eng
episciences.org
Logical Methods in Computer Science
1860-5974
2022-07-28
Volume 18, Issue 3
10.46298/lmcs-18(3:6)2022
8555
journal article
Fusible numbers and Peano Arithmetic
Jeff Erickson
Gabriel Nivasch
Junyan Xu
https://orcid.org/0000-0002-3789-2319
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 $\mathbb R$, is well-ordered,
with order type $\varepsilon_0$. 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
$[n,\infty)$, we have $g(n)^{-1} \ge F_{\varepsilon_0}(n-c)$ for some constant
$c$, where $F_\alpha$ 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."
https://lmcs.episciences.org/8555/pdf
Computer Science - Logic in Computer Science
Mathematics - Combinatorics
Mathematics - Logic
03F30, 03B70, 03F40