{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,29]],"date-time":"2022-03-29T10:39:20Z","timestamp":1648550360157},"reference-count":7,"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":12429,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1980,3]]},"abstract":"<jats:p>Determining the truth value of self-referential sentences is an interesting and often tricky problem. The G\u00f6del sentence, asserting its own unprovability in <jats:bold><jats:italic>P<\/jats:italic><\/jats:bold> (Peano arithmetic), is clearly true in <jats:italic>N<\/jats:italic>(the standard model of <jats:bold><jats:italic>P<\/jats:italic><\/jats:bold>), and L\u00f6b showed that a sentence asserting its own provability in <jats:bold><jats:italic>P<\/jats:italic><\/jats:bold> is also true in <jats:italic>N<\/jats:italic> (see Smorynski [Sm, 4.1.1]). The problem is more difficult, and still unsolved, for sentences of the kind constructed by Kreisel [K1], which assert their own falsity in some model <jats:italic>N<\/jats:italic>* of <jats:bold><jats:italic>P<\/jats:italic><\/jats:bold> whose complete diagram is arithmetically defined. Such a sentence \u03c7 has the property that <jats:italic>N<\/jats:italic> \u22a8 iff <jats:italic>N<\/jats:italic>* \u22ad \u03c7 (note that \u00ac\u03c7 has the same property).<\/jats:p><jats:p>We show in \u00a71 that the truth value in <jats:italic>N<\/jats:italic> of such a sentence \u03c7, after a certain normalization that breaks the symmetry between it and its negation, is determined by the parity of a natural number, called the rank of <jats:italic>N<\/jats:italic>, for the particular construction of <jats:italic>N<\/jats:italic>* used. The rank is the number of times the construction can be iterated starting from <jats:italic>N<\/jats:italic> and is finite for all the usual constructions. We also show that modifications of, e.g., Henkin's construction (in his completeness proof of predicate calculus) allow arbitrary finite values for the rank of <jats:italic>N<\/jats:italic>. Thus, on the one hand the truth value of \u03c7 in <jats:italic>N<\/jats:italic>, for a given \u201cnice\u201d construction of <jats:italic>N<\/jats:italic>*, is independent of the particular (normalized) choice of \u03c7, and we shall see that \u03c7 is unique up to (provable) equivalence in <jats:bold><jats:italic>P<\/jats:italic><\/jats:bold>. On the other hand, the truth value in question is sensitive to minor changes in the definition of <jats:italic>N<\/jats:italic>* and its determination seems to be largely a combinatorial problem.<\/jats:p>","DOI":"10.2307\/2273361","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:51:56Z","timestamp":1146952316000},"page":"144-154","source":"Crossref","is-referenced-by-count":1,"title":["operators and alternating sentences in arithmetic"],"prefix":"10.1017","volume":"45","author":[{"given":"Larry","family":"Manevitz","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jonathan","family":"Stavi","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200047228_ref003","doi-asserted-by":"publisher","DOI":"10.4064\/fm-37-1-265-285"},{"key":"S0022481200047228_ref002","first-page":"363","volume":"41","author":"Friedman","year":"1976","journal-title":"Uniformly defined descending sequences of degrees"},{"key":"S0022481200047228_ref005","first-page":"1","volume-title":"Dissertationes Mathematicae","author":"Kreisel"},{"key":"S0022481200047228_ref004","first-page":"321","volume":"33","author":"Kreisel","year":"1968","journal-title":"Survey of proof theory"},{"key":"S0022481200047228_ref007","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(08)71123-6"},{"key":"S0022481200047228_ref001","first-page":"107","article-title":"Strong models of arithmetic","volume":"14","author":"Ehrenfeucht","year":"1966","journal-title":"Bulletin de l'Acad\u00e9mie Polonaise des Sciences"},{"key":"S0022481200047228_ref006","doi-asserted-by":"publisher","DOI":"10.1090\/pspum\/005\/0141595"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200047228","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,26]],"date-time":"2019-05-26T19:36:32Z","timestamp":1558899392000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200047228\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1980,3]]},"references-count":7,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1980,3]]}},"alternative-id":["S0022481200047228"],"URL":"https:\/\/doi.org\/10.2307\/2273361","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1980,3]]}}}