# Lowness and avoidance

A gentle introduction to iterated jump control

This is a book project 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

**Introduction**
**Prerequisites** March 22, 2024.

*Basic definitions of computability theory, reverse mathematics and forcing.*

## First jump control

**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.*
**Lowness** April 24, 2024

*An effectivization of first-jump control constructions. Existence of low2 solutions for Ramsey's theorem.*
**Compactness avoidance** April 4, 2024

*Another important chapter: PA and DNC avoidance, constant-bound trace avoidance, Liu's theorems. Martin-Löf randomness.*
**Custom properties** August 23, 2024

*How to design custom preservation properties for separating problems when classical notions fail. Separation of EM from RT22, of ADS from CAC, and of CAC from RT22.*
**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

**Jump cone avoidance** July 25, 2024

*Introduction to second-jump control, jump cone avoidance for COH, partition regularity and strong jump cone avoidance of the pigeonhole principle.*
**Jump compactness avoidance**
**Higher jump cone avoidance** October 2, 2024

*Generalization to the levels of the arithmetic hierarchy and the hyperarithmetic hierarchy.*

**Bibliography**