Tracking chains of Σ2-elementarity☆

Abstract We apply the ordinal arithmetical tools that were developed in Wilken (2007) [10] and Carlson and Wilken (in press)  [4] in order to introduce tracking chains as the crucial means in the arithmetical analysis of (pure) elementary patterns of resemblance of order 2; see Carlson (2001) [2] , Carlson (2009) [3] , and Carlson and Wilken (in preparation)  [5] . Although underlying heuristics for an analysis of Σ 2 -elementarity within the structure R 2 is given in  [5] , this article is independent of  [5] and provides a complete arithmetical analysis of the structure R 2 below the least ordinal α such that any pure pattern of order 2 has a covering below α . α is shown to be the proof-theoretic ordinal of KP l 0 .