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On the positive side, Nelson embarks on a program of investigating how much mathematics can be interpreted in Raphael Robinson's theory of arithmetic Q. In the shadow of this program, some very nice logical investigations and results were produced by a number of people, not only regarding what can be interpreted in Q but also what cannot be so interpreted. We explain some of these results and rely on them to discuss Nelson's position.","publication_date":"2013,,","publication_name":"Bulletin of Symbolic Logic","grobid_abstract_attachment_id":"98823017"},"document_type":"paper","pre_hit_view_count_baseline":null,"quality":"high","language":"en","title":"Interpretability in Robinson's Q","broadcastable":false,"draft":null,"has_indexable_attachment":true,"indexable":true}}["work"]; window.loswp.workCoauthors = [11326288]; window.loswp.locale = "en"; window.loswp.countryCode = "SG"; window.loswp.cwvAbTestBucket = ""; window.loswp.designVariant = "ds_vanilla"; window.loswp.fullPageMobileSutdModalVariant = "control"; 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":98823017,"attachmentType":"pdf"}"><img alt="First page of “Interpretability in Robinson's Q”" class="ds-work-cover--cover-thumbnail" src="https://0.academia-photos.com/attachment_thumbnails/98823017/mini_magick20230218-1-1bu17l1.png?1676727569" /><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">Interpretability in Robinson's Q</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="11326288" href="https://lisboa.academia.edu/FernandoFerreira"><img alt="Profile image of Fernando Ferreira" class="ds-work-card--author-avatar" src="//a.academia-assets.com/images/s65_no_pic.png" />Fernando Ferreira</a></div><div class="ds-work-card--detail"><p class="ds-work-card--detail ds2-5-body-sm">2013, Bulletin of Symbolic Logic</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">28 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 = 97110182; 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if (!viewCountBody) { throw new Error('Failed to find work views element'); } viewCountBody.textContent = `${commaizedViewCount} views`; } catch (error) { // Remove the whole views element if there was some issue parsing. document.getElementById('work-metadata-view-count')?.parentNode?.remove(); throw new Error(`Failed to parse view count: ${viewCount}`, error); } }; // If the DOM is still loading, wait for it to be ready before updating the view count. if (document.readyState === "loading") { document.addEventListener('DOMContentLoaded', () => { updateViewCount(viewCount); }); // Otherwise, just update it immediately. } else { updateViewCount(viewCount); } })();</script></div><p class="ds-work-card--work-abstract ds-work-card--detail ds2-5-body-md">Edward Nelson published in 1986 a book defending an extreme formalist view of mathematics according to which there is an impassable barrier in the totality of exponentiation. On the positive side, Nelson embarks on a program of investigating how much mathematics can be interpreted in Raphael Robinson's theory of arithmetic Q. In the shadow of this program, some very nice logical investigations and results were produced by a number of people, not only regarding what can be interpreted in Q but also what cannot be so interpreted. We explain some of these results and rely on them to discuss Nelson's position.</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":98823017,"attachmentType":"pdf","workUrl":"https://www.academia.edu/97110182/Interpretability_in_Robinsons_Q"}">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":98823017,"attachmentType":"pdf","workUrl":"https://www.academia.edu/97110182/Interpretability_in_Robinsons_Q"}"><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="98823017" data-landing_url="https://www.academia.edu/97110182/Interpretability_in_Robinsons_Q" 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="48566791" 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/48566791/The_interpretability_logic_of_all_reasonable_arithmetical_theories">The interpretability logic of all reasonable arithmetical theories</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">Erkenntnis, 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 interpretability logic of all reasonable arithmetical theories","attachmentId":67111765,"attachmentType":"pdf","work_url":"https://www.academia.edu/48566791/The_interpretability_logic_of_all_reasonable_arithmetical_theories","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/48566791/The_interpretability_logic_of_all_reasonable_arithmetical_theories"><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="48566965" 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/48566965/The_Interpretability_Logic_of_all_Reasonable_Arithmetical_Theories_The_New_Conjecture">The Interpretability Logic of all Reasonable Arithmetical Theories. The New Conjecture</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">Logic Group Preprint Series, 2012</p><p class="ds-related-work--abstract ds2-5-body-sm">This paper is a presentation of a status qu stionis, to wit of the problem of the interpretability logic of all reasonable arithmetical theories. We present both the arithmetical side and the modal side of the question.</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 Interpretability Logic of all Reasonable Arithmetical Theories. The New Conjecture","attachmentId":67111544,"attachmentType":"pdf","work_url":"https://www.academia.edu/48566965/The_Interpretability_Logic_of_all_Reasonable_Arithmetical_Theories_The_New_Conjecture","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/48566965/The_Interpretability_Logic_of_all_Reasonable_Arithmetical_Theories_The_New_Conjecture"><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="117255565" 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/117255565/A_new_principle_in_the_interpretability_logic_of_all_reasonable_arithmetical_theories">A new principle in the interpretability logic of all reasonable arithmetical theories</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="51116229" href="https://independent.academia.edu/JoostJoosten">Joost Joosten</a></div><p class="ds-related-work--metadata ds2-5-body-xs">arXiv (Cornell University), 2020</p><p class="ds-related-work--abstract ds2-5-body-sm">The interpretability logic of a mathematical theory describes the structural behavior of interpretations over that theory. 