TY - JOUR

T1 - FUSIBLE NUMBERS AND PEANO ARITHMETIC

AU - Erickson, Jeff

AU - Nivasch, Gabriel

AU - Xu, Junyan

N1 - Publisher Copyright:
© J. Erickson, G. Nivasch, and J. Xu.

PY - 2022

Y1 - 2022

N2 - 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 interval [n, ∞), we have g(n)−1 ≥ Fε0 (n − c) for some constant c, where Fα 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.”.

AB - 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 interval [n, ∞), we have g(n)−1 ≥ Fε0 (n − c) for some constant c, where Fα 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.”.

KW - Peano Arithmetic

KW - fast-growing hierarchy

KW - ordinal

KW - well-ordering

UR - http://www.scopus.com/inward/record.url?scp=85135382601&partnerID=8YFLogxK

U2 - 10.46298/LMCS-18(3:6)2022

DO - 10.46298/LMCS-18(3:6)2022

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AN - SCOPUS:85135382601

SN - 1860-5974

VL - 18

SP - 6:1-6:26

JO - Logical Methods in Computer Science

JF - Logical Methods in Computer Science

IS - 3

ER -