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We give an axiomatization of the theory of sentences true in almost all finite models with addition. We also characterize its complete extensions and relate its infinite models to models of Presburger arithmetic.","publication_date":"2011,,","publication_name":"Theoretical Computer Science","grobid_abstract_attachment_id":"41270357"},"document_type":"paper","pre_hit_view_count_baseline":null,"quality":"high","language":"en","title":"Theories of initial segments of standard models of arithmetics and their complete extensions","broadcastable":true,"draft":null,"has_indexable_attachment":true,"indexable":true}}["work"]; window.loswp.workCoauthors = [41138865,40854177]; window.loswp.locale = "en"; window.loswp.countryCode = "SG"; window.loswp.cwvAbTestBucket = ""; window.loswp.designVariant = "ds_vanilla"; window.loswp.fullPageMobileSutdModalVariant = "full_page_mobile_sutd_modal"; window.loswp.useOptimizedScribd4genScript = false; window.loginModal = {}; window.loginModal.appleClientId = 'edu.academia.applesignon';</script><script defer="" src="https://accounts.google.com/gsi/client"></script><div class="ds-loswp-container"><div class="ds-work-card--grid-container"><div class="ds-work-card--container js-loswp-work-card"><div class="ds-work-card--cover"><div class="ds-work-cover--wrapper"><div class="ds-work-cover--container"><button class="ds-work-cover--clickable js-swp-download-button" data-signup-modal="{"location":"swp-splash-paper-cover","attachmentId":41270357,"attachmentType":"pdf"}"><img alt="First page of “Theories of initial segments of standard models of arithmetics and their complete extensions”" class="ds-work-cover--cover-thumbnail" src="https://0.academia-photos.com/attachment_thumbnails/41270357/mini_magick20190219-19329-bfh8us.png?1550617473" /><img alt="PDF Icon" class="ds-work-cover--file-icon" src="//a.academia-assets.com/images/single_work_splash/adobe_icon.svg" /><div class="ds-work-cover--hover-container"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span><p>Download Free PDF</p></div><div class="ds-work-cover--ribbon-container">Download Free PDF</div><div class="ds-work-cover--ribbon-triangle"></div></button></div></div></div><div class="ds-work-card--work-information"><h1 class="ds-work-card--work-title">Theories of initial segments of standard models of arithmetics and their complete extensions</h1><div class="ds-work-card--work-authors ds-work-card--detail"><a class="ds-work-card--author js-wsj-grid-card-author ds2-5-body-md ds2-5-body-link" data-author-id="41138865" href="https://independent.academia.edu/KonradZdanowski"><img alt="Profile image of Konrad Zdanowski" class="ds-work-card--author-avatar" src="//a.academia-assets.com/images/s65_no_pic.png" />Konrad Zdanowski</a><a class="ds-work-card--author js-wsj-grid-card-author ds2-5-body-md ds2-5-body-link" data-author-id="40854177" href="https://u-clermont1.academia.edu/JerzyTomasik"><img alt="Profile image of Jerzy Tomasik" class="ds-work-card--author-avatar" src="https://0.academia-photos.com/40854177/11112696/12401480/s65_jerzy.tomasik.jpg" />Jerzy Tomasik</a></div><div class="ds-work-card--detail"><p class="ds-work-card--detail ds2-5-body-sm">2011, Theoretical Computer Science</p><div class="ds-work-card--work-metadata"><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">visibility</span><p class="ds2-5-body-sm" id="work-metadata-view-count">…</p></div><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">description</span><p class="ds2-5-body-sm">17 pages</p></div><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">link</span><p class="ds2-5-body-sm">1 file</p></div></div><script>(async () => { const workId = 19973951; 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We give an axiomatization of the theory of sentences true in almost all finite models with addition. We also characterize its complete extensions and relate its infinite models to models of Presburger arithmetic.</p><div class="ds-work-card--button-container"><button class="ds2-5-button js-swp-download-button" data-signup-modal="{"location":"continue-reading-button--work-card","attachmentId":41270357,"attachmentType":"pdf","workUrl":"https://www.academia.edu/19973951/Theories_of_initial_segments_of_standard_models_of_arithmetics_and_their_complete_extensions"}">See full PDF</button><button class="ds2-5-button ds2-5-button--secondary js-swp-download-button" data-signup-modal="{"location":"download-pdf-button--work-card","attachmentId":41270357,"attachmentType":"pdf","workUrl":"https://www.academia.edu/19973951/Theories_of_initial_segments_of_standard_models_of_arithmetics_and_their_complete_extensions"}"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span>Download PDF</button></div></div></div></div><div data-auto_select="false" data-client_id="331998490334-rsn3chp12mbkiqhl6e7lu2q0mlbu0f1b" data-doc_id="41270357" data-landing_url="https://www.academia.edu/19973951/Theories_of_initial_segments_of_standard_models_of_arithmetics_and_their_complete_extensions" data-login_uri="https://www.academia.edu/registrations/google_one_tap" data-moment_callback="onGoogleOneTapEvent" id="g_id_onload"></div><div