Lowness and avoidance

A gentle introduction to iterated jump control

This is a book project about iterated jump control of combinatorial theorems. The current version of the draft is available here

Table of contents

1. Introduction November 29, 2024.
   From reverse mathematics to iterated jump control.
2. Prerequisites March 22, 2024.
   Basic definitions of computability theory, reverse mathematics and forcing.

First jump control

3. 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.
4. Lowness April 24, 2024
   An effectivization of first-jump control constructions. Existence of low2 solutions for Ramsey's theorem.
5. Compactness avoidance April 4, 2024
   Another important chapter: PA and DNC avoidance, constant-bound trace avoidance, Liu's theorems. Martin-Löf randomness.
6. 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.
7. 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.
8. Forcing design October 21, 2024
   How to design a notion of forcing with a good first-jump control, given a combinatorial theorem. Examples with the Erdos-Moser and free set theorems.

Higher jump control

9. 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.
10. Jump compactness avoidance November 15, 2024
    Product partition regularity and Monin and Patey separation of SRT22 from COH.
11. Higher jump cone avoidance October 2, 2024
    Generalization to the levels of the arithmetic hierarchy and the hyperarithmetic hierarchy.

12. Bibliography