Tracking chains of Σ2Σ2-elementarity

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 Σ2Σ2-elementarity within the structure R2R2 is given in [5], this article is independent of [5] and provides a complete arithmetical analysis of the structure R2R2 below the least ordinal αα such that any pure pattern of order 2 has a covering below αα. αα is shown to be the proof-theoretic ordinal of KPℓ0.

► The ordinal αα of pure patterns of resemblance of order 2 is determined.
► αα is shown to be the proof-theoretic ordinal of KPℓ0.
► Σ1Σ1 and Σ2Σ2 elementarity are characterized arithmetically below αα.
► Arithmetical means are developed in terms of notations familiar in proof theory.