\Lambda \Lambda

We say that a linear ordering L is extendible if every partial ordering that does not embed L can be extended to a linear ordering which does not embed L either. Jullien’s theorem is a complete classification of the countable extendible linear orderings. Fraïssé’s conjecture, which is actually a theorem, is the statement that says that the class of countable linear ordering, quasiordered by the relation of embeddablity, contains no infinite descending chain and no infinite antichain. In this paper we study the strength of these two theorems from the viewpoint of Reverse Mathematics and Effective Mathematics. As a result of our analysis we get that they are equivalent over the basic system of RCA0+Σ1 1-IND. We also prove that Fraïssé’s conjecture is equivalent, over RCA0, to two other interesting statements. One that says that the class of well founded labeled trees, with labels from {+, −}, and with a very natural order relation, is well quasiordered. The other statement says that every linear ordering which does not contain a copy of the rationals is equimorphic to a finite sum of indecomposable linear orderings. While studying the proof theoretic strength of Jullien’s theorem, we prove the extendibility of many linear orderings, including ω2 and η, using just ATR0+Σ1 1-IND. Moreover, for all these linear orderings, L, we prove that any partial ordering, P, which does not embed L has a linearization, hyperarithmetic (or equivalently ∆1 1) in P ⊕ L, which does not embed L.

Abstract. We answer a long-standing question of Rosenstein by exhibiting a complete theory of linear orderings with both a computable model and a prime model, but no computable prime model. The proof uses the relativized version of the concept of limitwise monotonic function. A linear ordering is computable if both its domain and its order relation are computable; it is computably presentable if it is isomorphic to a computable linear ordering. (There are natural generalizations of these notions to other kinds of structures; for instance, see [10] for details.) There is a large body of research on computable linear orderings ([4] gives an extensive overview). Much of this work has been focused on the relationship between classical and effective order types, but it is also interesting to take an approach inspired by classical model theory and study the relationship between effective order types and theories of linear orderings, asking, for instance, what kinds of computable linear orderings exist within the models of a given theory of linear orderings. Taking this approach, Rosenstein ended his book Linear Orderings [12] by asking