Fusible numbers and Peano Arithmetic

Abstract

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 an incredibly fast rate: Letting g(n) be the largest gap between consecutive fusible numbers in the interval [n, ∞), we have $g{(n)^{ - 1}} \geq {F_{{\varepsilon _0}}}(n - c)$ for some constant c, where Fα denotes the fast-growing hierarchy.Finally, we derive some true statements that can be formulated but not proven in Peano Arithmetic, of a different flavor than previously known such statements: PA cannot prove the true statement "For every natural number n there exists a smallest fusible number larger than n." Also, consider the algorithm "M(x): if x < 0 return −x, else return M(x − M(x − 1))/2." Then M terminates on real inputs, although PA cannot prove the statement "M terminates on all natural inputs."