## Abstract

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 ε_{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, ∞), we have g(n)^{−1} ≥ F_{ε0} (n − c) for some constant c, where F_{α} 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.”.

Original language | English (US) |
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Pages (from-to) | 6:1-6:26 |

Journal | Logical Methods in Computer Science |

Volume | 18 |

Issue number | 3 |

DOIs | |

State | Published - 2022 |

## Keywords

- Peano Arithmetic
- fast-growing hierarchy
- ordinal
- well-ordering

## ASJC Scopus subject areas

- Theoretical Computer Science
- General Computer Science