{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,9,20]],"date-time":"2023-09-20T05:04:37Z","timestamp":1695186277551},"reference-count":31,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,1,15]],"date-time":"2014-01-15T00:00:00Z","timestamp":1389744000000},"content-version":"unspecified","delay-in-days":5434,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Bull. symb. log."],"published-print":{"date-parts":[[1999,3]]},"abstract":"The fundamental question in reverse mathematics is to determine which set existence axioms are required to prove particular theorems of mathematics. In addition to being interesting in their own right, answers to this question have consequences in both effective mathematics and the foundations of mathematics. Before discussing these consequences, we need to be more specific about the motivating question.<\/jats:p>Reverse mathematics is useful for studying theorems of either countable or essentially countable mathematics. Essentially countable mathematics is a vague term that is best explained by an example. Complete separable metric spaces are essentially countable because, although the spaces may be uncountable, they can be understood in terms of a countable basis. Simpson (1985) gives the following list of areas which can be analyzed by reverse mathematics: number theory, geometry, calculus, differential equations, real and complex analysis, combinatorics, countable algebra, separable Banach spaces, computability theory, and the topology of complete separable metric spaces. Reverse mathematics is less suited to theorems of abstract functional analysis, abstract set theory, universal algebra, or general topology.<\/jats:p>Section 2 introduces the major subsystems of second order arithmetic used in reverse mathematics:RCA<\/jats:italic>0<\/jats:sub>,WKL<\/jats:italic>0<\/jats:sub>,ACA<\/jats:italic>0<\/jats:sub>,ATR<\/jats:italic>0<\/jats:sub>and\u2013CA<\/jats:italic>0<\/jats:sub>. Sections 3 through 7 consider various theorems of ordered group theory from the perspective of reverse mathematics. Among the results considered are theorems on ordered quotient groups (including an equivalent ofACA<\/jats:italic>0<\/jats:sub>), groups and semigroup conditions which imply orderability (WKL<\/jats:italic>0<\/jats:sub>), the orderability of free groups (RCA<\/jats:italic>0<\/jats:sub>), H\u00f6lder's Theorem (RCA<\/jats:italic>0<\/jats:sub>), Mal'tsev's classification of the order types of countable ordered groups (\u2013CA<\/jats:italic>0<\/jats:sub>)<\/jats:p>","DOI":"10.2307\/421140","type":"journal-article","created":{"date-parts":[[2006,5,7]],"date-time":"2006-05-07T07:12:24Z","timestamp":1146985944000},"page":"45-58","source":"Crossref","is-referenced-by-count":7,"title":["Ordered Groups: A Case Study in Reverse Mathematics"],"prefix":"10.1017","volume":"5","author":[{"given":"Reed","family":"Solomon","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,1,15]]},"reference":[{"key":"S1079898600007083_ref012","doi-asserted-by":"publisher","DOI":"10.1090\/conm\/106\/1057821"},{"key":"S1079898600007083_ref009","doi-asserted-by":"crossref","first-page":"167","DOI":"10.4064\/fm-46-2-167-174","article-title":"Note on ordered groups and rings","volume":"46","author":"Fuchs","year":"1958","journal-title":"Fundamenta Mathematicae"},{"key":"S1079898600007083_ref002","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(86)90066-7"},{"key":"S1079898600007083_ref024","volume-title":"Algebra and Order: Proceedings of the First International Symposium for Ordered Algebraic Structures Luminy\u2013Marseille 1984","author":"Revesz","year":"1986"},{"key":"S1079898600007083_ref021","first-page":"17","article-title":"Linear order on a group","volume":"4","author":"Ohnishi","year":"1952","journal-title":"Osaka Mathematical Journal"},{"key":"S1079898600007083_ref014","first-page":"33","article-title":"classes and degrees of theories","volume":"173","author":"Jockusch","year":"1972","journal-title":"Transactions of the American Mathematical Society"},{"key":"S1079898600007083_ref011","doi-asserted-by":"publisher","DOI":"10.1007\/BF01630810"},{"key":"S1079898600007083_ref027","doi-asserted-by":"publisher","DOI":"10.1017\/S0022481200028309"},{"key":"S1079898600007083_ref007","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(90)90014-S"},{"key":"S1079898600007083_ref026","doi-asserted-by":"publisher","DOI":"10.1090\/pspum\/042\/791071"},{"key":"S1079898600007083_ref019","first-page":"473","volume-title":"Izvestiya Akademii Nauk USSR Seriya Matematicheskaya","volume":"13","author":"Mal'tsev","year":"1949"},{"key":"S1079898600007083_ref030","doi-asserted-by":"crossref","unstructured":"Solomon D. 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