{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,3,8]],"date-time":"2023-03-08T17:51:02Z","timestamp":1678297862438},"reference-count":4,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":14256,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1975,3]]},"abstract":"<jats:p>It is long known that Lawvere's theory in <jats:italic>The category of categories as foundations of mathematics<\/jats:italic> A[1] does not work, as indicated in Ishell's review [0]. Isbell there gives a counterexample that CDT\u2014Category Description Theorem\u2014[1, p. 15] is in fact not a theorem of BT (the Basic Theory of [1]) and suggests adding CDT to the axioms.<\/jats:p><jats:p>Our starting point was the claim in [1] that \u201cthe basic theory needs no explicit axiom of infinity.\u201d We define a model \u2133 of BT in which all categories are finite. In particular, the \u201cmonoid of nonnegative integers <jats:italic>N<\/jats:italic>\u201d coincides in \u2133 with the terminal object 1. We study \u2133 in some detail in order to establish the true status of various \u201ctheorems\u201d or \u201cmetatheorems\u201d of BT: The metatheorem of [1, p. 11] saying that the discrete categories form a category of sets, CDT, the theorem on p. 15, and the theorem on p. 16 of [1] are all nontheorems. The remaining results indicated in [1] concerning BT are provable. However, as the Predicative Functor Construction Schema\u2014PFCS\u2014are justified in [1] by using the \u201cmetatheorem\u201d and CDT, we provide a proof of these two schemata by showing that the discrete categories of BT (or of convenient extensions of BT) form a two-valued Boolean topos.<\/jats:p>","DOI":"10.2307\/2272263","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T17:33:55Z","timestamp":1146936835000},"page":"14-18","source":"Crossref","is-referenced-by-count":3,"title":["Lawvere's basic theory of the category of categories"],"prefix":"10.1017","volume":"40","author":[{"given":"Georges","family":"Blanc","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Anne","family":"Preller","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200054190_ref001","article-title":"Review of [1]","volume":"34","author":"Isbell","year":"1967","journal-title":"Mathematical Reviews"},{"key":"S0022481200054190_ref003","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.52.6.1506"},{"key":"S0022481200054190_ref004","volume-title":"Categories for the working mathematician","author":"MacLane","year":"1971"},{"key":"S0022481200054190_ref002","first-page":"1","volume-title":"Proceedings of a Conference on Categorical Algebra","author":"Lawvere","year":"1966"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200054190","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,29]],"date-time":"2019-05-29T16:05:59Z","timestamp":1559145959000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200054190\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1975,3]]},"references-count":4,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1975,3]]}},"alternative-id":["S0022481200054190"],"URL":"https:\/\/doi.org\/10.2307\/2272263","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1975,3]]}}}