class="ds-top-related-works--grid-container"><div class="ds-related-content--container ds-top-related-works--container"><h2 class="ds-related-content--heading">Related papers</h2><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="0" data-entity-id="20105212" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/20105212/Theories_of_arithmetics_in_finite_models">Theories of arithmetics in finite models</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="41138865" href="https://independent.academia.edu/KonradZdanowski">Konrad Zdanowski</a></div><p class="ds-related-work--metadata ds2-5-body-xs">The Journal of Symbolic Logic, 2005</p><p class="ds-related-work--abstract ds2-5-body-sm">We investigate theories of initial segments of the standard models for arithmetics. It is easy to see that if the ordering relation is definable in the standard model then the decidability results can be transferred from the infinite model into the finite models. On the contrary we show that the Σ2-theory of multiplication is undecidable in finite models. We show that this result is optimal by proving that the Σ1-theory of multiplication and order is decidable in finite models as well as in the standard model. We show also that the exponentiation function is definable in finite models by a formula of arithmetic with multiplication and that one can define in finite models the arithmetic of addition and multiplication with the concatenation operation.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Theories of arithmetics in finite models","attachmentId":41164160,"attachmentType":"pdf","work_url":"https://www.academia.edu/20105212/Theories_of_arithmetics_in_finite_models","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/20105212/Theories_of_arithmetics_in_finite_models"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="1" data-entity-id="48462646" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/48462646/Completeness_theorems_incompleteness_theorems_and_models_of_arithmetic">Completeness theorems, incompleteness theorems and models of arithmetic</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="100282984" href="https://independent.academia.edu/KenMcAloon">Ken McAloon</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Transactions of the American Mathematical Society, 1978</p><p class="ds-related-work--abstract ds2-5-body-sm">Let & be a consistent extension of Peano arithmetic and let 6EJJ denote the set of TL°" consequences of &. Employing incompleteness theorems to generate independent formulas and completeness theorems to construct models, we build nonstandard models of SP"+2 m which the standard integers are A°+1-definable. We thus pinpoint induction axioms which are not provable in éE¡¡+2; in particular, we show that (parameter free) A?-induction is not provable in Primitive Recursive Arithmetic. Also, we give a solution of a problem of Gaifman on the existence of roots of diophantine equations in end extensions and answer questions about existentially complete models of 3^. Furthermore, it is shown that the proof of the Gödel Completeness Theorem cannot be formalized in 6E § and that the MacDowell-Specker Theorem fails for all truncated theories (£¡¡.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Completeness theorems, incompleteness theorems and models of arithmetic","attachmentId":67061152,"attachmentType":"pdf","work_url":"https://www.academia.edu/48462646/Completeness_theorems_incompleteness_theorems_and_models_of_arithmetic","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/48462646/Completeness_theorems_incompleteness_theorems_and_models_of_arithmetic"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="2" data-entity-id="102800636" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/102800636/On_Certain_Extentions_of_the_Arithmetic_of_Addition_of_Natural_Numbers">On Certain Extentions of the Arithmetic of Addition of Natural Numbers</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="62823989" href="https://moscowstate.academia.edu/AlexeySemenov">Alexey Semenov</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Mathematics of the USSR – Izvestia. 15:2, 1980</p><p class="ds-related-work--abstract ds2-5-body-sm">In this paper the problems of expressibility and decidability are studied for elementary theories obtained by extending the arithmetic of order and the arithmetic of addition of natural numbers. Results are obtained on the decidability and undecidability of elementary theories of concrete structures of the form ⟨N;+,P⟩, where P is a fixed monadic predicate, as well as results on the class of sets definable in the theory T⟨N;+,λx,∃y(x=dy)⟩.