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






  • 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{displaystyle Sigma _{n}^{0}}Sigma _{n}^{0} and Πn0{displaystyle Pi _{n}^{0}}Pi^0_n 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{displaystyle phi }phi is logically equivalent to a formula with only bounded quantifiers then ϕ{displaystyle phi }phi is assigned the classifications Σ00{displaystyle Sigma _{0}^{0}}Sigma^0_0 and Π00{displaystyle Pi _{0}^{0}}Pi^0_0.


The classifications Σn0{displaystyle Sigma _{n}^{0}}Sigma _{n}^{0} and Πn0{displaystyle Pi _{n}^{0}}Pi^0_n are defined inductively for every natural number n using the following rules:



  • If ϕ{displaystyle phi }phi is logically equivalent to a formula of the form ∃n1∃n2⋯∃nkψ{displaystyle exists n_{1}exists n_{2}cdots exists n_{k}psi }exists n_1 exists n_2cdots exists n_k psi, where ψ{displaystyle psi }psi is Πn0{displaystyle Pi _{n}^{0}}Pi^0_n, then ϕ{displaystyle phi }phi is assigned the classification Σn+10{displaystyle Sigma _{n+1}^{0}}Sigma^0_{n+1}.

  • If ϕ{displaystyle phi }phi is logically equivalent to a formula of the form ∀n1∀n2⋯∀nkψ{displaystyle forall n_{1}forall n_{2}cdots forall n_{k}psi }forall n_1 forall n_2cdots forall n_k psi, where ψ{displaystyle psi }psi is Σn0{displaystyle Sigma _{n}^{0}}Sigma _{n}^{0}, then ϕ{displaystyle phi }phi is assigned the classification Πn+10{displaystyle Pi _{n+1}^{0}}Pi^0_{n+1}.



References




  • 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..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output q{quotes:"""""""'""'"}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}


  • Moschovakis, Yiannis N. (1980). Descriptive Set Theory. North Holland. ISBN 0-444-70199-0.
    Second edition available online








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