10.46298/lmcs-18(3:6)2022 https://lmcs.episciences.org/8555 Erickson, Jeff Jeff Erickson Nivasch, Gabriel Gabriel Nivasch Xu, Junyan Junyan Xu 0000-0002-3789-2319 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 $\mathbb R$, is well-ordered, with order type $\varepsilon_0$. 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 $[n,\infty)$, we have $g(n)^{-1} \ge F_{\varepsilon_0}(n-c)$ for some constant $c$, where $F_\alpha$ 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." episciences.org Computer Science - Logic in Computer Science Mathematics - Combinatorics Mathematics - Logic 03F30, 03B70, 03F40 Attribution 4.0 International (CC BY 4.0) 2022-05-23 2022-07-28 2022-07-28 eng journal article arXiv:2003.14342 10.48550/arXiv.2003.14342 1860-5974 https://lmcs.episciences.org/8555/pdf VoR application/pdf Logical Methods in Computer Science Volume 18, Issue 3 Researchers Students