Patterns of resemblance and Bachmann-Howard fixed points

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. Montalban'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)-comprehension) is reduced to a previous result of the author, which is concerned with relativizations of the Bachmann-Howard ordinal.

Automated summary

This paper proves the equivalence between Timothy Carlson's patterns of resemblance and Pi(1)(1)-comprehension.