{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,19]],"date-time":"2025-11-19T09:26:18Z","timestamp":1763544378252},"reference-count":8,"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":8046,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1992,3]]},"abstract":"<jats:p>Let <jats:italic>I<\/jats:italic>\u22bf<jats:sub>0<\/jats:sub> denote the fragment of Peano arithmetic where induction is restricted only to bounded formulas (\u22bf<jats:sub>0<\/jats:sub>-formulas), i.e. formulas where all the quantifiers are bounded. We will examine the behaviour in <jats:italic>I<\/jats:italic>\u22bf<jats:sub>0<\/jats:sub> of Chebyshev's classical result on the distribution of primes. Chebyshev showed that each of the following functions on <jats:bold>N<\/jats:bold> is bounded by a polynomial in each of the others:<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200023148_eqnU1\" \/><\/jats:disp-formula><\/jats:p><jats:p>In his proof he used, as auxiliaries, factorials and binomial coefficients.<\/jats:p><jats:p>When we work in a weak system like <jats:italic>I<\/jats:italic>\u22bf<jats:sub>0<\/jats:sub> we have to deal with two problems:<\/jats:p><jats:p>(i) There is not an immediate meaning for symbols like <jats:italic>x<jats:sup>y<\/jats:sup><\/jats:italic>, \u03a0<jats:sub><jats:italic>p<\/jats:italic>\u2264<jats:italic>x<\/jats:italic><\/jats:sub><jats:italic>p<\/jats:italic>, and <jats:italic>x<\/jats:italic>!.<\/jats:p><jats:p>(ii) Some basic functions like exponentiation, factorial, and product of primes up to a certain element, are in general only partially definable.<\/jats:p><jats:p>The first problem is equivalent to finding \u22bf<jats:sub>0<\/jats:sub>-formulas representing the relations <jats:italic>z<\/jats:italic> = <jats:italic>x<jats:sup>y<\/jats:sup><\/jats:italic>, <jats:italic>y<\/jats:italic> = \u03a0<jats:sub><jats:italic>p<\/jats:italic>\u2264<jats:italic>x<\/jats:italic><\/jats:sub><jats:italic>p<\/jats:italic>, and <jats:italic>y<\/jats:italic> = <jats:italic>x<\/jats:italic>!. While we can easily express that <jats:italic>y<\/jats:italic> is the product of all primes less than or equal to <jats:italic>x<\/jats:italic> in a \u22bf<jats:sub>0<\/jats:sub>-way, it requires a subtle argument to define <jats:italic>z<\/jats:italic> = <jats:italic>x<jats:sup>y<\/jats:sup><\/jats:italic> by a \u22bf<jats:sub>0<\/jats:sub>-formula. The most natural way of expressing it is via the formalization of the Chinese remainder theorem coding the recursion procedure used in the definition of <jats:italic>x<jats:sup>y<\/jats:sup><\/jats:italic>. But this can be expressed only via a \u03a3<jats:sub>1<\/jats:sub>-formula. \u22bf<jats:sub>0<\/jats:sub>-definitions of exponentiation were given by Paris and Bennett (see [GD] and [B]), and all sensible definitions of this type are equivalent (see Remark 1). Later we will examine properties of such a definition <jats:italic>E<\/jats:italic><jats:sub>0<\/jats:sub>(<jats:italic>x, y, z<\/jats:italic>). The \u22bf<jats:sub>0<\/jats:sub>0-definition of <jats:italic>y<\/jats:italic> = \u03a0<jats:sub><jats:italic>p<\/jats:italic>\u2264<jats:italic>x<\/jats:italic><\/jats:sub><jats:italic>p<\/jats:italic> is much more straightforward, since both the relation of being a prime and the divisibility relation are \u22bf<jats:sub>0<\/jats:sub>-definable. We will give an explicit definition of it and denote it by \u03a8(<jats:italic>x, y<\/jats:italic>). In this way, by a \u22bf<jats:sub>0<\/jats:sub>-induction, we will be able to prove the uniqueness of the elements <jats:italic>y<\/jats:italic> and <jats:italic>z<\/jats:italic> satisfying \u03a8(<jats:italic>x, y<\/jats:italic>) and <jats:italic>E<\/jats:italic><jats:sub>0<\/jats:sub>(<jats:italic>x, y, z<\/jats:italic>), respectively.<\/jats:p>","DOI":"10.2307\/2275173","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:43:45Z","timestamp":1146941025000},"page":"12-27","source":"Crossref","is-referenced-by-count":13,"title":["Local behaviour of the Chebyshev theorem in models of <i>I<\/i>\u22bf<sub>0<\/sub>"],"prefix":"10.1017","volume":"57","author":[{"family":"Paola D'Aquino","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200023148_ref006","first-page":"317","volume-title":"Methods in mathematical logic (proceedings of the sixth Latin American symposium on mathematical logic, Caracas, 1983)","volume":"1130","author":"Paris","year":"1985"},{"key":"S0022481200023148_ref007","first-page":"1235","volume":"53","author":"Paris","year":"1988","journal-title":"Provability of the pigeonhole principle and the existence of infinitely many primes"},{"key":"S0022481200023148_ref005","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1016\/0168-0072(87)90066-2","article-title":"On the scheme of induction for bounded arithmetic formulas","volume":"35","author":"Paris","year":"1987","journal-title":"Annals of Pure and Applied Logic"},{"key":"S0022481200023148_ref001","unstructured":"Bennett J. H. , On spectra, Ph.D. thesis, Princeton University, Princeton, New Jersey, 1962."},{"key":"S0022481200023148_ref008","unstructured":"Woods A. , Some problems in logic and number theory and their connections, Ph.D. thesis, University of Manchester, Manchester, 1981."},{"key":"S0022481200023148_ref002","first-page":"187","volume-title":"Logic and algorithmic","author":"Gaifman","year":"1982"},{"key":"S0022481200023148_ref003","volume-title":"An introduction to the theory of numbers","author":"Hardy","year":"1954"},{"key":"S0022481200023148_ref004","first-page":"494","volume":"36","author":"Parikh","year":"1971","journal-title":"Existence and feasibility in arithmetic"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200023148","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,16]],"date-time":"2019-05-16T18:32:34Z","timestamp":1558031554000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200023148\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1992,3]]},"references-count":8,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1992,3]]}},"alternative-id":["S0022481200023148"],"URL":"https:\/\/doi.org\/10.2307\/2275173","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1992,3]]}}}