Patterns of resemblance and Bachmann-Howard fixed points

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. Montalbán'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.