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.