{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,29]],"date-time":"2025-09-29T12:08:41Z","timestamp":1759147721442,"version":"3.41.2"},"reference-count":0,"publisher":"Centre pour la Communication Scientifique Directe (CCSD)","license":[{"start":{"date-parts":[[2022,7,28]],"date-time":"2022-07-28T00:00:00Z","timestamp":1658966400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"abstract":"<jats:p>Inspired by a mathematical riddle involving fuses, we define the \"fusible\nnumbers\" as follows: $0$ is fusible, and whenever $x,y$ are fusible with\n$|y-x|&lt;1$, the number $(x+y+1)\/2$ is also fusible. We prove that the set of\nfusible numbers, ordered by the usual order on $\\mathbb R$, is well-ordered,\nwith order type $\\varepsilon_0$. Furthermore, we prove that the density of the\nfusible numbers along the real line grows at an incredibly fast rate: Letting\n$g(n)$ be the largest gap between consecutive fusible numbers in the interval\n$[n,\\infty)$, we have $g(n)^{-1} \\ge F_{\\varepsilon_0}(n-c)$ for some constant\n$c$, where $F_\\alpha$ denotes the fast-growing hierarchy. Finally, we derive\nsome true statements that can be formulated but not proven in Peano Arithmetic,\nof a different flavor than previously known such statements: PA cannot prove\nthe true statement \"For every natural number $n$ there exists a smallest\nfusible number larger than $n$.\" Also, consider the algorithm \"$M(x)$: if $x&lt;0$\nreturn $-x$, else return $M(x-M(x-1))\/2$.\" Then $M$ terminates on real inputs,\nalthough PA cannot prove the statement \"$M$ terminates on all natural inputs.\"<\/jats:p>","DOI":"10.46298\/lmcs-18(3:6)2022","type":"journal-article","created":{"date-parts":[[2022,7,29]],"date-time":"2022-07-29T13:10:37Z","timestamp":1659100237000},"source":"Crossref","is-referenced-by-count":0,"title":["Fusible numbers and Peano Arithmetic"],"prefix":"10.46298","volume":"Volume 18, Issue 3","author":[{"given":"Jeff","family":"Erickson","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Gabriel","family":"Nivasch","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3789-2319","authenticated-orcid":false,"given":"Junyan","family":"Xu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"25203","published-online":{"date-parts":[[2022,7,28]]},"container-title":["Logical Methods in Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/lmcs.episciences.org\/9850\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/lmcs.episciences.org\/9850\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,6,20]],"date-time":"2023-06-20T20:20:25Z","timestamp":1687292425000},"score":1,"resource":{"primary":{"URL":"https:\/\/lmcs.episciences.org\/8555"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,7,28]]},"references-count":0,"URL":"https:\/\/doi.org\/10.46298\/lmcs-18(3:6)2022","relation":{"has-preprint":[{"id-type":"arxiv","id":"2003.14342v7","asserted-by":"subject"},{"id-type":"arxiv","id":"2003.14342v5","asserted-by":"subject"}],"is-same-as":[{"id-type":"arxiv","id":"2003.14342","asserted-by":"subject"},{"id-type":"doi","id":"10.48550\/arXiv.2003.14342","asserted-by":"subject"}]},"ISSN":["1860-5974"],"issn-type":[{"type":"electronic","value":"1860-5974"}],"subject":[],"published":{"date-parts":[[2022,7,28]]},"article-number":"8555"}}