Patterns of resemblance and Bachmann-Howard fixed points

TL;DR

This paper proves a conjecture linking patterns of resemblance based on $ ext{Sigma}_1$-elementarity to $ ext{Pi}^1_1$-comprehension, using relativizations of the Bachmann-Howard ordinal.

Contribution

It establishes the equivalence between relativized patterns of resemblance and $ ext{Pi}^1_1$-comprehension, confirming a conjecture in reverse mathematics.

Findings

Proves the conjecture relating patterns of resemblance to $ ext{Pi}^1_1$-comprehension.

Reduces the main direction to a previous result on relativizations of the Bachmann-Howard ordinal.

Connects large computable ordinals with logical comprehension principles.

Abstract

Timothy Carlson's patterns of resemblance employ the notion of $\Sigma_1$-elementarity to describe large computable ordinals. It has been conjectured that a relativization of these patterns to dilators leads to an equivalence with $\Pi^1_1$-comprehension (Question 27 of A. Montalb\'an's "Open questions in reverse mathematics", Bull. Symb. Log. 17(3)2011, 431-454). In the present paper we prove this conjecture. The crucial direction (towards $\Pi^1_1$-comprehension) is reduced to a previous result of the author, which is concerned with relativizations of the Bachmann-Howard ordinal.