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The result is placed within two research themes regarding relational arithmetics." /> <meta name="twitter:image" content="http://a.academia-assets.com/images/twitter-card.jpeg" /> <meta property="fb:app_id" content="2369844204" /> <meta property="og:type" content="article" /> <meta property="og:url" content="https://www.academia.edu/8391573/A_Remark_on_Q" /> <meta property="og:title" content="A Remark on Q" /> <meta property="og:image" content="http://a.academia-assets.com/images/open-graph-icons/fb-paper.gif" /> <meta property="og:description" content="Robinson’s arithmetic Q is given a simple interpretation in Hajek’s weaker relational version QH that does not use Solovay’s technique of shortening cuts. The result is placed within two research themes regarding relational arithmetics." /> <meta property="article:author" content="https://utoronto.academia.edu/MihaiGanea" /> <meta name="description" content="Robinson’s arithmetic Q is given a simple interpretation in Hajek’s weaker relational version QH that does not use Solovay’s technique of shortening cuts. 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The result is placed within two research themes regarding relational arithmetics.</p></div></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="17065648" 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/17065648/Fragments_of_arithmetic">Fragments 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="196571" href="https://cmu.academia.edu/WilfriedSieg">Wilfried Sieg</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Annals of Pure and Applied Logic, 1985</p><p class="ds-related-work--abstract ds2-5-body-sm">We establish by elementary proof-theoretic means the conservativeness of two subsystems of analysis over primitive recursive arithmetic. The one subsystem was introduced by Friedman [6], the other is a strengthened version of a theory of Mint ; each has been shown to be of considerable interest for both mathematical practice and me&mathematical investigations. The foundational significance of such conservation results is clear: they provide a direct finitist justification of the part of mathematical practice formalizable in these subsystems.</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":"Fragments of arithmetic","attachmentId":42334579,"attachmentType":"pdf","work_url":"https://www.academia.edu/17065648/Fragments_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/17065648/Fragments_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="1" data-entity-id="9727886" 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/9727886/On_Arithmetic_Formulated_Connexively">On Arithmetic Formulated Connexively</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="9896443" href="https://rpi.academia.edu/ThomasFerguson">Thomas M Ferguson</a></div><p class="ds-related-work--abstract ds2-5-body-sm">One of the richest and most salient applications of a non-classical logic is the matter of how mathematics operates within its province. Historically, this is most evident in the case of intuitionism, insofar as the intuitionistic standpoints with respect to deduction and mathematical practice are tightly bound together. Yet even in the case of Robert Meyer's relevant arithmetic R#, that a robust and compelling theory of arithmetic can be erected on relevant foundations speaks to the maturity of relevant logics. Accordingly, as connexive logic matures as a field, the topography of mathematics against a connexive backdrop becomes more and more compelling. The contraclassicality of connexive logics entails that the development of connexive mathematics will be more complex---and, arguably, more interesting---than intuitionistic or relevant accounts. For example, although formally undecidable sentences in classical Peano arithmetic remain independent of its intuitionistic and relevant counterparts, there exist undecidable sentences of classical arithmetic that will become decidable modulo any reasonable connexive arithmetic. In, e.g., Peano arithmetic, the Gödel sentence G is undecidable. Classically, this entails that the sentence ~(G->~G) is likewise undecidable. Of course, in a connexive logic L and connexive arithmetic L#, L# will prove ~(G->~G), witnessing that some classically undecidable statements in number theory become decidable connexively. Although this example is extremely simple, it demonstrates that there are many subtle questions that uniquely arise in a connexive mathematics. In this paper, I wish to make a few comments on how mathematics---in particular, arithmetic---must behave if formulated connexively. We will first consider some relevant historical and philosophical topics, such as Łukasiewicz' number-theoretic argument against Aristotle's Thesis, before taking a foray into the formalization of modest subsystems of arithmetic in Richard Angell's PA1 and PA2, observing some of the pathologies that will greet arithmetic in these settings.