{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T15:46:40Z","timestamp":1774626400768,"version":"3.50.1"},"reference-count":15,"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":"In [5] G\u00f6del interpreted Peano arithmetic in Heyting arithmetic. In [8, p. 153], and [7, p. 344, (iii)], Kreisel observed that G\u00f6del's interpretation extended to second order arithmetic. In [11] (see [4, p. 92] for a correction) and [10] Myhill extended the interpretation to type theory. We will show that G\u00f6del's negative interpretation can be extended to Zermelo-Fraenkel set theory. We consider a set theory T<\/jats:italic> formulated in the minimal predicate calculus, which in the presence of the full law of excluded middle is the same as the classical theory of Zermelo and Fraenkel. Then, following Myhill, we define an inner model S<\/jats:italic> in which the axioms of Zermelo-Fraenkel set theory are true.<\/jats:p>More generally we show that any class X<\/jats:italic> that is (i) transitive in the negative sense, \u2200x<\/jats:italic> \u2208 X<\/jats:italic>\u2200y<\/jats:italic> \u2208 x<\/jats:italic> \u00ac \u00ac x<\/jats:italic> \u2208 X<\/jats:italic>, (ii) contained in the class St<\/jats:italic> = {x<\/jats:italic>: \u2200u<\/jats:italic>(\u00ac \u00ac u<\/jats:italic> \u2208 x<\/jats:italic>\u2192 u<\/jats:italic> \u2208 x<\/jats:italic>)} of stable sets, and (iii) closed in the sense that \u2200x<\/jats:italic>(x<\/jats:italic> \u2286 X<\/jats:italic> \u223c \u223c x<\/jats:italic> \u2208 X<\/jats:italic>), is a standard model of Zermelo-Fraenkel set theory. The class S<\/jats:italic> is simply the \u2286-least such class, and, hence, could be defined by S<\/jats:italic> = \u22c2{X<\/jats:italic>: \u2200x(x<\/jats:italic> \u2286 \u223c \u223c X<\/jats:italic>\u2192 \u223c \u223c x<\/jats:italic> \u2208 X<\/jats:italic>)}. However, since we can only conservatively extend T<\/jats:italic> to a class theory with \n\n\u03940<\/jats:sub>1<\/jats:sup>-comprehension, but not with \u03941<\/jats:sub>1<\/jats:sup>-comprehension, we will give a \u03940<\/jats:sub>1<\/jats:sup>-definition of S<\/jats:italic> within T<\/jats:italic>.<\/jats:p>","DOI":"10.2307\/2271902","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:35:13Z","timestamp":1146951313000},"page":"221-229","source":"Crossref","is-referenced-by-count":23,"title":["Extending G\u00f6del's negative interpretation to ZF"],"prefix":"10.1017","volume":"40","author":[{"given":"William C.","family":"Powell","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200053779_ref014","unstructured":"Powell W. , Heyting-valued models (in preparation)."},{"key":"S0022481200053779_ref011","first-page":"267","volume-title":"Proceedings of Symposia in Pure Mathematics, Part I","volume":"13","author":"Myhill"},{"key":"S0022481200053779_ref006","first-page":"185","volume-title":"Infinitistic methods","author":"Heyting"},{"key":"S0022481200053779_ref003","volume-title":"Bulletin of the American Mathematical Society","author":"Goodman"},{"key":"S0022481200053779_ref001","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0066773"},{"key":"S0022481200053779_ref007","first-page":"321","volume":"33","author":"Kreisel","year":"1968","journal-title":"A survey of proof theory"},{"key":"S0022481200053779_ref010","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19730190307"},{"key":"S0022481200053779_ref009","volume":"40","author":"Myhill","year":"1975","journal-title":"Constructive set theory"},{"key":"S0022481200053779_ref008","first-page":"143","volume-title":"Logic, Methodology and the Philosophy of Science III","author":"Kreisel","year":"1968"},{"key":"S0022481200053779_ref004","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0073966"},{"key":"S0022481200053779_ref002","first-page":"315","volume":"38","author":"Friedman","year":"1973","journal-title":"The consistency of classical set theory relative to a set theory with intuitionistic logic"},{"key":"S0022481200053779_ref013","unstructured":"Powell W. , A completeness theorem for Zermelo-Fraenkel set theory, this Journal (accepted)."},{"key":"S0022481200053779_ref012","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0066775"},{"key":"S0022481200053779_ref015","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0066744"},{"key":"S0022481200053779_ref005","first-page":"34","volume-title":"Ergebnisse eines mathematische Kolloquium","author":"G\u00f6del","year":"1933"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200053779","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,29]],"date-time":"2019-05-29T19:40:34Z","timestamp":1559158834000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200053779\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1975,6]]},"references-count":15,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1975,6]]}},"alternative-id":["S0022481200053779"],"URL":"https:\/\/doi.org\/10.2307\/2271902","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1975,6]]}}}