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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="6" data-entity-id="53576956" 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/53576956/Some_consequences_of_a_recursive_number_theoretic_relation_that_is_not_the_standard_interpretation_of_any_of_its_formal_representations">Some consequences of a recursive number-theoretic relation that is not the standard interpretation of any of its formal representations</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="319497" href="https://independent.academia.edu/BhupinderSinghAnand">Bhupinder Singh Anand</a></div><p class="ds-related-work--abstract ds2-5-body-sm">We give precise definitions of primitive and formal mathematical objects, and show: there is an elementary, recursive, number-theoretic relation that is not a formal mathematical object in Gödel's formal system P, since it is not the standard interpretation of any of its representations in P; the range of a recursive number-theoretic function does not always define a formal mathematical object (recursively enumerable set) consistently in any Axiomatic Set Theory that is a model for P; there is no P-formula, Con(P), whose standard interpretation is unambiguously equivalent to Gödel's number-theoretic definition of "P is consistent"; every recursive number-theoretic function is not strongly representable in P; Tarski's definitions of "satisfiability" and "truth" can be made constructive, and intuitionistically unobjectionable, by reformulating Church's Thesis constructively; the classical definition of Turing machines can be extended to include self-terminating, converging, and oscillating routines; a constructive Church's Thesis implies, firstly, that every partial recursive number-theoretic function has a unique, constructive, extension as a total function, and, secondly, that we can define effectively computable number-theoretic functions that are not classically Turing-computable; Turing's and Cantor's diagonal arguments do not necessarily define Cauchy sequences. recognised as true 9 under classical interpretation 10 , but which are not provable 11 within the system. Does this imply that such recognition, in some cases, cannot be duplicated in any artificially constructed and, more important, artificially controlled, mechanism or organism whose design is based on classical logic? 12 The scientific, and philosophical, dimensions of an affirmative answer to the last question have been broadly reviewed, and addressed, by Roger Penrose in [Pe90] and [Pe94]. Penrose's argument is based on a strongly Platonist thesis that sensory perceptions Arithmetic, and Mendelson's formal system S ([Me64], p102) as a standard formalisation of classical first order Peano Arithmetic. 7 We follow Mendelson's definition of an "undecidable sentence" ([Me64], p143). 8 When referring to a formal language, we assume the terms "sentence" and "proposition" are synonymous, and that they refer to a well-formed expression of the language that contains no free variables, and which translates, under an interpretation, as a proposition in the usual, linguistic, sense. 9 We note that the term "true" is used both in its familiar linguistic sense, and in a mathematically precise sense; the appropriate meaning is usually obvious from the context. Mathematically, we follow Mendelson's exposition of the truth of a formal sentence under an interpretation as determined by Tarski's definitions of satisfiability and truth ([Me64], p51). 10 We note that the term "interpretation" is also used both in its familiar linguistic sense, and in a mathematical sense; the appropriate meaning is usually obvious from the context. Mathematically, we follow Mendelson's definitions of "interpretation" ([Me64], §2, p49), and of "standard interpretation" ([Me64], p107). We note that the interpreted relation R(x) is obtained from the formula [R(x)] of a formal system P by replacing every primitive, undefined symbol of P in the formula [R(x)] by an interpreted mathematical symbol (i.e. a symbol that is a shorthand notation for some, semantically well-defined, concept of classical mathematics). So the P-formula [(Ax)R(x)] interprets as the sentence (Ax)R(x), and the P-formula [~(Ax)R(x)] as the sentence ~(Ax)R(x). We also note that the meta-assertions "[(Ax)R(x)] is a true sentence under the interpretation M of P", and "(Ax)R(x) is a true sentence of the interpretation M of P", are equivalent to the meta-assertion "R(x) is satisfied for any given value of x in the domain of the interpretation M of P" ([Me64], p51).</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":"Some consequences of a recursive number-theoretic relation that is not the standard interpretation of any of its formal representations","attachmentId":70356805,"attachmentType":"pdf","work_url":"https://www.academia.edu/53576956/Some_consequences_of_a_recursive_number_theoretic_relation_that_is_not_the_standard_interpretation_of_any_of_its_formal_representations","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/53576956/Some_consequences_of_a_recursive_number_theoretic_relation_that_is_not_the_standard_interpretation_of_any_of_its_formal_representations"><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="11399538" 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/11399538/On_Interpretations_of_Arithmetic_and_Set_Theory">On Interpretations of Arithmetic and Set Theory</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="27506117" href="https://independent.academia.edu/TinLokWong">Tin-Lok Wong</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Notre Dame Journal of Formal Logic, 2007</p><p class="ds-related-work--abstract ds2-5-body-sm">being 'inverse to each other' admits important variations. For example, Chang and Keisler [5, §A.31] specify one particular choice of axiomatisation of ZF; for this axiomatisation a weak form of interpretation-equivalence of 'ZF with infinity negated' and PA can be proved, but for stronger notions of interpretation-equivalence a different axiomatisation of ZF seems to be required.