episciences.org_8555_20230321085410629 20230321085410629 episciences.org raphael.tournoy+crossrefapi@ccsd.cnrs.fr episciences.org Logical Methods in Computer Science 1860-5974 07 28 2022 Volume 18, Issue 3 Fusible numbers and Peano Arithmetic Jeff Erickson Gabriel Nivasch Junyan Xu https://orcid.org/0000-0002-3789-2319 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." 07 28 2022 8555 https://creativecommons.org/licenses/by/4.0 arXiv:2003.14342 10.48550/arXiv.2003.14342 https://arxiv.org/abs/2003.14342v7 https://arxiv.org/abs/2003.14342v5 10.46298/lmcs-18(3:6)2022 https://lmcs.episciences.org/8555 https://lmcs.episciences.org/9850/pdf https://lmcs.episciences.org/9850/pdf