# Fusible numbers and Peano Arithmetic

### Wednesday, June 10th, 2020, 16:10

~~Checkpoint 480~~

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### Fusible numbers and Peano Arithmetic

### Gabriel Nivasch

### Abstract:

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 R, with order type eps_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_{eps_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.