{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,6]],"date-time":"2025-11-06T20:05:22Z","timestamp":1762459522329},"reference-count":14,"publisher":"Wiley","issue":"3-4","license":[{"start":{"date-parts":[[2018,10,30]],"date-time":"2018-10-30T00:00:00Z","timestamp":1540857600000},"content-version":"am","delay-in-days":365,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#am"},{"start":{"date-parts":[[2017,10,30]],"date-time":"2017-10-30T00:00:00Z","timestamp":1509321600000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"funder":[{"DOI":"10.13039\/100000121","name":"Division of Mathematical Sciences","doi-asserted-by":"publisher","award":["0943832","1101373"],"award-info":[{"award-number":["0943832","1101373"]}],"id":[{"id":"10.13039\/100000121","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematical Logic Qtrly"],"published-print":{"date-parts":[[2017,11]]},"abstract":"Abstract<\/jats:title>There are two well\u2010known ways of doing arithmetic with ordinal numbers: the \u201cordinary\u201d addition, multiplication, and exponentiation, which are defined by transfinite iteration; and the \u201cnatural\u201d (or \u201cHessenberg\u201d) addition and multiplication (denoted \u2295 and \u2297), each satisfying its own set of algebraic laws. In 1909, Jacobsthal considered a third, intermediate way of multiplying ordinals (denoted \u00d7 ), defined by transfinite iteration of natural addition, as well as the notion of exponentiation defined by transfinite iteration of his multiplication, which we denote . (Jacobsthal's multiplication was later rediscovered by Conway.) Jacobsthal showed these operations too obeyed algebraic laws. In this paper, we pick up where Jacobsthal left off by considering the notion of exponentiation obtained by transfinitely iterating natural multiplication instead; we shall denote this . We show that and that ; note the use of Jacobsthal's multiplication in the latter. We also demonstrate the impossibility of defining a \u201cnatural exponentiation\u201d satisfying reasonable algebraic laws.<\/jats:p>","DOI":"10.1002\/malq.201600006","type":"journal-article","created":{"date-parts":[[2017,10,30]],"date-time":"2017-10-30T12:18:56Z","timestamp":1509365936000},"page":"228-242","source":"Crossref","is-referenced-by-count":4,"title":["Intermediate arithmetic operations on ordinal numbers"],"prefix":"10.1002","volume":"63","author":[{"given":"Harry J.","family":"Altman","sequence":"first","affiliation":[{"name":"Department of Mathematics University of Michigan 530 Church St Ann Arbor MI 48109 United States of America"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2017,10,30]]},"reference":[{"key":"e_1_2_7_2_1","doi-asserted-by":"publisher","DOI":"10.4064\/fm-159-1-51-69"},{"key":"e_1_2_7_3_1","unstructured":"D.Asper\u00f3andK.Tsaprounis Long reals preprint (2015)."},{"key":"e_1_2_7_4_1","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-88514-3"},{"key":"e_1_2_7_5_1","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1942-07649-X"},{"key":"e_1_2_7_6_1","volume-title":"On Numbers and Games","author":"Conway J. 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