Fusible numbers and Peano Arithmetic
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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 ε_0. 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_ε_0(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. For example, PA cannot prove the true statement "For every natural number n there exists a smallest fusible number larger than n."
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