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(PDF) On Interpretability in the Theory of Concatenation | Vitezslav Svejdar - Academia.edu

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Since $\mathsf{Q}$ is known to be interpretable in that nontotal variant," /> <meta property="article:author" content="https://cuni.academia.edu/VitezslavSvejdar" /> <meta name="description" content="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," /> <title>(PDF) On Interpretability in the Theory of Concatenation | Vitezslav Svejdar - Academia.edu</title> <link rel="canonical" href="https://www.academia.edu/49462986/On_Interpretability_in_the_Theory_of_Concatenation" /> <script async src="https://www.googletagmanager.com/gtag/js?id=G-5VKX33P2DS"></script> <script> window.dataLayer = window.dataLayer || []; function gtag(){dataLayer.push(arguments);} gtag('js', new Date()); gtag('config', 'G-5VKX33P2DS', { cookie_domain: 'academia.edu', send_page_view: false, }); gtag('event', 'page_view', { 'controller': "single_work", 'action': "show", 'controller_action': 'single_work#show', 'logged_in': 'false', 'edge': 'unknown', // Send nil if there is no A/B test bucket, in case some records get logged // with missing data - that way we can distinguish between the two cases. // ab_test_bucket should be of the form <ab_test_name>:<bucket> 'ab_test_bucket': null, }) </script> <script> var $controller_name = 'single_work'; var $action_name = "show"; var $rails_env = 'production'; var $app_rev = 'd19b14b503aa9b51cb8adbd6e2077e0ec40bde90'; var $domain = 'academia.edu'; var $app_host = "academia.edu"; var $asset_host = "academia-assets.com"; var $start_time = new Date().getTime(); var $recaptcha_key = "6LdxlRMTAAAAADnu_zyLhLg0YF9uACwz78shpjJB"; var $recaptcha_invisible_key = "6Lf3KHUUAAAAACggoMpmGJdQDtiyrjVlvGJ6BbAj"; var $disableClientRecordHit = false; </script> <script> window.require = { config: function() { return function() {} } } </script> <script> window.Aedu = window.Aedu || {}; window.Aedu.hit_data = null; window.Aedu.serverRenderTime = new Date(1734711946000); window.Aedu.timeDifference = new Date().getTime() - 1734711946000; </script> <script type="application/ld+json">{"@context":"https://schema.org","@type":"ScholarlyArticle","abstract":"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 $\\mathsf{TC}$ .","author":[{"@context":"https://schema.org","@type":"Person","name":"Vitezslav Svejdar"}],"contributor":[],"dateCreated":"2021-06-29","dateModified":"2021-06-29","datePublished":"2009-01-01","headline":"On Interpretability in the Theory of Concatenation","image":"https://attachments.academia-assets.com/67808410/thumbnails/1.jpg","inLanguage":"en","keywords":["Philosophy","Pure Mathematics","Concatenation"],"publication":"Notre Dame Journal of Formal Logic","publisher":{"@context":"https://schema.org","@type":"Organization","name":"Duke University Press"},"sourceOrganization":[{"@context":"https://schema.org","@type":"EducationalOrganization","name":"cuni"}],"thumbnailUrl":"https://attachments.academia-assets.com/67808410/thumbnails/1.jpg","url":"https://www.academia.edu/49462986/On_Interpretability_in_the_Theory_of_Concatenation"}</script><link rel="stylesheet" media="all" 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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 $\\mathsf{TC}$ .","publication_date":"2009,,","publication_name":"Notre Dame Journal of Formal Logic"},"document_type":"paper","pre_hit_view_count_baseline":null,"quality":"high","language":"en","title":"On Interpretability in the Theory of Concatenation","broadcastable":true,"draft":null,"has_indexable_attachment":true,"indexable":true}}["work"]; window.loswp.workCoauthors = [114851681]; 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="{&quot;location&quot;:&quot;swp-splash-paper-cover&quot;,&quot;attachmentId&quot;:67808410,&quot;attachmentType&quot;:&quot;pdf&quot;}"><img alt="First page of “On Interpretability in the Theory of Concatenation”" class="ds-work-cover--cover-thumbnail" src="https://0.academia-photos.com/attachment_thumbnails/67808410/mini_magick20210629-21761-xvagn3.png?1625000432" /><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">On Interpretability in the Theory of Concatenation</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="114851681" href="https://cuni.academia.edu/VitezslavSvejdar"><img