The Proof-theoretic Strength of the Dushnik-miller Theorem for Countable Linear Orders

We show that the Dushnik-Miller Theorem for countable linear order-ings (stating that any countable linear ordering has a nontrivial self-embedding) is equivalent (over recursive comprehension (RCA 0)) to arithmetic comprehension (ACA 0). This paper presents a result in reverse mathematics, a program initiated by H. Friedman and S. Simpson, trying to determine the weakest possible \set-theoretical" axiom (system) to prove a given theorem of \ordinary" mathematics by trying to prove the axiom from the theorem (over a weaker \base system"). The \set-theoretical" axiom systems we will be concerned with are weak subsystems of second-order arithmetic. (We refer to Simpson Sita] for a detailed exposition of such systems.) In particular, we will use the axiom system RCA 0 of recursive comprehension (with 0 1-induction) as a base system and exhibit a theorem which is equivalent (over the base system RCA 0) to the axiom system ACA 0 of arithmetic comprehension (with 0 1-induction). The area of \ordinary" mathematics we study in the context of reverse mathematics is that of linear orderings. In particular, we will characterize the proof-theoretic strength of the 1991 Mathematics Subject Classiication. 03F35, 06A05.