Intrinsic Smallness

TL;DR

This paper explores the concept of intrinsic smallness in computability theory, focusing on sets with intrinsic density zero and their relationship to other notions of computational smallness.

Contribution

It introduces the study of intrinsically small sets, analyzes their preservation under computable functions, and examines their independence from hyperimmunity.

Findings

Intrinsic smallness is preserved by certain computable functions.

Intrinsic smallness and hyperimmunity are computationally independent.

Hyperimmune degrees contain sets that are not intrinsically small.

Abstract

Recent work in computability theory has focused on various notions of asymptotic computability, which capture the idea of a set being "almost computable." One potentially upsetting result is that all four notions of asymptotic computability admit "almost computable" sets in every Turing degree via coding tricks, contradicting the notion that "almost computable" sets should be computationally close to the computable sets. In response, Astor introduced the notion of intrinsic density: a set has defined intrinsic density if its image under any computable permutation has the same asymptotic density. Furthermore, Astor also introduced various notions of intrinsic computation in which the standard coding tricks cannot be used to embed intrinsically computable sets in every Turing degree. Our goal is to study the sets which are intrinsically small, i.e. those that have intrinsic density zero. We begin by studying which computable functions preserve intrinsic smallness. We also show that intrinsic smallness and hyperimmunity are computationally independent notions of smallness, i.e. any hyperimmune degree contains a Turing-equivalent hyperimmune set which is "as large as possible" and therefore not intrinsically small. Our discussion concludes by relativizing the notion of intrinsic smallness and discussing intrinsic computability as it relates to our study of intrinsic smallness.