Seminar 36. 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 $n$." Joint work with Jeff Erickson and Junyan Xu.
We will also briefly mention some recent progress on generalizations of fusible numbers, joint work with Alexander Bufetov and Fedor Pakhomov.
(This meeting of the seminar is joint with the Workshop "Fusible numbers" by CIRM and ERC ICHAOS project.)