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(PDF) How to Pick Out the Integers in the Rationals: An Application of Number Theory to Logic
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window.loswp.willEdgeCache = false; window.loswp.work = {"work":{"id":110331305,"created_at":"2023-12-01T17:42:33.510-08:00","from_world_paper_id":244545116,"updated_at":"2025-02-02T03:32:11.404-08:00","_data":{"publisher":"Mathematical Association of America","ai_abstract":"This paper explores the logical relationship between the integers and rational numbers, specifically addressing the definability of integers within the framework of rational numbers through tools of number theory and logic. Building on Julia Robinson's foundational work, it establishes a method to define integers in the rational number system and examines essential lemmas and theorems that show the connections between concepts in number theory and logic. The findings demonstrate that although the characterization of integers is complex, it can be effectively approached using existential quantifiers.","ai_title_tag":"Defining Integers in Rationals via Logic","publication_date":"1991,11,1","publication_name":"American Mathematical Monthly"},"document_type":"paper","pre_hit_view_count_baseline":null,"quality":"high","language":"en","title":"How to Pick Out the Integers in the Rationals: An Application of Number Theory to Logic","broadcastable":true,"draft":null,"has_indexable_attachment":true,"indexable":true,"seo_quality":null}}["work"]; window.loswp.workCoauthors = [33888294]; 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'; window.userInChina = "false";</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":108181304,"attachmentType":"pdf"}"><img alt="First page of “How to Pick Out the Integers in the Rationals: An Application of Number Theory to Logic”" class="ds-work-cover--cover-thumbnail" src="https://0.academia-photos.com/attachment_thumbnails/108181304/mini_magick20231202-1-2b2yea.png?1701481601" /><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">How to Pick Out the Integers in the Rationals: An Application of Number Theory to Logic</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="33888294" href="https://macalester.academia.edu/DanFlath"><img alt="Profile image of Dan Flath" class="ds-work-card--author-avatar" src="https://0.academia-photos.com/33888294/39630324/32729711/s65_dan.flath.jpg" />Dan Flath</a></div><div class="ds-work-card--detail"><p class="ds-work-card--detail ds2-5-body-sm">1991, American Mathematical Monthly</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">13 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 = 110331305; 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Building on Julia Robinson's foundational work, it establishes a method to define integers in the rational number system and examines essential lemmas and theorems that show the connections between concepts in number theory and logic. The findings demonstrate that although the characterization of integers is complex, it can be effectively approached using existential quantifiers.</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="97982632" 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/97982632/A_Substitutional_Framework_for_Arithmetical_Validity">A Substitutional Framework for Arithmetical Validity</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">Grazer Philosophische Studien, 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":"A Substitutional Framework for Arithmetical Validity","attachmentId":99457912,"attachmentType":"pdf","work_url":"https://www.academia.edu/97982632/A_Substitutional_Framework_for_Arithmetical_Validity","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/97982632/A_Substitutional_Framework_for_Arithmetical_Validity"><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="86388849" 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/86388849/A_History_of_Interactions_between_Logic_and_Number_Theory">A History of Interactions between Logic and Number 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="205178147" href="https://independent.academia.edu/Angusmacintyre">Angus macintyre</a></div><p class="ds-related-work--abstract ds2-5-body-sm">In an earlier period the relevant logical component was recursion theory (decidability and undecidability). 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The class will cover debates about metaphysics and the epistemology of numbers and arithmetic truths.</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":"Philosophy of Arithmetic","attachmentId":95681402,"attachmentType":"pdf","work_url":"https://www.academia.edu/92756599/Philosophy_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/92756599/Philosophy_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="4" 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="{"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-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="5" 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="6" data-entity-id="75285527" 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/75285527/On_Axiomatizability_of_the_Multiplicative_Theory_of_Numbers">On Axiomatizability of the Multiplicative Theory of 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="117308999" href="https://tabrizu.academia.edu/SaeedSalehi">Saeed Salehi</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 multiplicative theory of a set of numbers (which could be natural, integer, rational, real or complex numbers) is the first-order theory of the structure of that set with (solely) the multiplication operation (that set is taken to be multiplicative, i.e., closed under multiplication). In this paper we study the multiplicative theories of the complex, real and (positive) rational numbers. These theories (and also the multiplicative theories of natural and integer numbers) are known to be decidable (i.e., there exists an algorithm that decides whether a given sentence is derivable form the theory); here we present explicit axiomatizations for them and show that they are not finitely axiomatizable. For each of these sets (of complex, real and [positive] rational numbers) a language, including the multiplication operation, is introduced in a way that it allows quantifier elimination (for the theory of that set).</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 Axiomatizability of the Multiplicative Theory of Numbers","attachmentId":83115707,"attachmentType":"pdf","work_url":"https://www.academia.edu/75285527/On_Axiomatizability_of_the_Multiplicative_Theory_of_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/75285527/On_Axiomatizability_of_the_Multiplicative_Theory_of_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="7" 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="8" data-entity-id="21480115" 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/21480115/First_Order_Proof_Theory_of_Arithmetic">First-Order Proof Theory 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="5937540" href="https://ucsd.academia.edu/SamBuss">Sam Buss</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Studies in Logic and the Foundations of Mathematics, 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":"First-Order Proof Theory of Arithmetic","attachmentId":41904055,"attachmentType":"pdf","work_url":"https://www.academia.edu/21480115/First_Order_Proof_Theory_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/21480115/First_Order_Proof_Theory_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="9" data-entity-id="53141370" 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/53141370/On_Certain_Axiomatizations_of_Arithmetic_of_Natural_and_Integer_Numbers">On Certain Axiomatizations of Arithmetic of Natural and Integer 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="48364096" href="https://uksw.academia.edu/UrszulaWS">Urszula Wybraniec-Skardowska</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Axioms</p><p class="ds-related-work--abstract ds2-5-body-sm">The systems of arithmetic discussed in this work are non-elementary theories. In this paper, natural numbers are characterized axiomatically in two different ways. We begin by recalling the classical set P of axioms of Peano’s arithmetic of natural numbers proposed in 1889 (including such primitive notions as: set of natural numbers, zero, successor of natural number) and compare it with the set W of axioms of this arithmetic (including the primitive notions like: set of natural numbers and relation of inequality) proposed by Witold Wilkosz, a Polish logician, philosopher and mathematician, in 1932. The axioms W are those of ordered sets without largest element, in which every non-empty set has a least element, and every set bounded from above has a greatest element. We show that P and W are equivalent and also that the systems of arithmetic based on W or on P, are categorical and consistent. There follows a set of intuitive axioms PI of integers arithmetic, modelled on P and propos...</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 Axiomatizations of Arithmetic of Natural and Integer Numbers","attachmentId":70067696,"attachmentType":"pdf","work_url":"https://www.academia.edu/53141370/On_Certain_Axiomatizations_of_Arithmetic_of_Natural_and_Integer_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/53141370/On_Certain_Axiomatizations_of_Arithmetic_of_Natural_and_Integer_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></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":108181304,"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":108181304,"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_108181304" 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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href="https://www.academia.edu/112851703/The_logical_character_of_number_reply_to_abel_lasalle_casanave"><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="3" data-entity-id="7462101" 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/7462101/Logic_and_Arithmetic">Logic and 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="13153598" href="https://uwa.academia.edu/HartleySlater">Hartley Slater</a></div><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" 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