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"On Certain Extentions of the Arithmetic of Addition of Natural Numbers","attachmentId":102974238,"attachmentType":"pdf","work_url":"https://www.academia.edu/102800636/On_Certain_Extentions_of_the_Arithmetic_of_Addition_of_Natural_Numbers","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/102800636/On_Certain_Extentions_of_the_Arithmetic_of_Addition_of_Natural_Numbers"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="3" data-entity-id="106012637" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/106012637/Models_and_types_of_Peanos_arithmetic">Models and types of Peano's arithmetic</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="6964329" href="https://columbia.academia.edu/HGaifman">Haim Gaifman</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Annals of Mathematical Logic, 1976</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Models and types of Peano's arithmetic","attachmentId":105322539,"attachmentType":"pdf","work_url":"https://www.academia.edu/106012637/Models_and_types_of_Peanos_arithmetic","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/106012637/Models_and_types_of_Peanos_arithmetic"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="4" data-entity-id="115036275" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/115036275/Existentially_Closed_Models_in_the_Framework_of_Arithmetic">Existentially Closed Models in the Framework of Arithmetic</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="61300629" href="https://impan.academia.edu/ZofiaAdamowicz">Zofia Adamowicz</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Symbolic Logic, 2016</p><p class="ds-related-work--abstract ds2-5-body-sm">We prove that the standard cut is definable in each existentially closed model of I Δ 0 +exp by a (parameter free) Π 1-formula. This definition is optimal with respect to quantifier complexity and allows us to improve some previously known results on existentially closed models of fragments of arithmetic. 1. Introduction. This work was initially motivated by a gap in the proof of Corollary 1.3 of [2] providing a parameter free Π 1-definition of the standard cut, N, in each existentially closed (e.c.) model of I Δ 0 + exp. Our aim is to provide a correct proof of the above result and, use it to obtain an updated view of the theory of e.c. models of I Δ 0 +exp. Existentially closed models of arithmetic were investigated in the 1970's as a part of the efforts to get a full understanding of the model theory of existentially closed structures (existence of model completions and companion theories, finite and infinite forcing, etc.). The results obtained in the early 1970's by A. Robinson, J. Hirschfeld, D. C. Goldrei, A. Macintyre, and H. Simmons pointed out the most important property of e.c. models of sufficiently strong arithmetic theories: there exist formulas defining N in each such model. These results were not stated in their full generality. In the 1970's a systematic study of fragments of Peano arithmetic PA was still to come and the authors focused essentially on e.c. models of Π 2 (N) (thesetoftrueΠ 2-sentences) or of Π 2 (PA)(thesetofΠ 2 consequences of PA), and more generally on e.c. models of Π 2 (T B), where T B is any extension of Π 2 (PA). Regarding Π 2 (N), Robinson (see [14]) proved N to be Σ 3-definable in every e.c. model of Π 2 (N) and Hirschfeld (see [7]) improved Robinson's result obtaining a Σ 2-definition of N,or even aΠ 1-definition, if parameters are allowed. Hirschfeld also showed that these definitions are optimal (in terms of quantifier complexity) for e.c. models of Π 2 (N). As to Π 2 (T B), in [11] Macintyre and Simmons (see also [5]) extended Hirschfeld's Σ 2-definition of N to all e.c. models of Π 2 (T B) and showed that the parametric Π 1definition can be extended to those e.c. models of Π 2 (T B)inwhichtheΣ 1-definable elements are not cofinal. However, these definitions are not best possible, since there is no general result ruling out the possibility of a parameter free Π 1-definition of N valid in all e.c. models. As a matter of fact, such an optimal definition was Key words and phrases. fragments of Peano arithmetic, existentially closed models, turing degrees of arithmetic theories.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Existentially Closed Models in the Framework of Arithmetic","attachmentId":111561987,"attachmentType":"pdf","work_url":"https://www.academia.edu/115036275/Existentially_Closed_Models_in_the_Framework_of_Arithmetic","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/115036275/Existentially_Closed_Models_in_the_Framework_of_Arithmetic"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="5" data-entity-id="20105216" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/20105216/Finite_Arithmetics">Finite Arithmetics</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="41138865" href="https://independent.academia.edu/KonradZdanowski">Konrad Zdanowski</a><span>, </span><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="41271059" href="https://independent.academia.edu/MarcinMostowski">Marcin Mostowski</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Fundamenta Informaticae</p><p class="ds-related-work--abstract ds2-5-body-sm">The paper presents the current state of knowledge in the field of logical investigations of finite arithmetics. This is an attempt to summarize the ideas and results in this area. Some new results are presented - these are mainly generalizations of the earlier results related to properties of sl-theories and some nontrivial cases of FM-representability theorem.