definitions is one motivation for the study of hyperarithmetical theory. The first definition of the hyperarithmetic sets uses the analytical hierarchy. A set...

set theory is between set theory and recursion theory. It includes the study of lightface pointclasses, and is closely related to hyperarithmetical theory...

descriptive set theory combines the methods of descriptive set theory with those of generalized recursion theory (especially hyperarithmetical theory). In particular...

types as well as areas such as hyperarithmetical theory and α-recursion theory. Contemporary research in recursion theory includes the study of applications...

about f being computable in g by identifying g with its graph. Hyperarithmetical theory studies those sets that can be computed from a computable ordinal...

theory includes the study of generalized notions of this field such as arithmetic reducibility, hyperarithmetical reducibility and α-recursion theory...

are studied as part of hyperarithmetical theory. Effectively closed sets are a topic of study in classical computability theory. An effectively closed...

hierarchy that allows parameters). They include the arithmetical, hyperarithmetical, and analytical sets limit 1.  A (weak) limit cardinal is a cardinal...

classification of arithmetical reducibility. Hyperarithmetical reducibility: A set A {\displaystyle A} is hyperarithmetical in a set B {\displaystyle B} if A {\displaystyle...

the classes of provably recursive, hyperarithmetical, or Δ 2 1 {\displaystyle \Delta _{2}^{1}} functions of the theory. The field of ordinal analysis was...

called hyperarithmetical. An alternate classification of these sets by way of iterated computable functionals is provided by the hyperarithmetical theory. The...

classifications assigned to a formula and the set it defines. The hyperarithmetical hierarchy and the analytical hierarchy extend the arithmetical hierarchy...

(or, equivalently, all closed sets). Borel sets are important in measure theory, since any measure defined on the open sets of a space, or on the closed...

Recursive set Recursively enumerable set Arithmetical set Diophantine set Hyperarithmetical set Analytical set Analytic set, Coanalytic set Suslin set Projective...

ordinal is a limit ordinal. It is also the smallest ordinal that is not hyperarithmetical, and the smallest admissible ordinal after ω {\displaystyle \omega...

recursion as recursive comprehension is to weak Kőnig's lemma. It has the hyperarithmetical sets as minimal ω-model. Arithmetical transfinite recursion proves...

{\displaystyle } so it is in L ω + 2 {\displaystyle L_{\omega +2}} ). All hyperarithmetical subsets of ω {\displaystyle \omega } and relations on ω {\displaystyle...

hierarchy Chomsky hierarchy Analytical hierarchy Arithmetical hierarchy Hyperarithmetical hierarchy Abstract algebraic hierarchy Borel hierarchy Wadge hierarchy...

\omega ^{<\omega }} that have no arithmetical path, and indeed no hyperarithmetical path. However, every computable subtree of ω < ω {\displaystyle \omega...

{\displaystyle B} as a parameter. The set A {\displaystyle A} is hyperarithmetical in B {\displaystyle B} if there is a recursive ordinal α {\displaystyle...

and Gustav Hensel, he demonstrated how the Davis–Mostowski–Kleene hyperarithmetical hierarchy of arithmetical degrees can be naturally extended up to...

let S be ATR0 or another recursively axiomatizable theory that has an ω-model but no hyperarithmetical ω-models, and (if needed) conservatively extend S...

In the mathematical field of descriptive set theory, a subset A {\displaystyle A} of a Polish space X {\displaystyle X} is projective if it is Σ n 1 {\displaystyle...

subsets of an effective Polish space. It is closely related to the hyperarithmetical hierarchy. The lightface Borel hierarchy can be defined on any effective...

In computability theory, the Turing jump or Turing jump operator, named for Alan Turing, is an operation that assigns to each decision problem X a successively...

In the mathematical field of descriptive set theory, a pointclass is a collection of sets of points, where a point is ordinarily understood to be an element...

doi:10.2307/2267778, JSTOR 2267778, S2CID 34314018 "Hyperarithmetical Index Sets In Recursion Theory" by Steffen Lempp Hilbert Levitz, Transfinite Ordinals...

(North-Holland, 1980) Ash, Knight, *Computable Structures and the Hyperarithmetical Hierarchy* p.83. Studies in Logic and the Foundations of Mathematics...

1995), p. 728 C. J. Ash, J. Knight, Computable Structures and the Hyperarithmetical Hierarchy (Studies in Logic and the Foundation of Mathematics, 2000)...

Forcing in computability theory is a modification of Paul Cohen's original set-theoretic technique of forcing to deal with computability concerns. Conceptually...