{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,9,3]],"date-time":"2023-09-03T17:33:53Z","timestamp":1693762433753},"reference-count":15,"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":11699,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1982,3]]},"abstract":"<jats:p>NFP is the<jats:italic>predicative<\/jats:italic>fragment of NF. In this system we do not allow a set to exist if it cannot be defined without using quantifiers ranging over its type or parameters of a higher type. NFI is a less restrictive fragment located between NFP and NF.<\/jats:p><jats:p>We show that NFP is really weaker than NFI; similarly, NFI is weaker than NF. This result will be obtained in the following manner: on the one hand, we will show that NFP can be proved consistent in elementary arithmetic and that second order arithmetic is interpretable in NFI; on the other hand, we will prove the consistency of NFI in third order arithmetic, which is contained in NF.<\/jats:p><jats:p>The paper is divided in four sections. In \u00a71, we define the concepts needed and collect a few results together in such a way that they will be ready for later use. In \u00a72, we will present a model-theoretic (quick) proof of the consistency of NFI (and thus of NFP). The proof will be chosen (it is not the quickest!) so as to motivate in a natural manner the details of the proof-theoretical version of it that will be presented in \u00a73. \u00a74 will be devoted to the axiom of infinity in NFP and NFI.<\/jats:p>","DOI":"10.2307\/2273386","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:58:32Z","timestamp":1146952712000},"page":"131-136","source":"Crossref","is-referenced-by-count":11,"title":["On the consistency of an impredicative subsystem of Quine's NF"],"prefix":"10.1017","volume":"47","author":[{"given":"Marcel","family":"Crabb\u00e9","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200044753_ref015","volume-title":"Proof theory","author":"Takeuti","year":"1975"},{"key":"S0022481200044753_ref014","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1966-11611-7"},{"key":"S0022481200044753_ref012","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.39.9.972"},{"key":"S0022481200044753_ref011","doi-asserted-by":"crossref","DOI":"10.4159\/9780674042421","volume-title":"Set theory and its logic","author":"Quine","year":"1963"},{"key":"S0022481200044753_ref010","unstructured":"Oswald U. , Fragmente von \u201cNew-Foundations\u201d und Typentheorie, Thesis, E.T.H., Z\u00fcrich, 1976."},{"key":"S0022481200044753_ref009","unstructured":"Grishin V. N. , The method of stratification in set theory, Thesis, Academy of Science of U.S.S.R., Moscow, 1973 (Russian)."},{"key":"S0022481200044753_ref008","first-page":"1387","volume-title":"Consistency of a fragment of Quine's NF system","volume":"10","author":"Grishin","year":"1969"},{"key":"S0022481200044753_ref005","first-page":"255","volume-title":"Logique et Analyse","author":"Crabb\u00e9","year":"1976"},{"key":"S0022481200044753_ref002","first-page":"215","volume":"42","author":"Boffa","year":"1977","journal-title":"The consistency problem for NF"},{"key":"S0022481200044753_ref001","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0067644"},{"key":"S0022481200044753_ref003","first-page":"1657","article-title":"Les th\u00e9or\u00e8mes 3-stratifi\u00e9s de NF3","volume":"280","author":"Boffa","year":"1975","journal-title":"Comptes Rendus de l'Acad\u00e9mie des Sciences de Paris"},{"key":"S0022481200044753_ref013","first-page":"116","volume-title":"Logic, methodology and philosophy of science, Proceedings of the 1960 International Congress, Stanford","author":"Specker","year":"1962"},{"key":"S0022481200044753_ref006","doi-asserted-by":"publisher","DOI":"10.4064\/fm-101-1-11-17"},{"key":"S0022481200044753_ref007","volume-title":"Foundations of set theory","author":"Fraenkel","year":"1973"},{"key":"S0022481200044753_ref004","first-page":"1","article-title":"Types ambigus","volume":"280","author":"Crabb\u00e9","year":"1975","journal-title":"Comptes Rendus de l'Acad\u00e9mie des Sciences de Paris"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200044753","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,7,26]],"date-time":"2021-07-26T20:21:04Z","timestamp":1627330864000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200044753\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1982,3]]},"references-count":15,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1982,3]]}},"alternative-id":["S0022481200044753"],"URL":"https:\/\/doi.org\/10.2307\/2273386","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1982,3]]}}}