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Finite Arithmetics","attachmentId":41990674,"attachmentType":"pdf","work_url":"https://www.academia.edu/20105216/Finite_Arithmetics","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/20105216/Finite_Arithmetics"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="6" data-entity-id="79913768" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/79913768/Substructure_lattices_of_models_of_arithmetic">Substructure lattices of models of arithmetic</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="216453850" href="https://independent.academia.edu/GeorgeMills23">George Mills</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Annals of Mathematical Logic, 1979</p><p class="ds-related-work--abstract ds2-5-body-sm">We completely characterize those distributive lattices which can be obtained as elementary substructure lattices of models of Peano arithmetic. Stated concisely: every plausible distributive Mice occurs abundantly. Our proof employs the notion of a strongly definable type in many variables. With slight modifications the method also yields a characterization of those distributive lattices which can be obtained uniformly hy Gaifman's methods oi definable and end extensional l-types. As :J special case this gives another proof of two conjectures involving finite distributive lattices and models of arithmetic posed by Gaifman and initially proved by Schmurl. We also show that every minimal type (in the sense of Gaifman) satisfies a strong partitton property which we will call being "uniformly Ramsey". (2) VMbPA 3N> M Lt (N/M)= D. (3) D is complete, compactly generated, and eacf~ compact element of D hots C X,, compact predecessors.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Substructure lattices of models of arithmetic","attachmentId":86469413,"attachmentType":"pdf","work_url":"https://www.academia.edu/79913768/Substructure_lattices_of_models_of_arithmetic","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/79913768/Substructure_lattices_of_models_of_arithmetic"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="7" data-entity-id="58396494" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/58396494/On_end_extensions_of_models_of_subsystems_of_peano_arithmetic">On end extensions of models of subsystems of peano arithmetic</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="30022870" href="https://independent.academia.edu/CostasDimitrakopoulos">Costas Dimitrakopoulos</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Theoretical Computer Science, 2001</p><p class="ds-related-work--abstract ds2-5-body-sm">We survey results and problems concerning subsystems of Peano Arithmetic. In particular, we deal with end extensions of models of such theories. First, we discuss the results of Paris</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"On end extensions of models of subsystems of peano arithmetic","attachmentId":72828053,"attachmentType":"pdf","work_url":"https://www.academia.edu/58396494/On_end_extensions_of_models_of_subsystems_of_peano_arithmetic","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/58396494/On_end_extensions_of_models_of_subsystems_of_peano_arithmetic"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="8" data-entity-id="81438523" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/81438523/Models_of_Bounded_Arithmetic_Theories_and_Some_Related_Complexity_Questions">Models of Bounded Arithmetic Theories and Some Related Complexity Questions</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="12461242" href="https://sbu-ir.academia.edu/MortezaMoniri">Morteza Moniri</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Bulletin of the Section of Logic</p><p class="ds-related-work--abstract ds2-5-body-sm">In this paper, we study bounded versions of some model-theoretic notions and results. We apply these results to the context of models of bounded arithmetic theories as well as some related complexity questions. As an example, we show that if the theory $\rm S_2 ^1(PV)$ has bounded model companion then $\rm NP=coNP$. We also study bounded versions of some other related notions such as Stone topology.