{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,18]],"date-time":"2026-08-18T03:46:29Z","timestamp":1787024789093,"version":"3.56.0"},"reference-count":12,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,1,15]],"date-time":"2014-01-15T00:00:00Z","timestamp":1389744000000},"content-version":"unspecified","delay-in-days":685,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Bull. symb. log"],"published-print":{"date-parts":[[2012,3]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    We try to answer the question which is the \u201cright\u201d foundation of mathematics, second order logic or set theory. Since the former is usually thought of as a formal language and the latter as a first order theory, we have to rephrase the question. We formulate what we call\n                    <jats:italic>the second order view<\/jats:italic>\n                    and a competing\n                    <jats:italic>set theory view<\/jats:italic>\n                    , and then discuss the merits of both views. On the surface these two views seem to be in manifest conflict with each other. However, our conclusion is that it is very difficult to see any real difference between the two. We analyze a phenomenonwe call\n                    <jats:italic>internal categoricity<\/jats:italic>\n                    which extends the familiar categoricity results of second order logic to Henkin models and show that set theory enjoys the same kind of internal categoricity. Thus the existence of non-standard models, which is usually taken as a property of first order set theory, and categoricity, which is usually taken as a property of second order axiomatizations, can coherently coexist when put into their proper context. We also take a fresh look at complete second order axiomatizations and give a hierarchy result for second order characterizable structures. Finally we consider the problem of existence in mathematics from both points of view and find that second order logic depends on what we call\n                    <jats:italic>large domain assumptions<\/jats:italic>\n                    , which come quite close to the meaning of the axioms of set theory.\n                  <\/jats:p>","DOI":"10.2178\/bsl\/1327328440","type":"journal-article","created":{"date-parts":[[2012,1,23]],"date-time":"2012-01-23T10:38:21Z","timestamp":1327315101000},"page":"91-121","source":"Crossref","is-referenced-by-count":23,"title":["Second Order Logic or Set Theory?"],"prefix":"10.1017","volume":"18","author":[{"given":"Jouko","family":"V\u00e4\u00e4n\u00e4nen","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2014,1,15]]},"reference":[{"key":"S107989860000024X_ref007","unstructured":"[] Keskinen Lauri , Characterizing all models in infinite cardinalities, Ph.D. thesis, University of Amsterdam, 2011."},{"key":"S107989860000024X_ref005","unstructured":"[] Hyttinen Tapani , Kangas Kaisa , and V\u00e4\u00e4n\u00e4nen Jouko , On second order characterizability, to appear."},{"key":"S107989860000024X_ref009","volume-title":"Theory of recursive functions and effective computability","author":"Rogers","year":"1987"},{"key":"S107989860000024X_ref002","doi-asserted-by":"publisher","DOI":"10.1080\/01445340210146889"},{"key":"S107989860000024X_ref001","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(79)90001-9"},{"key":"S107989860000024X_ref010","unstructured":"[] Solovay Robert , FOM posting http:\/\/cs.nyu.edu\/pipermail\/fom\/2006-May\/010561.html."},{"key":"S107989860000024X_ref008","first-page":"76","article-title":"A hierarchy of formulas in set theory","volume":"57","author":"L\u00e9vy","year":"1965","journal-title":"Memoirs of the American Mathematical Society"},{"key":"S107989860000024X_ref012","unstructured":"[] V\u00e4\u00e4n\u00e4nen Jouko , Second-order logic and foundations of mathematics, this Bulletin, vol. 7 (2001), no. 4, pp. 504\u2013520."},{"key":"S107989860000024X_ref004","volume-title":"Grundz\u00fcge der theoretischen Logik","volume":"27","author":"Hilbert","year":"1972"},{"key":"S107989860000024X_ref006","volume-title":"Set theory","author":"Jech","year":"2003"},{"key":"S107989860000024X_ref003","first-page":"127","volume-title":"Axiomatic set theory (Proceedings of the Symposium on Pure Mathematics, Vol. XIII, Part II)","author":"Garland","year":"1974"},{"key":"S107989860000024X_ref011","first-page":"391","volume-title":"Logic Colloquium '78 (Mons, 1978)","volume":"97","author":"V\u00e4\u00e4n\u00e4nen","year":"1979"}],"container-title":["The Bulletin of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S107989860000024X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,25]],"date-time":"2019-04-25T20:30:24Z","timestamp":1556224224000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S107989860000024X\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,3]]},"references-count":12,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2012,3]]}},"alternative-id":["S107989860000024X"],"URL":"https:\/\/doi.org\/10.2178\/bsl\/1327328440","relation":{},"ISSN":["1079-8986","1943-5894"],"issn-type":[{"value":"1079-8986","type":"print"},{"value":"1943-5894","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,3]]}}}