Interesting Esoterica

Fusible numbers and Peano Arithmetic

Article by Jeff Erickson and Gabriel Nivasch and Junyan Xu
  • Published in 2020
  • Added on
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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. For example, PA cannot prove the true statement "For every natural number $n$ there exists a smallest fusible number larger than $n$."

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BibTeX entry

@article{FusiblenumbersandPeanoArithmetic,
	title = {Fusible numbers and Peano Arithmetic},
	author = {Jeff Erickson and Gabriel Nivasch and Junyan Xu},
	url = {https://arxiv.org/abs/2003.14342v1 https://arxiv.org/pdf/2003.14342v1},
	urldate = {2020-04-01},
	year = 2020,
	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 {\$}\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. For example, PA
cannot prove the true statement "For every natural number {\$}n{\$} there exists a
smallest fusible number larger than {\$}n{\$}."},
	comment = {},
	archivePrefix = {arXiv},
	eprint = {2003.14342},
	primaryClass = {cs.LO},
	collections = {Unusual arithmetic}
}