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Models of Bounded Arithmetic Theories and Some Related Complexity Questions","attachmentId":87481717,"attachmentType":"pdf","work_url":"https://www.academia.edu/81438523/Models_of_Bounded_Arithmetic_Theories_and_Some_Related_Complexity_Questions","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/81438523/Models_of_Bounded_Arithmetic_Theories_and_Some_Related_Complexity_Questions"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="9" data-entity-id="27771727" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/27771727/Internal_End_Extensions_of_Peano_Arithmetic_and_a_Problem_of_Gaifman">Internal End-Extensions of Peano Arithmetic and a Problem of Gaifman</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="34940590" href="https://anekdoty.academia.edu/LarryManevitz">Larry Manevitz</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of the London Mathematical Society, 1976</p><p class="ds-related-work--abstract ds2-5-body-sm">A well known result of M. Rabin states that the only existentially complete model of full arithmetic is the standard one. H. Gaifman [1], raised the parallel question for end-extensions of full arithmetic, i.e. does every non-standard model of full arithmetic have an end-extension in which a diophantine equation unsolvable in the original model has a solution. A. Wilkie provided a partial answer [4] when he proved that every countable model of P, Peano Arithmetic, has such an end-extension (which is in fact isomorphic to the original model).</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Internal End-Extensions of Peano Arithmetic and a Problem of Gaifman","attachmentId":48049579,"attachmentType":"pdf","work_url":"https://www.academia.edu/27771727/Internal_End_Extensions_of_Peano_Arithmetic_and_a_Problem_of_Gaifman","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/27771727/Internal_End_Extensions_of_Peano_Arithmetic_and_a_Problem_of_Gaifman"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div></div></div><div class="ds-sticky-ctas--wrapper js-loswp-sticky-ctas hidden"><div class="ds-sticky-ctas--grid-container"><div class="ds-sticky-ctas--container"><button class="ds2-5-button js-swp-download-button" data-signup-modal="{"location":"continue-reading-button--sticky-ctas","attachmentId":41270357,"attachmentType":"pdf","workUrl":null}">See full PDF</button><button class="ds2-5-button ds2-5-button--secondary js-swp-download-button" data-signup-modal="{"location":"download-pdf-button--sticky-ctas","attachmentId":41270357,"attachmentType":"pdf","workUrl":null}"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span>Download PDF</button></div></div></div><div class="ds-below-fold--grid-container"><div class="ds-work--container js-loswp-embedded-document"><div class="attachment_preview" data-attachment="Attachment_41270357" style="display: none"><div class="js-scribd-document-container"><div class="scribd--document-loading js-scribd-document-loader" style="display: block;"><img alt="Loading..." src="//a.academia-assets.com/images/loaders/paper-load.gif" /><p>Loading Preview</p></div></div><div style="text-align: center;"><div class="scribd--no-preview-alert js-preview-unavailable"><p>Sorry, preview is currently unavailable. 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1983</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Submodels and definable points in models of Peano arithmetic","attachmentId":73891899,"attachmentType":"pdf","work_url":"https://www.academia.edu/60450514/Submodels_and_definable_points_in_models_of_Peano_arithmetic","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/60450514/Submodels_and_definable_points_in_models_of_Peano_arithmetic"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div 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class="ds-related-work--metadata ds2-5-body-xs">Archive for Mathematical Logic, 1990</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"On some formalized conservation results in arithmetic","attachmentId":101956619,"attachmentType":"pdf","work_url":"https://www.academia.edu/101406073/On_some_formalized_conservation_results_in_arithmetic","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/101406073/On_some_formalized_conservation_results_in_arithmetic"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" 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href="https://www.academia.edu/448790/Classical_and_Intuitionistic_Models_of_Arithmetic">Classical and Intuitionistic Models of Arithmetic</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="337209" href="https://uci.academia.edu/KaiWehmeier">Kai Wehmeier</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Notre Dame Journal of Formal Logic 37, 1996, 452–461, 1996</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Classical and Intuitionistic Models of Arithmetic","attachmentId":2007411,"attachmentType":"pdf","work_url":"https://www.academia.edu/448790/Classical_and_Intuitionistic_Models_of_Arithmetic","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span 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data-entity-id="113731321" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/113731321/Predicative_Logic_and_Formal_Arithmetic">Predicative Logic and Formal Arithmetic</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="117828974" href="https://independent.academia.edu/JohnBurgess31">John Burgess</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Notre Dame Journal of Formal Logic, 1998</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Predicative Logic and Formal Arithmetic","attachmentId":110619161,"attachmentType":"pdf","work_url":"https://www.academia.edu/113731321/Predicative_Logic_and_Formal_Arithmetic","alternativeTracking":true}"><span 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