WQO and BQO theory in subsystems of second order arithmetic

Abstract

We consider the reverse mathematics of wqo and bqo theory. We survey the literature on the subject, which deals mainly with the more advanced results about wqos and bqos, and prove some new results about the elementary properties of these combinatorial structures. We state several open problems about the axiomatic strength of both elementary and advanced results. A quasi-ordering (i.e. a reflexive and transitive binary relation) is a wqo (well quasi-ordering) if it contains no infinite descending chains and no infinite sets of pairwise incomparable elements. This concept is very natural, and has been introduced several times, as documented in (19). The usual working definition of wqo is obtained from the one given above with an application of Ramsey's theorem: a quasi-orderingon the set Q is wqo if for every sequence {xn | n ∈ N} of elements of Q there exist m<n such that xmxn. The notion of bqo (better quasi-ordering) is a strengthening of wqo which was introduced by Nash-Williams in the 1960's in a sequence of papers culminating in (30) and (31). This notion has proved to be very useful in showing that specific quasi- orderings are indeed wqo. Moreover the property of being bqo is preserved by a much wider class of operations than those that preserve the property of being wqo, the general pattern being that when wqos are closed under a finitary operation, bqos are closed under its infinitary generalization (see e.g. Higman's and Nash-Williams' theorems in Section 2). (28) is a survey of wqo and bqo theory, while (38) (see (6) for a simplification in that approach) and (32) are alternative introductions to bqo theory. We postpone the precise (and rather technical) definition of bqo to Section 1. Wqo and bqo theory represents an area of combinatorics which has always interested logicians. From the viewpoint of reverse mathematics ((42) is the basic reference on the subject) one of the reasons for this interest stems from the fact that these theories appear to use axioms that are within the realm of second order arithmetic, yet are much stronger than those necessary to develop other areas of ordinary mathematics (as defined in the introduction of (42)).