alt="Profile image of Vitezslav Svejdar" class="ds-work-card--author-avatar" src="//a.academia-assets.com/images/s65_no_pic.png" />Vitezslav Svejdar</a></div><div class="ds-work-card--detail"><p class="ds-work-card--detail ds2-5-body-sm">2009, Notre Dame Journal of Formal 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">9 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 = 49462986; 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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">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-work-card--button-container"><button class="ds2-5-button js-swp-download-button" data-signup-modal="{&quot;location&quot;:&quot;continue-reading-button--work-card&quot;,&quot;attachmentId&quot;:67808410,&quot;attachmentType&quot;:&quot;pdf&quot;,&quot;workUrl&quot;:&quot;https://www.academia.edu/49462986/On_Interpretability_in_the_Theory_of_Concatenation&quot;}">See full PDF</button><button class="ds2-5-button ds2-5-button--secondary js-swp-download-button" data-signup-modal="{&quot;location&quot;:&quot;download-pdf-button--work-card&quot;,&quot;attachmentId&quot;:67808410,&quot;attachmentType&quot;:&quot;pdf&quot;,&quot;workUrl&quot;:&quot;https://www.academia.edu/49462986/On_Interpretability_in_the_Theory_of_Concatenation&quot;}"><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="67808410" data-landing_url="https://www.academia.edu/49462986/On_Interpretability_in_the_Theory_of_Concatenation" 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="97110182" 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/97110182/Interpretability_in_Robinsons_Q">Interpretability in Robinson&#39;s 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="11326288" href="https://lisboa.academia.edu/FernandoFerreira">Fernando Ferreira</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Bulletin of Symbolic Logic, 2013</p><p class="ds-related-work--abstract ds2-5-body-sm">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. 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Different theories have different logics. This paper revolves around the question what logic describes the behavior that is present in all theories with a minimum amount of arithmetic; the intersection over all such theories so to say. We denote this target logic by IL(All). In this paper we present a new principle R in IL(All). We show that R does not follow from the logic ILP0W * that contains all previously known principles. This is done by providing a modal incompleteness proof of ILP0W * : showing that R follows semantically but not syntactically from ILP0W *. Apart from giving the incompleteness proof by elementary methods, we also sketch how to work with so-called Generalized Veltman Semantics as to establish incompleteness. To this extent, a new version of this Generalized Veltman Semantics is defined and studied. Moreover, for the important principles the frame correspondences are calculated. After the modal results it is shown that the new principle R is indeed valid in any arithmetically theory. The proof employs some elementary results on definable cuts in arithmetical theories. 1 Technically speaking the property of so-called essential reflexivity is sufficient. A theory is</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="{&quot;location&quot;:&quot;wsj-grid-card-download-pdf-modal&quot;,&quot;work_title&quot;:&quot;A new principle in the interpretability logic of all reasonable arithmetical theories&quot;,&quot;attachmentId&quot;:113160838,&quot;attachmentType&quot;:&quot;pdf&quot;,&quot;work_url&quot;:&quot;https://www.academia.edu/117255565/A_new_principle_in_the_interpretability_logic_of_all_reasonable_arithmetical_theories&quot;,&quot;alternativeTracking&quot;: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/117255565/A_new_principle_in_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="5" data-entity-id="5149149" 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/5149149/Arithmetical_definability_and_computational_complexity">Arithmetical definability and computational complexity</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="6951880" href="https://esti.academia.edu/HachaichiYassine">Hachaichi Yassine</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Theoretical Computer Science, 2004</p><p class="ds-related-work--abstract ds2-5-body-sm">In this paper, we introduce and study some syntactical fragments of monadic second-order and ÿrst-order (PEANO) arithmetic which we will prove the connection to famous complexity classes. Starting from descriptive complexity results, and giving an e ective method for translating formulas between di erent logical structures representing encodings of integers, we give some new arithmetical characterizations of NP, PH, NL, and P.