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 an incredibly fast rate: Letting g(n) be the largest gap between consecutive fusible numbers in the interval [n, ∞), we have g(n)⁻¹ ≥ F_ε0 (n − c) for some constant c, where F_α 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.”.