealing with sets of reals having lightface definitions; that is, definitions that do not require an arbitrary real parameter (Moschovakis 1980). Thus effective descriptive set theory combines descriptive set theory with recursion theory.

Contents

ConstructionsEdit

Effective Polish spaceEdit

An effective Polish space is a complete separable metric space that has a computable presentation. Such spaces are studied in both effective descriptive set theory and in constructive analysis. In particular, standard examples of Polish spaces such as the real line, the Cantor set and the Baire space are all effective Polish spaces.

Arithmetical hierarchyEdit

The arithmetical hierarchy, arithmetic hierarchy or Kleene–Mostowski hierarchy classifies certain sets based on the complexity of formulas that define them. Any set that receives a classification is called "arithmetical".

More formally, the arithmetical hierarchy assigns classifications to the formulas in the language of first-order arithmetic. The classifications are denoted   and   for natural numbers n (including 0). The Greek letters here are lightface symbols, which indicates that the formulas do not contain set parameters.

If a formula   is logically equivalent to a formula with only bounded quantifiers then   is assigned the classifications   and  .

The classifications   and   are defined inductively for every natural number n using the following rules:

  • If   is logically equivalent to a formula of the form  , where   is  , then   is assigned the classification  .
  • If   is logically equivalent to a formula of the form  , where   is  , then   is assigned the classification  .

ReferencesEdit

  • Mansfield, Richard; Weitkamp, Galen (1985). Recursive Aspects of Descriptive Set Theory. Oxford University Press. pp. 124–38. ISBN 978-0-19-503602-2. MR 0786122.
  • Moschovakis, Yiannis N. (1980). Descriptive Set Theory. North Holland. ISBN 0-444-70199-0. Second edition available online