# Fusible numbers and Peano Arithmetic

- Published in 2020
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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 $\mathbb R$, 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. 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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### BibTeX entry

@article{FusiblenumbersandPeanoArithmetic, title = {Fusible numbers and Peano Arithmetic}, author = {Jeff Erickson and Gabriel Nivasch and Junyan Xu}, url = {https://arxiv.org/abs/2003.14342v1 https://arxiv.org/pdf/2003.14342v1}, urldate = {2020-04-01}, year = 2020, abstract = {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. For example, PA cannot prove the true statement "For every natural number {\$}n{\$} there exists a smallest fusible number larger than {\$}n{\$}."}, comment = {}, archivePrefix = {arXiv}, eprint = {2003.14342}, primaryClass = {cs.LO}, collections = {Unusual arithmetic} }