Effective descriptive set theory

Short description: Branch of mathematics

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

* 1 Constructions
* 1.1 Effective Polish space
* 1.2 Arithmetical hierarchy
* 2 References

## Constructions

### Effective Polish space

Main article: Effective Polish space

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 hierarchy

Main article: Arithmetical hierarchy

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 Σn0 and Πn0 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 ϕ is logically equivalent to a formula with only bounded quantifiers then ϕ is assigned the classifications Σ00 and Π00.

The classifications Σn0 and Πn0 are defined inductively for every natural number n using the following rules:

* If ϕ is logically equivalent to a formula of the form ∃n1∃n2⋯∃nkψ, where ψ is Πn0, then ϕ is assigned the classification Σn+10.
* If ϕ is logically equivalent to a formula of the form ∀n1∀n2⋯∀nkψ, where ψ is Σn0, then ϕ is assigned the classification Πn+10.

## References

* Mansfield, Richard; Weitkamp, Galen (1985). Recursive Aspects of Descriptive Set Theory. Oxford University Press. pp. 124–38. ISBN 978-0-19-503602-2. https://archive.org/details/recursiveaspects0000mans/page/124.
* Moschovakis, Yiannis N. (1980). Descriptive Set Theory. North Holland. ISBN 0-444-70199-0. https://archive.org/details/descriptivesetth0000mosc. Second edition available online

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