Long games on braids

This note is an exposition of a connection between the well-ordering of braids and ramseyan unprovabe statements PH 2 and PH 3 , the ParisHarrigton principles for pairs and triples. 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 a combinatorial 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 I 2, 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 insipred by Dehornoy’s well-ordering of positive braids of order-type ! ! ! . The n-strand braid group Bn is a group with the following presentation: Bn = h 1,..., n 1; i j = j i for |i j| 2, i j i = j i j for |i j| = 1i.