</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="{&quot;location&quot;:&quot;wsj-grid-card-download-pdf-modal&quot;,&quot;work_title&quot;:&quot;Arithmetical definability and computational complexity&quot;,&quot;attachmentId&quot;:49428968,&quot;attachmentType&quot;:&quot;pdf&quot;,&quot;work_url&quot;:&quot;https://www.academia.edu/5149149/Arithmetical_definability_and_computational_complexity&quot;,&quot;alternativeTracking&quot;: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/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 class="ds-related-work--container js-wsj-grid-card" data-collection-position="6" data-entity-id="80112869" 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/80112869/The_interpretability_logic_of_Peano_arithmetic">The interpretability logic 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="18337933" href="https://independent.academia.edu/AlessandroBerarducci">Alessandro Berarducci</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Symbolic Logic, 1990</p><p class="ds-related-work--abstract ds2-5-body-sm">PA is Peano arithmetic. The formula InterpPA(α, β) is a formalization of the assertion that the theory PA + α interprets the theory PA + β (the variables α and β are intended to range over codes of sentences of PA). We extend Solovay&amp;#39;s modal analysis of the formalized provability predicate of PA, PrPA(x), to the case of the formalized interpretability relation InterpPA(x, y). The relevant modal logic, in addition to the usual provability operator ‘□’, has a binary operator ‘⊳’ to be interpreted as the formalized interpretability relation. We give an axiomatization and a decision procedure for the class of those modal formulas that express valid interpretability principles (for every assignment of the atomic modal formulas to sentences of PA). Our results continue to hold if we replace the base theory PA with Zermelo-Fraenkel set theory, but not with Gödel-Bernays set theory. This sensitivity to the base theory shows that the language is quite expressive. Our proof uses in an ess...</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="{&quot;location&quot;:&quot;wsj-grid-card-download-pdf-modal&quot;,&quot;work_title&quot;:&quot;The interpretability logic of Peano arithmetic&quot;,&quot;attachmentId&quot;:86603412,&quot;attachmentType&quot;:&quot;pdf&quot;,&quot;work_url&quot;:&quot;https://www.academia.edu/80112869/The_interpretability_logic_of_Peano_arithmetic&quot;,&quot;alternativeTracking&quot;: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/80112869/The_interpretability_logic_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="7" data-entity-id="8584745" 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/8584745/The_Semantic_Tableaux_Version_of_the_Second_Incompleteness_Theorem_Extends_Almost_to_Robinsons_Arithmetic_Q">The Semantic Tableaux Version of the Second Incompleteness Theorem Extends Almost to Robinson&#39;s Arithmetic 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="17729655" href="https://neu-vn.academia.edu/LinhDan">Linh Dan</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2000</p><p class="ds-related-work--abstract ds2-5-body-sm">We will generalize the Second Incompleteness Theorem almost to the level of Robinson&#39;s System Q. We will prove there exists a Π1 sentence V , such that if α is any finite consistent extension of Q+V then α will be unable to prove its Semantic Tableaux consistency.