Tracking Chains Revisited

Abstract

The structure C2 := (1 ∞ , ≤, ≤1, ≤2), introduced and first analyzed in [5], is shown to be elementary recursive. Here, 1 ∞ denotes the proof-theoretic ordinal of the fragment Π 1 1 -CA0 of second order number theory, or equivalently the set theory KPℓ0, which axiomatizes limits of models of Kripke-Platek set theory with infinity. The partial orderings ≤1 and ≤2 denote the relations of Σ1-and Σ2-elementary substructure, respectively. In a subsequent article [11] we will show that the structure C2 comprises the core of the structure R2 of pure elementary patterns of resemblance of order 2. In [5] the stage has been set by showing that the least ordinal containing a cover of each pure pattern of order 2 is 1 ∞ . However, it is not obvious from [5] that C2 is an elementary recursive structure. This is shown here through a considerable disentanglement in the description of connectivity components of ≤1 and ≤2. The key to and starting point of our analysis is the apparatus of ordinal arithmetic developed in [7] and in Section 5 of [4], which was enhanced in [5], specifically for the analysis of C2.