{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,8,1]],"date-time":"2024-08-01T22:08:59Z","timestamp":1722550139293},"reference-count":91,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2014,1,15]],"date-time":"2014-01-15T00:00:00Z","timestamp":1389744000000},"content-version":"unspecified","delay-in-days":2420,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Bull. symb. log."],"published-print":{"date-parts":[[2007,6]]},"abstract":"<jats:p>Kurt G\u00f6del (1906\u20131978) with his work on the constructible universe<jats:italic>L<\/jats:italic>established the relative consistency of the Axiom of Choice (AC) and the Continuum Hypothesis (CH). More broadly, he ensured the ascendancy of first-order logic as the framework and a matter of method for set theory and secured the cumulative hierarchy view of the universe of sets. G\u00f6del thereby transformed set theory and launched it with structured subject matter and specific methods of proof. In later years G\u00f6del worked on a variety of set theoretic constructions and speculated about how problems might be settled with new axioms. We here chronicle this development from the point of view of the evolution of set theory as a field of mathematics. Much has been written, of course, about G\u00f6del's work in set theory, from textbook expositions to the introductory notes to his collected papers. The present account presents an integrated view of the historical and mathematical development as supported by his recently published lectures and correspondence. Beyond the surface of things we delve deeper into the mathematics. What emerges are the roots and anticipations in work of Russell and Hilbert, and most prominently the sustained motif of truth as formalizable in the \u201cnext higher system\u201d. We especially work at bringing out how transforming G\u00f6del's work was for set theory. It is difficult now to see what conceptual and technical distance G\u00f6del had to cover and how dramatic his re-orientation of set theory was.<\/jats:p>","DOI":"10.2178\/bsl\/1185803804","type":"journal-article","created":{"date-parts":[[2007,12,13]],"date-time":"2007-12-13T14:55:58Z","timestamp":1197557758000},"page":"153-188","source":"Crossref","is-referenced-by-count":14,"title":["G\u00f6del and Set Theory"],"prefix":"10.1017","volume":"13","author":[{"given":"Akihiro","family":"Kanamori","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,1,15]]},"reference":[{"key":"S1079898600002201_ref064","first-page":"271","volume-title":"Proceedings of symposia in pure mathematics","volume":"13","author":"Myhill","year":"1971"},{"key":"S1079898600002201_ref077","volume-title":"Collected papers","author":"Tarski","year":"1986"},{"key":"S1079898600002201_ref034","unstructured":"G\u00f6del Kurt [1970a], Some considerations leading to the probable conclusion that the true power of the continuum is \u21352, handwritten document, in Feferman [1995], pp. 420\u2013422."},{"key":"S1079898600002201_ref003","unstructured":"Brendle J\u00f6rg , Larson Paul , and Todor\u010devi\u0107 Stevo [\u221e], Rectangular axioms, perfect set properties and decomposition, to appear."},{"key":"S1079898600002201_ref052","doi-asserted-by":"publisher","DOI":"10.1093\/philmat\/nkj009"},{"key":"S1079898600002201_ref066","first-page":"338","article-title":"The foundations of mathematics","volume":"25","author":"Ramsey","year":"1925","journal-title":"Proceedings of the London Mathematical Society"},{"key":"S1079898600002201_ref049","volume-title":"The higher infinite","author":"Kanamori","year":"2003"},{"key":"S1079898600002201_ref053","doi-asserted-by":"crossref","first-page":"197","DOI":"10.4099\/jjm1924.15.0_197","article-title":"Sur l'uniformisation des compl\u00e9mentaires analytiques et les ensembles projectifs de la seconde classe","volume":"15","author":"Kond\u00f4","year":"1939","journal-title":"Japanese Journal of Mathematics"},{"key":"S1079898600002201_ref081","first-page":"199","article-title":"Zur Einf\u00fchrung der transfiniten Zahlen","volume":"1","author":"von Neumann","year":"1923","journal-title":"Acta Litterarum ac Scientiarum Regiae Universitatis Hungaricae Francisco-Josephinae (Szeged), sectio scientiarum mathematicarum"},{"key":"S1079898600002201_ref070","first-page":"137","volume-title":"Matematikerkongressen i Helsingfors den 4\u20137 Juli 1922, Den femte skandinaviska matematikerkongressen, Redog\u00f6relse","author":"Skolem","year":"1923"},{"key":"S1079898600002201_ref073","doi-asserted-by":"publisher","DOI":"10.4064\/fm-17-1-210-239"},{"key":"S1079898600002201_ref009","volume-title":"Kurt G\u00f6del, Collected works, Publications 1929\u20131936","volume":"I","author":"Feferman","year":"1986"},{"key":"S1079898600002201_ref018","doi-asserted-by":"publisher","DOI":"10.1007\/BF01457986"},{"key":"S1079898600002201_ref040","doi-asserted-by":"publisher","DOI":"10.4064\/fm-26-1-241-255"},{"key":"S1079898600002201_ref050","first-page":"487","volume":"10","author":"Kanamori","year":"2004","journal-title":"Zermelo