Force a set model of $Z_3$ + Harrington's Principle

TL;DR

The paper proves that, under the assumption of a remarkable cardinal with a weakly inaccessible above it, one can force a model of $Z_3 + {\sf HP}$ via set forcing without reshaping, establishing the consistency of Harrington's Principle at the level of third-order arithmetic. The construction proceeds in four steps: (1) force to obtain a club of $L$-cardinals with strong reflecting properties, (2) build intermediate parameters $B_0,A$ ensuring a stationary set $S$ with reflecting features, (3) shoot a club through $S$ using Baumgartner forcing to align $Lim(C)$ with $L$-cardinals, and (4) code a real via almost disjoint forcing to witness HP in the final model. The result shows $Z_3 + {\sf HP}$ does not imply $0^{\sharp}$ exists, while $Z_4 + {\sf HP}$ does, clarifying the boundary between HP and $0^{\sharp}$ in higher-order arithmetic. The methods hinge on the interplay of large-cardinal reflecting properties, controlled forcing (including Harrington and Baumgartner forcings), and almost disjoint coding to produce the witness real, all within a reshaping-free framework. The work also highlights that the needed strong reflecting properties can be achieved from a relatively weakly stronger large-cardinal assumption, informing the relative strength of HP across higher-order theories.

Abstract

Let $Z_3$ denote $3^{rd}$ order arithmetic. Let Harrington's Principle, HP, denote the statement that there is a real $x$ such that every $x$--admissible ordinal is a cardinal in $L$. In this paper, assuming there exists a remarkable cardinal with a weakly inaccessible cardinal above it, we force a set model of $Z_3\, + \, {\sf HP}$ via set forcing without reshaping.

Theorems & Definitions (62)

• Theorem 1.1
• Definition 1.2
• Theorem 1.3
• Definition 1.4
• Definition 2.1
• Lemma 2.2
• Lemma 2.3
• proof
• Definition 2.4
• Definition 2.5