</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 Arithmetic Formulated Connexively","attachmentId":35913338,"attachmentType":"pdf","work_url":"https://www.academia.edu/9727886/On_Arithmetic_Formulated_Connexively","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/9727886/On_Arithmetic_Formulated_Connexively"><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="2743345" 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/2743345/On_an_arithmetic_in_a_set_theory_within_Lukasiewicz_logic">On an arithmetic in a set theory within ̷Lukasiewicz logic</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="392902" href="https://aist-go.academia.edu/ShunsukeYatabe">Shunsuke Yatabe</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2007</p><p class="ds-related-work--abstract ds2-5-body-sm">• CL 0 proves a general form of recursive definition:(∃ X)(∀ x) x∈ X≡ ϕ (x, X).–For example, any partial recursive functions can be represented in CL 0.• It has been conjectured that CL 0 is enough strong to develop an arithmetic.–Skolem:“it may be possible to derive a significant amount of mathematics”[S57].–Hajek once suggested that crisp Peano arithmetic can be developed in CL 0.</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 an arithmetic in a set theory within ̷Lukasiewicz logic","attachmentId":30837523,"attachmentType":"pdf","work_url":"https://www.academia.edu/2743345/On_an_arithmetic_in_a_set_theory_within_Lukasiewicz_logic","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/2743345/On_an_arithmetic_in_a_set_theory_within_Lukasiewicz_logic"><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="48566747" 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/48566747/Cardinal_Arithmetic_in_the_Style_of_Baron_Von_M%C3%BCnchhausen">Cardinal Arithmetic in the Style of Baron Von Münchhausen</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="80365520" href="https://uu.academia.edu/albertvisser">albert visser</a></div><p class="ds-related-work--metadata ds2-5-body-xs">The Review of Symbolic Logic, 2009</p><p class="ds-related-work--abstract ds2-5-body-sm">In this paper we show how to interpret Robinson's Arithmetic Q and the theory R of Tarski, Mostowski and Robinson as theories of cardinals in very weak theories of relations over a domain. Bei der Verfolgung eines Hasen wollte ich mit meinem Pferdüber einen Morast setzen. Mitten im Sprung musste ich erkennen, dass der Morast viel breiter war, als ich anfänglich eingeschätzt hatte. Schwebend in der Luft wendete ich daher wieder um, wo ich hergekommen war, um einen größeren Anlauf zu nehmen. Gleichwohl sprang ich zum zweiten Mal noch zu kurz und fiel nicht weit vom anderen Ufer bis an den Hals in den Morast. Hier hätte ich unfehlbar umkommen müssen, wenn nicht die Stärke meines Armes mich an meinem eigenen Haarzopf, samt dem Pferd, welches ich fest zwischen meine Knie schloss, wieder herausgezogen hätte. Baron von Münchhausen.</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":"Cardinal Arithmetic in the Style of Baron Von Münchhausen","attachmentId":67111742,"attachmentType":"pdf","work_url":"https://www.academia.edu/48566747/Cardinal_Arithmetic_in_the_Style_of_Baron_Von_M%C3%BCnchhausen","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/48566747/Cardinal_Arithmetic_in_the_Style_of_Baron_Von_M%C3%BCnchhausen"><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="113731321" 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/113731321/Predicative_Logic_and_Formal_Arithmetic">Predicative Logic and Formal 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="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><p class="ds-related-work--abstract ds2-5-body-sm">After a summary of earlier work it is shown that elementary or Kalmar arithmetic can be interpreted within the system of Russell's Principia Mathematica with the axiom of infinity but without the axiom of reducibility. 1 Historical introduction After discovering the inconsistency in Frege's Grundgesetze der Arithmetik, Russell proposed two changes: first, dropping the assumption that to every higher-order entity there corresponds a first-order entity; and second, restricting the assumptions on the existence of higher-order entities, so that instead of a simple hierarchy of first-order, second-order, third-order, and so on, one has a ramified hierarchy in which each order is subdivided into various types in such a way that a condition involving quantification over all entities of one type is never assumed to determine another entity of the same type, but only of a higher type. But Russell found that with these two changes he could not derive classical mathematics, so in Principia Mathematica he partially compensated for the first change by assuming the axiom of infinity and, for all mathematical purposes, wholly undid the second change by assuming his axiom of reducibility. The predicativist tradition from Weyl [21] to Feferman [2] and beyond accepts infinity but rejects reducibility and is willing to give up parts of classical mathematics. However, predicativists have been unable to derive classical arithmetic and unwilling to give it up and so have simply assumed it as axiomatic. This assumption has its defenders, as with Feferman and Hellman [3], and also its detractors, as with C. Parsons [15]. It is, therefore, of some philosophical as well as historical interest to ask how large a fragment of classical arithmetic can be developed within the Russellian system of Principia Mathematica with infinity but without reducibility. Now many subsystems of classical or Peano arithmetic have been recognized since the work of Skolem [18], Kalmar [9], Grzegorczyk [4], and other pioneers. Among these the most studied have been the subprimitive or Grzegorczyk arithmetics n. These agree in allowing definitions by primitive recursion, but only when the function F being defined recursively is bounded by some function already given; or</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 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/113731321/Predicative_Logic_and_Formal_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="156497" 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/156497/A_Note_on_Logical_PERs_and_Reducibility_Logical_Relations_strike_again_">A Note on Logical PERs and Reducibility. Logical Relations strike again!