Patterns of resemblance of order 2

We will investigate patterns of resemblance of order 2 over a family of arithmetic structures on the ordinals. In particular, we will show that they determine a computable well ordering under appropriate assumptions. This paper continues the investigation into the patterns formed by embeddings between initial segments of structures on the ordinals. The goals are twofold: to eventually find an ultrafinestructure for large cardinal axioms based on embeddings and to derive computable well orderings which can be used for proof theoretic analysis. The former goal appears to be some distance away. Significant progress has been made on the latter: even when restricting to embeddings which do not move ordinals the resulting analysis yields notation systems for large ordinals. Patterns of resemblance of order one, which were defined and investigated in Carlson [6,7,8], lead to an ordinal notation system for the theory KP`0 (see Wilken [13,14,15,16]). This paper develops the notatations corresponding to the next level of embedding: patterns of resemblance of order two. While traditional methods of proof theoretic analysis (see [3,4,5,9,10,11,12]) can be used to provide an analysis of KP`0, a calculus for β-logic is being developed that provides a more natural fit with notations based on embeddings. Our study of patterns of resemblance of order 2 in this paper will be analagous to the study of patterns of resemblance of order 1 in Sections 1-6 of [8]. Consider the structure R0 = (ORD, 0,+,≤) on the collection of ordinals ORD, where ≤ is the usual ordering of ordinals, and + is the usual operation