{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,14]],"date-time":"2026-05-14T13:13:19Z","timestamp":1778764399437,"version":"3.51.4"},"reference-count":16,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":10054,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1986,9]]},"abstract":"<jats:p>Myhill [12] extended the ideas of Shapiro [15], and proposed a system of epistemic set theory IST (based on modal <jats:italic>S<\/jats:italic>4 logic) in which the meaning of the necessity operator is taken to be the intuitive provability, as formalized in the system itself. In this setting one works in classical logic, and yet it is possible to make distinctions usually associated with intuitionism, e.g. a constructive existential quantifier can be expressed as (\u2203<jats:italic>x<\/jats:italic>) \u25a1 \u2026. This was first confirmed when Goodman [7] proved that Shapiro's epistemic first order arithmetic is conservative over intuitionistic first order arithmetic via an extension of G\u00f6del's modal interpretation [6] of intuitionistic logic.<\/jats:p><jats:p>Myhill showed that whenever a sentence \u25a1<jats:italic>A<\/jats:italic> \u2228 \u25a1<jats:italic>B<\/jats:italic> is provable in IST, then <jats:italic>A<\/jats:italic> is provable in IST or <jats:italic>B<\/jats:italic> is provable in IST (the disjunction property), and that whenever a sentence \u2203<jats:italic>x<\/jats:italic>.\u25a1<jats:italic>A(x)<\/jats:italic> is provable in IST, then so is A(<jats:italic>t<\/jats:italic>) for some closed term <jats:italic>t<\/jats:italic> (the existence property). He adapted the Friedman slash [4] to epistemic systems.<\/jats:p><jats:p>Goodman [8] used Epistemic Replacement to formulate a ZF-like strengthening of IST, and proved that it was a conservative extension of ZF and that it had the disjunction and existence properties. It was then shown in [13] that a slight extension of Goodman's system with the Epistemic Foundation (ZFE<jats:sub>R<\/jats:sub>, cf. \u00a71) suffices to interpret intuitionistic ZF set theory with Replacement (ZFI<jats:sub>R<\/jats:sub>, [10]). This is obtained by extending G\u00f6del's modal interpretation [6] of intuitionistic logic. ZFE<jats:sub>R<\/jats:sub> still had the properties of Goodman's system mentioned above.<\/jats:p>","DOI":"10.2307\/2274028","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:18:32Z","timestamp":1146953912000},"page":"748-754","source":"Crossref","is-referenced-by-count":2,"title":["Some properties of epistemic set theory with collection"],"prefix":"10.1017","volume":"51","author":[{"given":"Andre","family":"Scedrov","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200030917_ref016","volume-title":"Intensional mathematics","author":"Shapiro","year":"1985"},{"key":"S0022481200030917_ref015","doi-asserted-by":"crossref","unstructured":"Shapiro S. , Epistemic and intuitionistic arithmetic, in [16], pp. 11\u201346.","DOI":"10.1016\/S0049-237X(08)70138-1"},{"key":"S0022481200030917_ref014","volume-title":"Annals of Pure and Applied Logic","author":"Scedrov"},{"key":"S0022481200030917_ref013","doi-asserted-by":"crossref","unstructured":"Scedrov A. , Extending G\u00f6del's modal interpretation to type theory and set theory, in [16], pp. 81\u2013119.","DOI":"10.1016\/S0049-237X(08)70141-1"},{"key":"S0022481200030917_ref012","doi-asserted-by":"crossref","unstructured":"Myhill J. , Intensional set theory, in [16], pp. 47\u201361.","DOI":"10.1016\/S0049-237X(08)70139-3"},{"key":"S0022481200030917_ref010","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0066775"},{"key":"S0022481200030917_ref006","first-page":"39","article-title":"Eine Interpretation des intuitionistischen Aussagenkalk\u00fcls","volume":"4","author":"G\u00f6del","year":"1932","journal-title":"Ergebnisse eines Mathematischen Kolloquiums"},{"key":"S0022481200030917_ref004","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0066773"},{"key":"S0022481200030917_ref002","unstructured":"Flagg R. , Integrating classical and intuitionistic mathematics, Ph.D. Thesis, State University of New York at Buffalo, Buffalo, New York, 1984."},{"key":"S0022481200030917_ref001","first-page":"1","volume-title":"Logic Colloquium '78","author":"Beeson","year":"1979"},{"key":"S0022481200030917_ref011","first-page":"347","volume":"40","author":"Myhill","year":"1975","journal-title":"Constructive set theory"},{"key":"S0022481200030917_ref009","unstructured":"Goodman N. , Boolean-valued models of epistemic set theory (in preparation)."},{"key":"S0022481200030917_ref003","first-page":"895","volume":"50","author":"Flagg","year":"1985","journal-title":"Epistemic set theory is a conservative extension of intuitionistic set theory"},{"key":"S0022481200030917_ref008","doi-asserted-by":"crossref","unstructured":"Goodman N. , A genuinely intensional set theory, in [16], pp. 63\u201379.","DOI":"10.1016\/S0049-237X(08)70140-X"},{"key":"S0022481200030917_ref005","volume-title":"Advances in Mathematics","author":"Friedman"},{"key":"S0022481200030917_ref007","first-page":"192","volume":"49","author":"Goodman","year":"1984","journal-title":"Epistemic arithmetic is a conservative extension of intuitionistic arithmetic"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200030917","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,22]],"date-time":"2019-05-22T04:41:04Z","timestamp":1558500064000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200030917\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1986,9]]},"references-count":16,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1986,9]]}},"alternative-id":["S0022481200030917"],"URL":"https:\/\/doi.org\/10.2307\/2274028","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1986,9]]}}}