{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,8,25]],"date-time":"2025-08-25T11:10:09Z","timestamp":1756120209214,"version":"3.44.0"},"reference-count":0,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2025,2,5]],"date-time":"2025-02-05T00:00:00Z","timestamp":1738713600000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["The Review of Symbolic Logic"],"published-print":{"date-parts":[[2025,6]]},"abstract":"Abstract<\/jats:title>We explore general notions of consistency<\/jats:italic>. These notions are sentences \n$\\mathcal {C}_{\\alpha }$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> (they depend on numerations \n$\\alpha $\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> of a certain theory) that generalize the usual features of consistency statements. The following forms of consistency fit the definition of general notions of consistency (\n${\\texttt {Pr}}_{\\alpha }$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> denotes the provability predicate for the numeration \n$\\alpha $\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>): \n$\\neg {\\texttt {Pr}}_{\\alpha }(\\ulcorner \\perp \\urcorner )$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, \n$\\omega \\text {-}{\\texttt {Con}}_{\\alpha }$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> (the formalized \n$\\omega $\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-consistency), \n$\\neg {\\texttt {Pr}}_{\\alpha }(\\ulcorner {\\texttt {Pr}}_{\\alpha }(\\ulcorner \\cdots {\\texttt {Pr}}_{\\alpha }(\\ulcorner \\perp \\urcorner )\\cdots \\urcorner )\\urcorner )$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, and \n$n\\text {-}{\\texttt {Con}}_{\\alpha }$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> (the formalized n<\/jats:italic>-consistency of Kreisel).<\/jats:p>We generalize the former notions of consistency while maintaining two important features, to wit: G\u00f6del\u2019s Second Incompleteness Theorem, i.e., (with \n$\\xi $\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> some standard \n$\\Delta _0(T)$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-numeration of the axioms of T<\/jats:italic>), and a result by Feferman that guarantees the existence of a numeration \n$\\tau $\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> such that \n$T\\vdash \\mathcal {C}_\\tau $\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>.<\/jats:p>We encompass slow consistency into our framework. To show how transversal and natural our approach is, we create a notion of provability<\/jats:italic> from a given \n$\\mathcal {C}_{\\alpha }$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, we call it \n$\\mathcal {P}_{\\mathcal {C}_{\\alpha }}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, and we present sufficient conditions on \n$\\mathcal {C}_{\\alpha }$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> for the notion \n$\\mathcal {P}_{\\mathcal {C}_{\\alpha }}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> to satisfy the standard derivability conditions. Moreover, we also develop a notion of interpretability<\/jats:italic> from a given \n$\\mathcal {C}_{\\alpha }$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, we call it \n$\\rhd _{\\mathcal {C}_{\\alpha }}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, and we study some of its properties. All these new notions\u2014of provability and interpretability\u2014serve primarily to emphasize the naturalness of our notions, not necessarily to give insights on these topics.<\/jats:p>","DOI":"10.1017\/s1755020325000024","type":"journal-article","created":{"date-parts":[[2025,2,5]],"date-time":"2025-02-05T03:41:06Z","timestamp":1738726866000},"page":"616-635","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":0,"title":["GENERAL NOTIONS OF CONSISTENCY"],"prefix":"10.1017","volume":"18","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6686-005X","authenticated-orcid":false,"given":"PAULO GUILHERME","family":"SANTOS","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2025,2,5]]},"container-title":["The Review of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S1755020325000024","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,8,25]],"date-time":"2025-08-25T10:47:49Z","timestamp":1756118869000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S1755020325000024\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,2,5]]},"references-count":0,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2025,6]]}},"alternative-id":["S1755020325000024"],"URL":"https:\/\/doi.org\/10.1017\/s1755020325000024","relation":{},"ISSN":["1755-0203","1755-0211"],"issn-type":[{"type":"print","value":"1755-0203"},{"type":"electronic","value":"1755-0211"}],"subject":[],"published":{"date-parts":[[2025,2,5]]},"assertion":[{"value":"\u00a9 The Author(s), 2025. Published by Cambridge University Press on behalf of The Association for Symbolic Logic","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}}]}}