and set theory"},{"key":"S1079898600002201_ref015","volume-title":"Kurt G\u00f6del, Collected works, Correspondence H-Z","volume":"V","author":"Feferman","year":"2003"},{"key":"S1079898600002201_ref012","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780195072556.001.0001","volume-title":"Kurt G\u00f6del, Collected works, Unpublished essays and lectures","volume":"III","author":"Feferman","year":"1995"},{"key":"S1079898600002201_ref082","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1515\/crll.1925.154.219","article-title":"Eine Axiomatisierung der Mengenlehre","volume":"154","author":"von Neumann","year":"1925","journal-title":"Journal f\u00fcr die reine und angewandte Mathematik"},{"key":"S1079898600002201_ref045","volume-title":"Set theory","author":"Jech","year":"2002"},{"key":"S1079898600002201_ref014","volume-title":"Kurt G\u00f6del, Collected works, Correspondence A\u2013G","volume":"IV","author":"Feferman","year":"2003"},{"key":"S1079898600002201_ref033","volume-title":"Philosophy of mathematics. 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From G\u00f6del to Philosophy","author":"Wang","year":"1996"},{"key":"S1079898600002201_ref069","first-page":"103","volume-title":"Videnskaps-selskapets Skrifter, I","author":"Skolem","year":"1920"},{"key":"S1079898600002201_ref024","unstructured":"G\u00f6del Kurt [1933], The present situation in the foundations of mathematics, handwritten text for an invited lecture, in Feferman [1995], pp. 45\u201353, and the page references are to these."},{"key":"S1079898600002201_ref020","unstructured":"G\u00f6del Kurt [1929], \u00dcber die Vollst\u00e4ndigkeit der Logikkalk\u00fcls, doctoral dissertation, University of Vienna, reprinted and translated in Feferman [1986], pp. 60\u2013101."},{"key":"S1079898600002201_ref058","doi-asserted-by":"publisher","DOI":"10.2140\/pjm.1960.10.223"},{"key":"S1079898600002201_ref068","first-page":"521","article-title":"Measurable cardinals and constructible sets","volume":"9","author":"Scott","year":"1961","journal-title":"Bulletin de l'Academie Polonaise des Sciences, S\u00e9rie des Sciences Mathematiques, Astronomiques et Physiques"},{"key":"S1079898600002201_ref083","doi-asserted-by":"publisher","DOI":"10.1007\/BF01459102"},{"key":"S1079898600002201_ref001","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0098398"},{"key":"S1079898600002201_ref035","doi-asserted-by":"crossref","unstructured":"G\u00f6del Kurt [1970b], A proof of Cantor's continuum hypothesis from a highly plausible axiom about orders of growth, handwritten document, in Feferman [1995], pp. 422\u2013423.","DOI":"10.1093\/oso\/9780195072556.003.0019"},{"key":"S1079898600002201_ref062","volume-title":"Descriptive set theory","author":"Moschovakis","year":"1980"},{"key":"S1079898600002201_ref063","doi-asserted-by":"publisher","DOI":"10.1080\/01445349808837306"},{"key":"S1079898600002201_ref006","doi-asserted-by":"publisher","DOI":"10.1007\/BF01459233"},{"key":"S1079898600002201_ref061","first-page":"91","volume-title":"Essays on the foundations of mathematics","author":"Montague","year":"1961"},{"key":"S1079898600002201_ref005","volume-title":"Logical dilemmas: The life and work of Kurt G\u00f6del","author":"Dawson","year":"1997"},{"key":"S1079898600002201_ref041","first-page":"529","volume":"12","author":"Hauser","year":"2006","journal-title":"G\u00f6del's program revisited, part I: The turn to phenomenology"},{"key":"S1079898600002201_ref004","volume-title":"The undecidable: Basic papers on undecidable propositions, unsolvable problems and computable functions","author":"Davis","year":"1965"},{"key":"S1079898600002201_ref072","first-page":"75","volume":"1","author":"Steel","year":"1995","journal-title":"HODL(R) is a core model below"},{"key":"S1079898600002201_ref065","doi-asserted-by":"publisher","DOI":"10.1090\/hmath\/025"},{"key":"S1079898600002201_ref087","doi-asserted-by":"crossref","first-page":"459","DOI":"10.3917\/rip.234.0459","article-title":"G\u00f6del, Tarski and truth","volume":"59","author":"Wole\u0144ski","year":"2005","journal-title":"Revue International de Philosophie"},{"key":"S1079898600002201_ref016","volume-title":"Thoralf Skolem, Selected works in logic","author":"Fenstad","year":"1970"},{"key":"S1079898600002201_ref090","doi-asserted-by":"publisher","DOI":"10.4064\/fm-16-1-29-47"},{"key":"S1079898600002201_ref051","doi-asserted-by":"publisher","DOI":"10.5840\/gfpj200425217"},{"key":"S1079898600002201_ref027","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.25.4.220"},{"key":"S1079898600002201_ref085","volume-title":"Popular lectures on mathematical logic","author":"Wang","year":"1981"},{"key":"S1079898600002201_ref002","doi-asserted-by":"crossref","first-page":"65","DOI":"10.2307\/2268862","article-title":"A system of axiomatic set theory. 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