Iterated jump noncomputability and compactness

TL;DR

This paper employs reverse mathematics to analyze the logical strength and relationships of iterated jump versions of principles like AST, DNR, WWKL, and WKL, revealing complex non-linear provability structures.

Contribution

It introduces a detailed reverse mathematics analysis of iterated jump principles, highlighting their intricate relationships and non-linear provability hierarchy.

Findings

Identifies an infinite chain of principles in the provability hierarchy.

Discovers an infinite antichain indicating strong non-linearity.

Maps the logical relationships among iterated jump principles.

Abstract

We use reverse mathematics to analyze "iterated jump" versions of the following four principles: the atomic model theorem with subenumerable types (AST), the diagonally noncomputable principle (DNR), weak weak K\H{o}nig's lemma (WWKL), and weak K\H{o}nig's lemma (WKL). The logical relationships between these principles are summarized in Figure 1 and include, among other things, an infinite chain and an infinite antichain, the latter of which represents a strong form of non-linearity in terms of provability strength among "natural" combinatorial principles.