{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,5]],"date-time":"2022-04-05T22:21:09Z","timestamp":1649197269304},"reference-count":11,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":14164,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1975,6]]},"abstract":"<jats:p>The concept of a topos as a system, or world in which mathematics could be defined and interpreted, was developed by F. W. Lawvere and M. Tierney. Much of their work is embodied in the lecture notes <jats:italic>Elementary toposes<\/jats:italic> by A. Kock and G. C. Wraith [6].<\/jats:p><jats:p>In an early paper Lawvere set forth a set of axioms for approximately such a system [8]. The topos constructed there is a set-like category that includes among its axioms an axiom of infinity and an axiom of choice.<\/jats:p><jats:p>In its final form an elementary topos is freed from any such axioms.<\/jats:p><jats:p>The most prominent example of an elementary topos is a set theory with the usual Zermelo-Fraenkel or Godel-Bernays set of axioms.<\/jats:p><jats:p>In this paper I have tried to determine what, if any, is the effect of an axiom of choice introduced in a topos, and how are some of the set-theoretic equivalents of such an axiom related in topos theory.<\/jats:p><jats:p>Since the set-theoretic membership relation \u2208, the notions of an \u201cempty\u201d set and a \u201cpower\u201d set are definable in topos theory, it makes sense to talk about a \u201cchoice map\u201d that picks a single element out of every nonempty object, provided that these objects can be somehow collected into a single object or \u201cfamily.\u201d In other words, an analogue of the usual choice axiom can be formulated in elementary topos language; this is axiom AC2 of the text.<\/jats:p>","DOI":"10.2307\/2271900","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T17:35:13Z","timestamp":1146936913000},"page":"197-212","source":"Crossref","is-referenced-by-count":2,"title":["Two forms of the axiom of choice for an elementary topos"],"prefix":"10.1017","volume":"40","author":[{"given":"Anna Michaelides","family":"Penk","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200053755_ref011","volume-title":"Theory of categories","author":"Mitchell","year":"1965"},{"key":"S0022481200053755_ref008","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.52.6.1506"},{"key":"S0022481200053755_ref005","volume-title":"Topos theoretic factorization of nonstandard extensions","author":"Kock","year":"1972"},{"key":"S0022481200053755_ref004","volume-title":"Introduction to functorial semantics","author":"Kock","year":"1971"},{"key":"S0022481200053755_ref001","first-page":"1501","article-title":"Cat\u00e9gories exactes","volume":"272","author":"Barr","year":"1971","journal-title":"Comptes Rendus Hebdomadaires des S\u00e9ances de l'Acad\u00e9mie des Sciences"},{"key":"S0022481200053755_ref010","unstructured":"Mikkelsen C. J. , Colimits in toposes (in preparation)."},{"key":"S0022481200053755_ref009","doi-asserted-by":"crossref","unstructured":"Lawvere F. W. , Functorial semantics of algebraic theories, Dissertation, Columbia University, 1963.","DOI":"10.1073\/pnas.50.5.869"},{"key":"S0022481200053755_ref002","volume-title":"Abelian categories","author":"Freyd","year":"1964"},{"key":"S0022481200053755_ref006","volume-title":"Elementary toposes","author":"Kock","year":"1971"},{"key":"S0022481200053755_ref007","doi-asserted-by":"publisher","DOI":"10.1111\/j.1746-8361.1969.tb01194.x"},{"key":"S0022481200053755_ref003","doi-asserted-by":"publisher","DOI":"10.1017\/S0004972700044828"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200053755","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,29]],"date-time":"2019-05-29T15:40:05Z","timestamp":1559144405000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200053755\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1975,6]]},"references-count":11,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1975,6]]}},"alternative-id":["S0022481200053755"],"URL":"https:\/\/doi.org\/10.2307\/2271900","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1975,6]]}}}