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
- 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
How to design custom preservation properties for separating problems.
- Conservation theorems
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
- Jump PA and DNC avoidance