Elementary patterns of resemblance

We will study patterns which occur when considering how Sigma (1)-elementar y substructures arise within hierarchies of structures. The order in which such patterns evolve will be seen to be independent of the hierarchy of str uctures provided the hierarchy satisfies some mild conditions. These patter ns form the lowest level of what we call patterns of resemblance e. They we re originally used by the author to verify a conjecture of W. Reinhardt con cerning epistemic theories (see Carlson, Arch. Math. Logic 38 (1999) 449-46 0; Ann. Pure Appl. Logic, to appear), but their relationship to axioms of i nfinity and usefulness for ordinal analysis were manifest from the beginnin g. This paper is the first part of a series which provides an introduction to an extensive program including the ordinal analysis of set theories. Fut ure 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, Pi (1)( 1)-CA(0) (as KPl(0) is a conservative extension of Pi (1)(1)-CA(0)). (C) 20 01 Elsevier Science B.V. All rights reserved.