Fusible numbers and Peano Arithmetic

Inspired by a mathematical riddle involving fuses, we define the fusible numbers as follows: 0 is fusible, and whenever x, y are fusible with |y − x| < 1, the number (x + y + 1)/2 is also fusible. We prove that the set of fusible numbers, ordered by the usual order on ℝ, is well-ordered, with order type ε0. Furthermore, we prove that the density of the fusible numbers along the real line grows at ...