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Martin's conjecture for regressive functions on the hyperarithmetic degrees

Patrick Lutz

Abstract

We answer a question of Slaman and Steel by showing that a version of Martin's conjecture holds for all regressive functions on the hyperarithmetic degrees. A key step in our proof, which may have applications to other cases of Martin's conjecture, consists of showing that we can always reduce to the case of a continuous function.

Martin's conjecture for regressive functions on the hyperarithmetic degrees

Abstract

We answer a question of Slaman and Steel by showing that a version of Martin's conjecture holds for all regressive functions on the hyperarithmetic degrees. A key step in our proof, which may have applications to other cases of Martin's conjecture, consists of showing that we can always reduce to the case of a continuous function.
Paper Structure (12 sections, 18 theorems, 8 equations)

This paper contains 12 sections, 18 theorems, 8 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Background on Hyperarithmetic Reducibility
  4. Background on Martin's Conjecture
  5. Determinacy Lemmas
  6. Pointed Perfect Tree Lemmas
  7. Computable Linear Orders
  8. Proof of the Main Theorem
  9. Replacing $f$ with an injective, computable function
  10. Proving that $f$ preserves $\omega_1^x$
  11. Coding argument
  12. The Case of Borel Functions

Key Result

Theorem 1.1

If $f \colon 2^\omega \to 2^\omega$ is a Turing-invariant function such that $f(x) \le_T x$ for all $x$ then either $f$ is constant on a cone or $f(x) \equiv_T x$ on a cone.

Theorems & Definitions (31)

  • Theorem 1.1: $\mathsf{ZF} + \mathsf{AD}$; Slaman and Steel
  • Theorem 1.2: $\mathsf{ZF} + \mathsf{AD}$
  • Theorem 2.1: $\mathsf{ZF} + \mathsf{AD}$; Martin
  • Conjecture 2.2: Part 1 of Martin's Conjecture
  • Definition 2.3
  • Definition 2.4
  • Definition 2.6
  • Lemma 2.7: $\mathsf{ZF} + \mathsf{AD}$
  • Lemma 2.8: $\mathsf{ZF} + \mathsf{AD}$
  • proof
  • ...and 21 more