Showing papers on "Hierarchy (mathematics) published in 1971"

The next admissible set

Abstract: In this paper we describe generalizations of several approaches to the hyperarithmetic hierarchy, show how they are related to the Kripke-Platek theory of admissible ordinals and sets, and study conditions under which the various approaches remain equivalent.To put matters in some perspective, let us first review various approaches to the theory of hyperarithmetic sets. For most purposes, it is convenient to first define the semi-hyperarithmetic (semi-HA) subsets of N. A set is then said to be hyperarithmetic (HA) if both it and its complement are semi-HA. A total number-theoretic function is HA if its graph is HA.

An Intrinsic Characterization of the Hierarchy of Constructible Sets of Integers1)

Abstract: Publisher Summary This chapter provides an intrinsic characterization of the hierarchy of constructible sets of integers. Some of the results are in second-order number theory, and the others are in set theory; standard (incompatible) notations for each are used. The hierarchy of arithmetical degrees is identical with the ramified analytic hierarchy; it is identical with the hyperarithmetic hierarchy where that hierarchy is defined - taking into account the difference between Turing degrees and arithmetical degrees. The division into cases is motivated by the relativized hyperarithmetical hierarchy. Forcing for unlimited statements is defined in terms of forcing for limited statements, and by induction on complexity.

A Countable Hierarchy for the Superjump

Abstract: Publisher Summary This chapter discusses the countable hierarchy for the superjump. The notion of a recursive functional of finite type was introduced and the usefulness of this concept was illustrated by showing that the hyperarithmetical sets of integers were exactly the sets recursive in the type 2 object 2 E . There can be no countable hierarchy that generates the class of sets of integers recursive in a type 3 object that is at least as strong as 3 E . The chapter reviews Shoenfield's construction and discusses why it works. The superjump is described and the modified countable computation is given, which makes use of Shoenfield's hierarchy. The chapter presents a countable hierarchy of type 2 jumps, and uses its constructed hierarchy to derive some results about recursiveness.

Error and Reject Rates in a Hierarchical Pattern Recognizer

Abstract: The reject and error rates of a certain hierarchical decision structure are derived under assumptions of statistical independence among the members of the hierarchy. It is shown that high decision reliability can be obtained with much more relaxed requirements on the individual recognizers in the hierarchy.