Fusible numbers and Peano Arithmetic

@article{Erickson2021FusibleNA,
  title={Fusible numbers and Peano Arithmetic},
  author={Jeff Erickson and Gabriel Nivasch and Junyan Xu},
  journal={2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)},
  year={2021},
  pages={1-13}
}
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 ℝ, is well-ordered, with order type ε0. 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… 

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Generalized fusible numbers and their ordinals
. Erickson defined the fusible numbers as a set F of reals generated by repeated application of the function x ` y ` 1 2 . Erickson, Nivasch, and Xu showed that F is well ordered, with order type ε 0

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