</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="{&quot;location&quot;:&quot;wsj-grid-card-download-pdf-modal&quot;,&quot;work_title&quot;:&quot;The Semantic Tableaux Version of the Second Incompleteness Theorem Extends Almost to Robinson&#39;s Arithmetic Q&quot;,&quot;attachmentId&quot;:34953292,&quot;attachmentType&quot;:&quot;pdf&quot;,&quot;work_url&quot;:&quot;https://www.academia.edu/8584745/The_Semantic_Tableaux_Version_of_the_Second_Incompleteness_Theorem_Extends_Almost_to_Robinsons_Arithmetic_Q&quot;,&quot;alternativeTracking&quot;: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/8584745/The_Semantic_Tableaux_Version_of_the_Second_Incompleteness_Theorem_Extends_Almost_to_Robinsons_Arithmetic_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 class="ds-related-work--container js-wsj-grid-card" data-collection-position="8" data-entity-id="85680722" 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/85680722/Arithmetic_Formulated_in_a_Logic_of_Meaning_Containment">Arithmetic Formulated in a Logic of Meaning Containment</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="693815" href="https://illinois.academia.edu/RossBrady">Ross Brady</a></div><p class="ds-related-work--metadata ds2-5-body-xs">The Australasian Journal of Logic, 2021</p><p class="ds-related-work--abstract ds2-5-body-sm">We assess Meyer’s formalization of arithmetic in his [21], based on the strong relevant logic R and compare this with arithmetic based on a suitable logic of meaning containment, which was developed in Brady [7]. We argue in favour of the latter as it better captures the key logical concepts of meaning and truth in arithmetic. We also contrast the two approaches to classical recapture, again favouring our approach in [7]. We then consider our previous development of Peano arithmetic including primitive recursive functions, finally extending this work to that of general recursion.</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="{&quot;location&quot;:&quot;wsj-grid-card-download-pdf-modal&quot;,&quot;work_title&quot;:&quot;Arithmetic Formulated in a Logic of Meaning Containment&quot;,&quot;attachmentId&quot;:90304942,&quot;attachmentType&quot;:&quot;pdf&quot;,&quot;work_url&quot;:&quot;https://www.academia.edu/85680722/Arithmetic_Formulated_in_a_Logic_of_Meaning_Containment&quot;,&quot;alternativeTracking&quot;: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/85680722/Arithmetic_Formulated_in_a_Logic_of_Meaning_Containment"><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="3292839" 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/3292839/Interpretability_over_Peano_Arithmetic">Interpretability over 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="3804648" href="https://gu-se.academia.edu/ClaesStranneg%C3%A5rd">Claes Strannegård</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Symbolic Logic, 1999</p><p class="ds-related-work--abstract ds2-5-body-sm">We investigate the modal logic of interpretability over Peano arithmetic (PA). Our main result is an extension of the arithmetical completeness theorem for the interpretability logic ILM ! . This extension concerns recursively enumerable sets of formulas of interpretability logic (rather than single formulas). As corollaries we obtain a uniform arithmetical completeness theorem for the interpretability logic ILM and a theorem answering a question of Orey from 1961. All these results also hold for Zermelo-Fraenkel set theory (ZF).</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="{&quot;location&quot;:&quot;wsj-grid-card-download-pdf-modal&quot;,&quot;work_title&quot;:&quot;Interpretability over Peano Arithmetic&quot;,&quot;attachmentId&quot;:50362267,&quot;attachmentType&quot;:&quot;pdf&quot;,&quot;work_url&quot;:&quot;https://www.academia.edu/3292839/Interpretability_over_Peano_Arithmetic&quot;,&quot;alternativeTracking&quot;: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/3292839/Interpretability_over_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></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="{&quot;location&quot;:&quot;continue-reading-button--sticky-ctas&quot;,&quot;attachmentId&quot;:67808410,&quot;attachmentType&quot;:&quot;pdf&quot;,&quot;workUrl&quot;:null}">See full PDF</button><button class="ds2-5-button ds2-5-button--secondary js-swp-download-button" data-signup-modal="{&quot;location&quot;:&quot;download-pdf-button--sticky-ctas&quot;,&quot;attachmentId&quot;:67808410,&quot;attachmentType&quot;:&quot;pdf&quot;,&quot;workUrl&quot;: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_67808410" 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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