Long games on positive braids

We introduce two games played on positive braids. The first game is played on 3-strand positive braids and its termination time is Ackermannian in the input. We give an ordinal-theoretic and a model-theoretic proof (using semi-regular cuts in models of arithmetic). The second game is played on arbitrary positive braids and its termination is unprovable in IΣ2, the two-quantifier induction arithmetic. We provide proofs of the result using ordinals and the method of indicators (building a 2-extendible cut in a model of arithmetic). The results were inspired by Dehornoy’s well-ordering of positive braids of order-type ωω ω . The n-strand braid group Bn is a group with the following presentation: Bn = 〈σ1, . . . , σn−1;σiσj = σjσi for |i−j| ≥ 2, σiσjσi = σjσiσj for |i−j| = 1〉. A braid is called positive if it has a representation without σ i for any i. We denote the set of positive n-strand braids by B n . Many phenomena about braids were explained by the discovery by Dehornoy in 1992 of a left-invariant linear ordering of positive braids (see [8]). Laver later showed that Dehornoy’s ordering is a well-ordering when rescricted to positive braids [17]. Burckel showed that the order-type is ω ω (the reason for this ordertype of braids essentially coming from Higman’s Lemma). This well-ordering led us to our unprovable statements about braids in this article. For each of our theorems we provide two proofs: model-theoretic and ordinaltheoretic to cater for mathematicians’ different tastes and intuitions. The model-theoretic part of the article was inspired by the treatment of α-large sets by Paris in [18], the ordinal-theoretic side follows the approach uses standard methods such as fundamental sequences and fast-growing hiearchies. Many experts will see connections with Paris-Kirby Hydra Games from [16], Beklemishev’s Worm Principle [3, 7] and reductions of α-large sets. At the dawn of logic, early logicians, most notably K. Gödel and R. Goodstein wrote about ordinal descent through ω ω with justification why this is an acceptable mathematical principle. This ordinal was perceived as the biggest ordinal that allows for a convincing verbal “justification” of why the corresponding ordinal descent principle is true. The fact that all existing concrete mathematics can be conducted in IΣ2 can be seen as one reason why the well-ordering of positive braids looked so mysterious. The bijection between braids and ordinals can be done constructively but the proof of well-orderedness necessarily requires means stronger than those formalizable in IΣ2. Our first game terminates in ∗The authors were supported by grant number 613080000 of the NWO to participate in a workshop in Utrecht University in July 2006.