{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,9]],"date-time":"2026-01-09T06:53:52Z","timestamp":1767941632765,"version":"3.49.0"},"reference-count":12,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2026,1,7]],"date-time":"2026-01-07T00:00:00Z","timestamp":1767744000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100000038","name":"Natural Sciences and Engineering Research Council of Canada","doi-asserted-by":"crossref","award":["502710N"],"award-info":[{"award-number":["502710N"]}],"id":[{"id":"10.13039\/501100000038","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Kolmogorov complexity asks whether a string can be outputted by a Turing Machine (TM) whose description is shorter. Analogously, a real number is considered computable if a Turing machine can generate its decimal expansion. The modern \u03f5-approximation definition of computability, widely used in practical computation, ensures that computable reals are constructively closed under addition. However, Turing\u2019s original 1936 digit-by-digit notion, which demands the direct output of the n-th digit, presents a stark divergence. Though the set of Turing-computable reals is not constructively closed under addition, we prove that a Turing machine capable of computing x+y non-constructively exists. The core constructive computational barrier arises from determining the ones digit of a sum like 0.333\u00af+0.666\u00af=0.999\u00af=1.000\u00af. This particular example is ambiguous because both 0.999\u00af and 1.000\u00af are legitimate decimal representations of the same number. However, if any of the infinite number of 3s in the first term is changed to a 2 (e.g., 0.33\u202632\u2026+0.666\u00af), the sum\u2019s leading digit is definitely zero. Conversely, if it is changed to a 4 (e.g., 0.33\u202634\u2026+0.666\u00af), the leading digit is definitely one. This implies an inherent undecidability in determining these digits. Recent papers and our work address this issue. Hamkins provides an informal argument, while Berthelette et al. present more complicated formal proof, and our contribution offers a simple reduction to the Halting Problem. We demonstrate that determining when carry propagation stops can be resolved with a single query to an oracle that tells if and when a given TM halts. Because a concrete answer to this query exists, so does a TM computing the digits of x+y, though the proof is non-constructive. As far as we know, the analogous question for multiplication remains open. This, we feel, is an interesting addition to the story. This reveals a subtle but significant difference between the modern \u03f5-approximation definition and Turing\u2019s original 1936 digit-by-digit notion of a computable number, as well as between constructive and non-constructive proof. This issue of computability and numerical precision ties into algorithmic information and Kolmogorov complexity.<\/jats:p>","DOI":"10.3390\/e28010071","type":"journal-article","created":{"date-parts":[[2026,1,7]],"date-time":"2026-01-07T08:26:56Z","timestamp":1767774416000},"page":"71","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Why Turing\u2019s Computable Numbers Are Only Non-Constructively Closed Under Addition"],"prefix":"10.3390","volume":"28","author":[{"given":"Jeff","family":"Edmonds","sequence":"first","affiliation":[{"name":"Department of Electrical Engineering and Computer Science, York University, Toronto, ON M3J 1P3, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2026,1,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1016\/0003-4843(79)90021-4","article-title":"A computable ordinary differential equation which possesses no computable solution","volume":"17","author":"Richards","year":"1979","journal-title":"Ann. Math. Log."},{"key":"ref_2","unstructured":"Hamkins, J.D. (2025, June 15). Alan Turing, On Computable Numbers. Available online: https:\/\/jdh.hamkins.org\/alan-turing-on-computable-numbers\/."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"114573","DOI":"10.1016\/j.tcs.2024.114573","article-title":"On computable numbers, with an application to the Druckproblem","volume":"1002","author":"Berthelette","year":"2024","journal-title":"Theor. Comput. Sci."},{"key":"ref_4","first-page":"433","article-title":"Le calcul des nombres","volume":"14","author":"Borel","year":"1912","journal-title":"Revue du Mois"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"230","DOI":"10.1112\/plms\/s2-42.1.230","article-title":"On Computable Numbers, with an Application to the Entscheidungsproblem","volume":"s2-42","author":"Turing","year":"1936","journal-title":"Proc. Lond. Math. Soc."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"145","DOI":"10.2307\/2267043","article-title":"Nicht konstruktiv beweisbare S\u00e4tze der Analysis","volume":"14","author":"Specker","year":"1949","journal-title":"J. Symb. Log."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"514","DOI":"10.2307\/2271358","article-title":"Some undecidable problems involving elementary functions of a real variable","volume":"33","author":"Richardson","year":"1968","journal-title":"J. Symb. Log."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Pour-El, M.B., and Richards, J.I. (1989). Computability in Analysis and Physics, Springer.","DOI":"10.1007\/978-3-662-21717-7"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Weihrauch, K. (2000). Computable Analysis: An Introduction, Springer.","DOI":"10.1007\/978-3-642-56999-9"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1145\/103162.103163","article-title":"What every computer scientist should know about floating-point arithmetic","volume":"23","author":"Goldberg","year":"1991","journal-title":"ACM Comput. Surv."},{"key":"ref_11","first-page":"5","article-title":"On the computability of real functions","volume":"37","author":"Braverman","year":"2006","journal-title":"SIGACT News"},{"key":"ref_12","unstructured":"Kleene, S.C. (1952). Introduction to Metamathematics, Van Nostrand."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/1\/71\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,9]],"date-time":"2026-01-09T05:16:12Z","timestamp":1767935772000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/1\/71"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,1,7]]},"references-count":12,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2026,1]]}},"alternative-id":["e28010071"],"URL":"https:\/\/doi.org\/10.3390\/e28010071","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,1,7]]}}}