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A $Π^0_2$ Singleton of Minimal Arithmetic Degree

Peter Gerdes

TL;DR

This paper demonstrates that some promising approaches inspired by the analogy with the r.e.e sets fail to show that no $\omega$-REA set is arithmetically minimal, and constructs a $\Pi^0_2$ singleton of minimal arithmetic degree.

Abstract

In the study of the arithmetic degrees (the degree structure induced by relative arithmetic definability, ($\leq_{a}$) the $ω$-REA sets play a role analogous to the role the r.e. degrees play in the study of the Turing degrees. However, much less is known about the arithmetic degrees and the role of the $ω$-REA sets in that structure than about the Turing degrees. Indeed, even basic questions such as the existence of a $ω$-REA set of minimal arithmetic degree are open. This paper makes progress on this question by demonstrating that some promising approaches inspired by the analogy with the r.e sets fail to show that no $ω$-REA set is arithmetically minimal. Finally, it constructs a $Π^0_2$ singleton of minimal arithmetic degree. Not only is this a result of considerable interest in it's own right, constructions of $Π^0_2$ singletons often pave the way for constructions of $ω$-REA sets with similar properties. Along the way, a number of interesting results relating arithmetic reducibility and rates of growth are established.

A $Π^0_2$ Singleton of Minimal Arithmetic Degree

TL;DR

This paper demonstrates that some promising approaches inspired by the analogy with the r.e.e sets fail to show that no -REA set is arithmetically minimal, and constructs a singleton of minimal arithmetic degree.

Abstract

In the study of the arithmetic degrees (the degree structure induced by relative arithmetic definability, () the -REA sets play a role analogous to the role the r.e. degrees play in the study of the Turing degrees. However, much less is known about the arithmetic degrees and the role of the -REA sets in that structure than about the Turing degrees. Indeed, even basic questions such as the existence of a -REA set of minimal arithmetic degree are open. This paper makes progress on this question by demonstrating that some promising approaches inspired by the analogy with the r.e sets fail to show that no -REA set is arithmetically minimal. Finally, it constructs a singleton of minimal arithmetic degree. Not only is this a result of considerable interest in it's own right, constructions of singletons often pave the way for constructions of -REA sets with similar properties. Along the way, a number of interesting results relating arithmetic reducibility and rates of growth are established.
Paper Structure (29 sections, 36 theorems, 4 equations)

This paper contains 29 sections, 36 theorems, 4 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Computations, Strings and Degrees
  4. Trees and Forcing
  5. ⁰n Classes and -REA sets
  6. Ordinal Notations
  7. Rates of Growth
  8. Fast Growing Functions and Minimality
  9. Motivation
  10. An Arithmetically Hyperimmune Minimal Degree
  11. Fast Growth and Definability
  12. A Arithmetically Minimal ⁰₂ Singleton
  13. Towers of Trees
  14. Verifying the Main Theorem
  15. Constructing the Tower
  16. ...and 14 more sections

Key Result

Lemma 2.1

Every ${\Pi}^{0}_{1}$ function class is homeomorphic with a ${\Pi}^{0}_{2}$ class via a computable (hence degree preserving) homeomorphism and vice versa. This holds with all possible uniformity.

Theorems & Definitions (81)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 3.1
  • proof
  • Theorem 3.2
  • Corollary 3.3
  • proof
  • Definition 3.4
  • ...and 71 more