Lowness and avoidance

A gentle introduction to iterated jump control

This is a project book about iterated jump control of combinatorial theorems. I plan to add chapters over time, with no commitment to finish it one day.

Table of contents

  1. Introduction
  2. Prerequisites March 22, 2024.
    Basic definitions of computability theory, reverse mathematics and forcing.

First jump control

  1. Cone avoidance March 19, 2024
    A few well-known theorems about cone avoidance, with an emphasis on the forcing question. Cone avoidance basis theorem, Seetapun's theorem, equivalence with other notions of preservations.
  2. Lowness April 24, 2024
    An effectivization of first-jump control constructions. Existence of low2 solutions for Ramsey's theorem.
  3. Compactness avoidance April 4, 2024
    Another important chapter: PA and DNC avoidance, constant-bound trace avoidance, Liu's theorems. Martin-Löf randomness.
  4. Custom properties
    How to design custom preservation properties for separating problems.
  5. Conservation theorems May 30, 2024
    Conservation theorems also use properties of the forcing question to propagate theories from the ground model to the extended model.

Higher jump control

  1. Jump cone avoidance
  2. Jump PA and DNC avoidance