Fusible numbers and Peano Arithmetic

Inspired by a mathematical riddle involving fuses, we define a set of rational numbers which we call "fusible numbers". We prove that the set of fusible numbers is well-ordered in $\mathbb{R}$, with order type $\varepsilon_0$. We prove that the density of the fusible numbers along the real line grows at an incredibly fast rate, namely at least like the function $F_{\varepsilon_0}$ of 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, "For every natural number $n$ there exists a smallest fusible number larger than $n$."