Fusible numbers and Peano Arithmetic
Jeff Erickson, Gabriel Nivasch, Junyan Xu · 2021
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≥ Fε0(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."