</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="10506" href="https://upenn.academia.edu/JeanGallier">Jean Gallier</a></div><p class="ds-related-work--abstract ds2-5-body-sm">We prove a general theorem for establishing properties expressed by binary relations on typed ( rst-order) -terms, using a variant of the reducibility method and logical PERs. As an application, we prove simultaneously that -reduction in the simply-typed -calculus is strongly normalizing, and that the Church-Rosser property holds (and similarly for -reduction).</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":"A Note on Logical PERs and Reducibility. Logical Relations strike again!","attachmentId":40269,"attachmentType":"pdf","work_url":"https://www.academia.edu/156497/A_Note_on_Logical_PERs_and_Reducibility_Logical_Relations_strike_again_","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/156497/A_Note_on_Logical_PERs_and_Reducibility_Logical_Relations_strike_again_"><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="516850" 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/516850/A_note_on_H%C3%A1jek_Paris_and_Shepherdsons_theorem">A note on Hájek, Paris and Shepherdson's theorem</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="392902" href="https://aist-go.academia.edu/ShunsukeYatabe">Shunsuke Yatabe</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Logic Journal of IGPL, 2005</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":"A note on Hájek, Paris and Shepherdson's theorem","attachmentId":2480120,"attachmentType":"pdf","work_url":"https://www.academia.edu/516850/A_note_on_H%C3%A1jek_Paris_and_Shepherdsons_theorem","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/516850/A_note_on_H%C3%A1jek_Paris_and_Shepherdsons_theorem"><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="27862073" 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/27862073/A_Characterisation_of_the_Relations_Definable_in_Presburger_Arithmetic">A Characterisation of the Relations Definable in Presburger 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="15998172" href="https://ahus.academia.edu/MathiasBarra">Mathias Barra</a></div><p class="ds-related-work--abstract ds2-5-body-sm">Four sub-recursive classes of functions, B, D, BD and BDD are defined, and compared to the classes G 0 , G 1 and G 2 , originally defined by Grzegorczyk, based on bounded minimalisation, and charac-terised by Harrow in [5]. B is essentially G 0 with predecessor substituted for successor ; BD is G 1 with (truncated) difference substituted for addition. We prove that the induced relational classes are preserved (G 0 = B and G 1 = BD). We also obtain D = PrA qf (the quantifier free fragment of Presburger Arithmetic), and BD = PrA , and BDD = G 2 , where BDD is G 2 with integer division and remainder substituted for multiplication , and where G 2 is known to be equal to the predicates definable by a bounded formula in Peano Arithmetic.</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":"A Characterisation of the Relations Definable in Presburger Arithmetic","attachmentId":48146572,"attachmentType":"pdf","work_url":"https://www.academia.edu/27862073/A_Characterisation_of_the_Relations_Definable_in_Presburger_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 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js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/58886653/The_Arithmetic_Hierarchy_Parikhs_Theorem_and_Related_Matters">The Arithmetic Hierarchy, Parikh's Theorem and Related Matters</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="46358647" href="https://helsinki.academia.edu/JulietteKennedy">Juliette Kennedy</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2000</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":"The Arithmetic Hierarchy, Parikh's Theorem and Related Matters","attachmentId":73079808,"attachmentType":"pdf","work_url":"https://www.academia.edu/58886653/The_Arithmetic_Hierarchy_Parikhs_Theorem_and_Related_Matters","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/58886653/The_Arithmetic_Hierarchy_Parikhs_Theorem_and_Related_Matters"><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":34788887,"attachmentType":"pdf","workUrl":null}">See full PDF</button><button class="ds2-5-button ds2-5-button--secondary js-swp-download-button" 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The Reasoner, Vol (1) 8 p6-7</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="319497" href="https://independent.academia.edu/BhupinderSinghAnand">Bhupinder Singh Anand</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2007</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":"A c onstructive definition of the intuitive truth of the Axioms and Rules of Inference of Peano Arithmetic. 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