Patterns of resemblance and Bachmann-Howard fixed points

Abstract

Timothy Carlson’s patterns of resemblance employ the notion of $$\Sigma _1$$ Σ 1 -elementarity to describe large computable ordinals. It has been conjectured that a relativization of these patterns to dilators leads to an equivalence with $$\Pi ^1_1$$ Π 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 of the equivalence (towards $$\Pi ^1_1$$ Π 1 1 -comprehension) is reduced to a previous result of the author, which is concerned with relativizations of the Bachmann-Howard ordinal.