Ordinal arithmetic and Sigma(1)-elementarity

We will introduce a partial ordering less than or equal to(1) on the class of ordinals which will serve as a foundation for an approach to ordinal not ations for formal systems of set theory and second-order arithmetic. In thi s paper we use less than or equal to(1) to provide a new characterization o f the ubiquitous ordinal epsilon(0). less than or equal to(1) is the partial ordering on the class of ordinals d efined by induction so that alpha less than or equal to(1)beta iff (alpha, <, less than or equal to(1)) is a Sigma(1)-elementary substructure of (beta, <, less than or equal to(1 )) To be more precise, by induction on beta we define the set of a such that a lpha less than or equal to(1)beta (note that we have taken some liberty in writing (alpha, <, less than or equal to(1)) where we should have restricte d the relations to alpha). Future papers will use less than or equal to(1) along with generalizations to "higher levels" to give proof theoretic analy ses of various formal systems. The original use of less than or equal to(1) was as a tool in confirming Re inhardt's conjecture that the Strong Mechanistic Thesis is consistent with Epistemic Arithmetic (see [I]). Theorem 2.5 of this paper implies the prope rties of less than or equal to(1) needed for that result by giving an effec tive description of less than or equal to(1) in terms of ordinal arithmetic .