Elementary patterns of resemblance

Abstract

We will study patterns which occur when considering how Σ 1 -elementary substructures arise within hierarchies of structures. The order in which such patterns evolve will be seen to be independent of the hierarchy of structures provided the hierarchy satisfies some mild conditions. These patterns form the lowest level of what we call patterns of resemblance. They were originally used by the author to verify a conjecture of W. Reinhardt concerning epistemic theories (see Carlson, Arch. Math. Logic 38 (1999) 449–460; Ann. Pure Appl. Logic, to appear), but their relationship to axioms of infinity and usefulness for ordinal analysis were manifest from the beginning. This paper is the first part of a series which provides an introduction to an extensive program including the ordinal analysis of set theories. Future papers will conclude the introduction and establish, among other things, that notations we will derive from the patterns considered here represent the proof-theoretic ordinal of the theory KPℓ 0 or, equivalently, Π 1 1−CA 0 (as KPℓ 0 is a conservative extension of Π 1 1−CA 0 ).