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 $|y-x|<1$, number $(x+y+1)/2$ also fusible. We prove that set of numbers, ordered usual order on $\mathbb R$, well-ordered, type $\varepsilon_0$. Furthermore, density numbers along real line grows at an incredibly fast rate: Letting $g(n)$ be largest gap between consecutive in interval $[n,\infty)$, have $g(n)^{-1} \ge F_{\varepsilon_0}(n-c)$ for some constant $c$, where $F_\alpha$ denotes fast-growing hierarchy. Finally, derive true statements can formulated but not proven Peano Arithmetic, different flavor than previously known such statements: PA cannot statement "For every natural $n$ there exists smallest larger $n$." Also, consider algorithm "$M(x)$: if $x<0$ return $-x$, else $M(x-M(x-1))/2$." Then $M$ terminates inputs, although "$M$ all inputs."