</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 Interpretations of Arithmetic and Set Theory","attachmentId":46729721,"attachmentType":"pdf","work_url":"https://www.academia.edu/11399538/On_Interpretations_of_Arithmetic_and_Set_Theory","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/11399538/On_Interpretations_of_Arithmetic_and_Set_Theory"><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="49462986" 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/49462986/On_Interpretability_in_the_Theory_of_Concatenation">On Interpretability in the Theory of Concatenation</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="114851681" href="https://cuni.academia.edu/VitezslavSvejdar">Vitezslav Svejdar</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Notre Dame Journal of Formal Logic, 2009</p><p class="ds-related-work--abstract ds2-5-body-sm">We prove that a variant of Robinson arithmetic $\mathsf{Q}$ with nontotal operations is interpretable in the theory of concatenation $\mathsf{TC}$ introduced by A. Grzegorczyk. Since $\mathsf{Q}$ is known to be interpretable in that nontotal variant, our result gives a positive answer to the problem whether $\mathsf{Q}$ is interpretable in $\mathsf{TC}$ . An immediate consequence is essential undecidability of $\mathsf{TC}$ .</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 Interpretability in the Theory of Concatenation","attachmentId":67808410,"attachmentType":"pdf","work_url":"https://www.academia.edu/49462986/On_Interpretability_in_the_Theory_of_Concatenation","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/49462986/On_Interpretability_in_the_Theory_of_Concatenation"><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="8391573" 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/8391573/A_Remark_on_Q">A Remark on Q</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="8585079" href="https://utoronto.academia.edu/MihaiGanea">Mihai Ganea</a></div><p class="ds-related-work--abstract ds2-5-body-sm">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.</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 Remark on Q","attachmentId":34788887,"attachmentType":"pdf","work_url":"https://www.academia.edu/8391573/A_Remark_on_Q","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/8391573/A_Remark_on_Q"><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":98823017,"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":98823017,"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_98823017" 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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ds2-5-body-xs">Theoretical Computer Science, 2004</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":"Arithmetical definability and computational complexity","attachmentId":49428968,"attachmentType":"pdf","work_url":"https://www.academia.edu/5149149/Arithmetical_definability_and_computational_complexity","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/5149149/Arithmetical_definability_and_computational_complexity"><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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Arithmetical Hierarchy</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="36769883" href="https://independent.academia.edu/BorutRobic">Borut Robic</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Springer eBooks, 2015</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 Arithmetical Hierarchy","attachmentId":109950760,"attachmentType":"pdf","work_url":"https://www.academia.edu/112843929/The_Arithmetical_Hierarchy","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" 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class="ds-related-work--metadata ds2-5-body-xs">2002</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":"Some consequences of defining mathematical objects constructively and mathematical truth effectively -- A lay perspective","attachmentId":80176337,"attachmentType":"pdf","work_url":"https://www.academia.edu/70412185/Some_consequences_of_defining_mathematical_objects_constructively_and_mathematical_truth_effectively_A_lay_perspective","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" 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Logic, 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":"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-related-work-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-related-work-sidebar-card" data-collection-position="12" data-entity-id="488584" 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/488584/A_c_onstructive_definition_of_the_intuitive_truth_of_the_Axioms_and_Rules_of_Inference_of_Peano_Arithmetic_The_Reasoner_Vol_1_8_p6_7">A c onstructive definition of the intuitive truth of the Axioms and Rules of Inference of Peano Arithmetic. 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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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/63455901/Decidability_of_the_Multiplicative_and_Order_Theory_of_Numbers">Decidability of the Multiplicative and Order Theory of Numbers</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="161759955" href="https://independent.academia.edu/ZibaAssadi">Ziba Assadi</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2020</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":"Decidability of the Multiplicative and Order Theory of 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