{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,5,7]],"date-time":"2023-05-07T10:40:05Z","timestamp":1683456005657},"reference-count":25,"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":7316,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1994,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In the first part of this paper we discuss some aspects of Detlefsen's attempt to save Hilbert's Program from the consequences of G\u00f6del's Second Incompleteness Theorem. His arguments are based on his interpretation of the long standing and well-known controversy on what, exactly, finitistic means are. In his paper [1] Detlefsen takes the position that there is a form of the<jats:italic>\u03c9<\/jats:italic>-rule which is a finitistically valid means of proof, sufficient to prove the consistency of elementary number theory<jats:italic>Z<\/jats:italic>. On the other hand, he claims that<jats:italic>Z<\/jats:italic>with its first-order logic is not strong enough to allow a formalization of such an<jats:italic>\u03c9<\/jats:italic>-rule. This would explain why the unprovability of Con(<jats:italic>Z<\/jats:italic>) in<jats:italic>Z<\/jats:italic>does not imply that the consistency of<jats:italic>Z<\/jats:italic>cannot be proved by finitistic means. We show that Detlefsen's proposal is unacceptable as originally formulated in [1], but that a reasonable modification of the rule he suggest leads to a partial program already studied for many years. We investigate the scope of such a program in terms of proof-theoretic reducibilities. We also show that this partial program encompasses mathematically important theories studied in the \u201cReverse Mathematics\u201d program. In order to investigate the provability with such a modified rule, we define new consistency and provability predicates which are weaker than the usual ones. We then investigate their properties, including a few that have no apparent philosophical significance but compare interestingly with the properties of the program based on the iteration of our<jats:italic>\u03c9<\/jats:italic>-rule. We determine some of the limitations of such programs, pointing out that these limitations partly explain why partial programs that have been successfully carried out use quite different and substantially more radical extensions of finitistic methods with more general forms of restricted reasoning.<\/jats:p>","DOI":"10.2307\/2275269","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:51:08Z","timestamp":1146955868000},"page":"322-343","source":"Crossref","is-referenced-by-count":11,"title":["Hilbert's program and the omega-rule"],"prefix":"10.1017","volume":"59","author":[{"given":"Aleksandar","family":"Ignjatovi\u0107","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200020429_ref025","doi-asserted-by":"publisher","DOI":"10.2307\/2026089"},{"key":"S0022481200020429_ref024","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(08)71123-6"},{"key":"S0022481200020429_ref023","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1007\/BF00485379","article-title":"Relative consistency and accessible domains","volume":"84","author":"Sieg","year":"1990","journal-title":"Synthese"},{"key":"S0022481200020429_ref020","doi-asserted-by":"publisher","DOI":"10.1007\/BF00485460"},{"key":"S0022481200020429_ref018","first-page":"349","volume":"53","author":"Simpson","year":"1988","journal-title":"Partial realization of Hilbert's program"},{"key":"S0022481200020429_ref016","first-page":"405","article-title":"On a restricted \u03c9-rule","volume":"VII","author":"Shoenfield","year":"1959","journal-title":"Bulletin De L'Academie Polonaise Des Sciences"},{"key":"S0022481200020429_ref014","first-page":"129","volume":"2","author":"Rosser","year":"1937","journal-title":"G\u00f6del theorems for nonconstructive logics"},{"key":"S0022481200020429_ref012","unstructured":"Ignjatovi\u0107 A. , Hilbert's program and the omega-rule, unpublished manuscript, 12 1988."},{"key":"S0022481200020429_ref010","article-title":"\u00dcber das Unendliche","volume":"95","author":"Hilbert","journal-title":"Mathematische Annalen"},{"key":"S0022481200020429_ref005","first-page":"383","volume":"27","author":"Feferman","year":"1962","journal-title":"Incompleteness along paths in progressions of theories"},{"key":"S0022481200020429_ref003","article-title":"Introductory note to 1931c","author":"Feferman","year":"1986","journal-title":"Kurt G\u00f6del: Collected works, vol. I"},{"key":"S0022481200020429_ref002","first-page":"259","volume":"27","author":"Feferman","year":"1962","journal-title":"Transfinite recursive progressions of axiomatic theories"},{"key":"S0022481200020429_ref019","unstructured":"Simpson S. G. , Subsystems of second-order arithmetic (in preparation)."},{"key":"S0022481200020429_ref004","first-page":"354","volume":"53","author":"Feferman","year":"1988","journal-title":"Hilbert's program relativized: proof-theoretical and foundational reductions"},{"key":"S0022481200020429_ref017","first-page":"783","volume":"49","author":"Simpson","year":"1984","journal-title":"Which set existence axioms are needed to prove the Cauchy-Peano theorem for ordinary differential equations?"},{"key":"S0022481200020429_ref007","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(83)90012-X"},{"key":"S0022481200020429_ref022","first-page":"338","volume":"53","author":"Sieg","year":"1988","journal-title":"Hilbert's program sixty years later"},{"key":"S0022481200020429_ref013","unstructured":"Ignjatovi\u0107 A. , Fragments of first- and second-order arithmetic and length of proofs, Ph.D. Thesis, University of California at Berkeley, Berkeley, California (1990)."},{"key":"S0022481200020429_ref006","first-page":"557","volume":"41","author":"Friedman","year":"1976","journal-title":"Systems of second-order arithmetic with restricted induction. I, II"},{"key":"S0022481200020429_ref001","first-page":"297","article-title":"On interpreting G\u00f6del's second theorem","volume":"8","author":"Detlefsen","year":"1979","journal-title":"Journal of Philosphieal Logic"},{"key":"S0022481200020429_ref021","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(85)90030-2"},{"key":"S0022481200020429_ref008","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-010-3072-4"},{"key":"S0022481200020429_ref009","unstructured":"Harrington L. , private communication to H. Friedman."},{"key":"S0022481200020429_ref015","first-page":"721","volume":"47","author":"Schmerl","year":"1982","journal-title":"Iterated reflection principles and the \u03c9-rule"},{"key":"S0022481200020429_ref011","doi-asserted-by":"publisher","DOI":"10.1007\/BF01457953"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200020429","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,5,7]],"date-time":"2023-05-07T10:09:59Z","timestamp":1683454199000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200020429\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1994,3]]},"references-count":25,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1994,3]]}},"alternative-id":["S0022481200020429"],"URL":"https:\/\/doi.org\/10.2307\/2275269","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1994,3]]}}}