Feedback computability studies oracles that provide convergence and divergence information about computability relative to that same oracle. We consider strict and loose forms of feedback computability relative to Kleene’s 2 E , the Type-2 object embodying quantification over natural numbers. We prove that the sets of integers semi-computable this way are precisely those Σ 1 -definable over L α I and L σ 1 1 , respectively, where α I denotes the least recursively inaccessible ordinal and σ 1 1 the least Σ 1 1 -reflecting ordinal. Furthermore, our construction of higher-type feedback computability through ordinal inductive fix-points serves as a natural and general framework for the study of feedback computability.
In this article, we investigate the degree structures of various hyperspaces. One of our goals is to gain a topological understanding of higher-type computability using negative information, and to this end, we mainly explore the degree structures of hyperspaces whose ground types are endowed with cofinite topology or its relatives.
We show three natural approaches to defining when a subfactor N of a II 1 factor M is oracle computable relative to a presentation of M , namely in terms of the Jones basic construction, Pimsner–Popa bases, and the conditional expectation map, that are equivalent and thereby determine a robust notion of subfactor computability.
The aim of this paper is to search for new subrecursive degrees of representations of irrational numbers which lie outside the cone of Cauchy sequences. The main tool for producing such degrees is a combination of Weihrauch intersections with trace functions from below and from above, without resorting to Dedekind cuts. Further results on disjunctions of representations yield four distinct non-zero degrees that lie strictly subrecursively below the degree of Cauchy sequences.
We explore numbered Boolean algebras over classes Ξ of arithmetical and analytical hierarchies. We show the existence and uniqueness (up to computable isomorphism) of universal Boolean Ξ -algebras, determine the classes in which such algebras exist, and classify the universal algebras up to isomorphism. As applications, we characterize Tarski–Lindenbaum algebras of four semantic classes of models of a given finite rich signature, namely, the class of all countable saturated models having decidable ω -stable theories, the class of prime models with decidable theories and first-order definable elements, the class of models with decidable non-finitely axiomatizable theories, and the class of models with finitely axiomatizable theories; it is shown that the first three of these algebras are universal Boolean algebras over hierarchy classes Σ 1 1 , Σ 2 0 , and Σ 3 0 , respectively, and the fourth one is a countable atomic Boolean Π 3 0 -algebra whose quotient modulo the Frèchet ideal is a Σ 4 0 -universal Boolean Σ 4 0 -algebra.
