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 $g{(n)^{ - 1}} geq {F_{{arepsilon _0}}}(n - c)$ 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."
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机译：灵感来自涉及熔断器的数学谜语，我们定义了令人震惊的数字，如下所示：0是可熔，无论x，y都是可融合的吗 - x | <1，数字（x + y + 1）/ 2也是可熔。我们证明，通过常规订单订购的令人融合号码的集合是齐全的，订单类型ε 0 。此外，我们证明了沿着实线的熔断数的密度以令人难以置信的快速增长：让G（n）是间隔中连续熔数之间的最大差距[n，∞），我们有$ g {（ n）^ { - 1}} geq {f _ {{ varepsilon _0}}（n - c）$ for for f and contand c，其中fα 表示快速增长的层次结构。最后，我们派生了一些真正的陈述，可以在PEANO算术中制定但不能证明，不同的味道而不是先前已知的这种陈述：PA不能证明存在的真实陈述“为每个自然数量存在最小的羽毛数大于n。“此外，考虑算法“m（x）：如果x <0 return-x，则返回m（x - m（x - 1））/ 2。”然后M终止实际输入，尽管PA不能证明语句“M终止对所有自然输入。”
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