Priority Arguments and Epsilon Substitutions

While the priority argument has been one of the main techniques of recursion theory, it has seen only a few applications to other areas of mathematics [Mar75, Sol84]. One possibility for another such application was pointed out by Kreisel: Hilbert’s ǫsubstitution method, a technique for proving the 1-consistency of theories. Kreisel’s observation was that the proof that the method works [Ack40] bears a striking resemblence to the structure of a traditional finite injury priority argument. Such a connection might have benefits for both fields. The ǫ-substitution method has powerful extensions [Ara05b, Ara05a, Ara06] which might provide new tools for solving difficult recursion theoretic problems. In the other direction, the most popular proof theoretic technique for proving 1-consistency results, cut-elimination, has bogged down in technical details, and new ideas are neeed to make ordinal analytic results more accessible. Unfortunately, Kreisel’s observation has been difficult to turn into a concrete argument. After Yang [Yan95], the reason is clear: the success of all finite injury priority arguments is exactly enough to prove the 1-consistency of the weak theory IΣ1, and therefore finite injury arguments cannot be sufficient to prove the consistency of stronger theories. Using a general framework for priority arguments developed by Lerman and Lempp [LL90, LL92, LL97], Yang goes on to show that arguments on the n-th level of their hierarchy of priority arguments are equivalent to the 1-consistency of IΣn, and so it requires the full ω levels of that hierarchy to give 1-consistency for all of first-order arithmetic. The better known infinite injury and monster injury priority arguments belong to the second and third levels of this hierarchy, and, as the name “monster” suggests, going to higher levels becomes impractical without some kind of general framework. The Lerman-Lempp framework is one of several that have been proposed [Ash86, Ash90, Kni90, GS]. One technique, usually described using “workers on many levels,” originally developed by Harrington, has been extended to hyperarithmetic levels. We show in this paper that, if one is prepared to use one of these frameworks to describe the necessary priority argument, that the ǫ-substitution method can be proven to work using a priority argument. We follow Yang in using the Lerman-Lempp framework, although we know of no reason that other frameworks would not work just as well.