{"id":{"repo_id":"ohiolink","oai_identifier":"oai:etd.ohiolink.edu:osu1366149288"},"canonical_url":"https://search.dev.ndltd.org/etd/ohiolink/oai:etd.ohiolink.edu:osu1366149288","repository":{"repo_id":"ohiolink","name":"OhioLINK","base_url":"https://etd.ohiolink.edu/acprod/odb_etd/ws/oai/oai"},"display":{"title":"Internal Set Theory and Euler's Introductio in Analysin Infinitorum","abstract":"In Leonhard Euler's seminal work <em>Introductio in Analysin Infinitorum</em> (1748), he readily used infinite numbers and infinitesimals in many of his proofs. We aim to reformulate a group of proofs from the <em>Introductio</em> using concepts and techniques from Abraham Robinson's celebrated Nonstandard Analysis (NSA); in particular, we will use Internal Set Theory, Edward Nelson's distinctive version of NSA. We will specifically examine Euler's proofs of the Euler formula, the Euler product, the Wallis product and the divergence of the harmonic series. All of these results have been proved in subsequent centuries using epsilontic arguments. In some cases, the epsilontic arguments differ significantly from Euler's original proofs. We will compare and contrast the epsilontic proofs with those we have developed by following Euler more closely through NSA. We claim that NSA possesses the tools to provide appropriate proxies of some--but certainly not all--of the inferential moves found in the <em>Introductio</em>.","abstract_html":"In Leonhard Euler&#x27;s seminal work &lt;em&gt;Introductio in Analysin Infinitorum&lt;/em&gt; (1748), he readily used infinite numbers and infinitesimals in many of his proofs. We aim to reformulate a group of proofs from the &lt;em&gt;Introductio&lt;/em&gt; using concepts and techniques from Abraham Robinson&#x27;s celebrated Nonstandard Analysis (NSA); in particular, we will use Internal Set Theory, Edward Nelson&#x27;s distinctive version of NSA. We will specifically examine Euler&#x27;s proofs of the Euler formula, the Euler product, the Wallis product and the divergence of the harmonic series. All of these results have been proved in subsequent centuries using epsilontic arguments. In some cases, the epsilontic arguments differ significantly from Euler&#x27;s original proofs. We will compare and contrast the epsilontic proofs with those we have developed by following Euler more closely through NSA. We claim that NSA possesses the tools to provide appropriate proxies of some--but certainly not all--of the inferential moves found in the &lt;em&gt;Introductio&lt;/em&gt;.","abstract_has_math":false,"creators":["Reeder, Patrick F."],"institution":"The Ohio State University","degree_name":"Master of Science","degree_level":"masters","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Sinnott, Warren"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2013,"date_issued":"2013-08-08","date_published":"2013-08-08","updated_at":"2026-07-24T03:37:46Z","subjects":["Mathematics","Logic","Euler","Nonstandard analysis","Internal Set Theory","Infinitesimal","Infinite"],"languages":["English"],"rights":["unrestricted","This thesis or dissertation is protected by copyright: all rights reserved. It may not be copied or redistributed beyond the terms of applicable copyright laws."],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://rave.ohiolink.edu/etdc/view?acc_num=osu1366149288","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Sinnott, Warren"]},{"key":"dc:creator","label":"Author","values":["Reeder, Patrick F."]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2013-08-08"]},{"key":"dc:publisher","label":"Institution","values":["The Ohio State University / OhioLINK"]},{"key":"dc:type","label":"Dc Type","values":["Electronic Thesis or Dissertation"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["masters"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["The Ohio State University"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics","Logic","Euler","Nonstandard analysis","Internal Set Theory","Infinitesimal","Infinite"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["English"]},{"key":"dc:rights","label":"Dc Rights","values":["unrestricted","This thesis or dissertation is protected by copyright: all rights reserved. It may not be copied or redistributed beyond the terms of applicable copyright laws."]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://rave.ohiolink.edu/etdc/view?acc_num=osu1366149288"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["In Leonhard Euler's seminal work <em>Introductio in Analysin Infinitorum</em> (1748), he readily used infinite numbers and infinitesimals in many of his proofs. We aim to reformulate a group of proofs from the <em>Introductio</em> using concepts and techniques from Abraham Robinson's celebrated Nonstandard Analysis (NSA); in particular, we will use Internal Set Theory, Edward Nelson's distinctive version of NSA. We will specifically examine Euler's proofs of the Euler formula, the Euler product, the Wallis product and the divergence of the harmonic series. All of these results have been proved in subsequent centuries using epsilontic arguments. In some cases, the epsilontic arguments differ significantly from Euler's original proofs. We will compare and contrast the epsilontic proofs with those we have developed by following Euler more closely through NSA. We claim that NSA possesses the tools to provide appropriate proxies of some--but certainly not all--of the inferential moves found in the <em>Introductio</em>."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf","p.54","510.8 KB"]},{"key":"dc:title","label":"Title","values":["Internal Set Theory and Euler's Introductio in Analysin Infinitorum"]}]}],"canonical_facts":{"dc:contributor":["Sinnott, Warren"],"dc:creator":["Reeder, Patrick F."],"dc:date":["2013-08-08"],"dc:description":["In Leonhard Euler's seminal work <em>Introductio in Analysin Infinitorum</em> (1748), he readily used infinite numbers and infinitesimals in many of his proofs. We aim to reformulate a group of proofs from the <em>Introductio</em> using concepts and techniques from Abraham Robinson's celebrated Nonstandard Analysis (NSA); in particular, we will use Internal Set Theory, Edward Nelson's distinctive version of NSA. We will specifically examine Euler's proofs of the Euler formula, the Euler product, the Wallis product and the divergence of the harmonic series. All of these results have been proved in subsequent centuries using epsilontic arguments. In some cases, the epsilontic arguments differ significantly from Euler's original proofs. We will compare and contrast the epsilontic proofs with those we have developed by following Euler more closely through NSA. We claim that NSA possesses the tools to provide appropriate proxies of some--but certainly not all--of the inferential moves found in the <em>Introductio</em>."],"dc:format":["application/pdf","p.54","510.8 KB"],"dc:identifier":["http://rave.ohiolink.edu/etdc/view?acc_num=osu1366149288"],"dc:language":["English"],"dc:publisher":["The Ohio State University / OhioLINK"],"dc:rights":["unrestricted","This thesis or dissertation is protected by copyright: all rights reserved. It may not be copied or redistributed beyond the terms of applicable copyright laws."],"dc:subject":["Mathematics","Logic","Euler","Nonstandard analysis","Internal Set Theory","Infinitesimal","Infinite"],"dc:title":["Internal Set Theory and Euler's Introductio in Analysin Infinitorum"],"dc:type":["Electronic Thesis or Dissertation"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["masters"],"thesis:degree_name":["Master of Science"],"thesis:institution_name":["The Ohio State University"]},"updated_at":"2026-07-24T03:37:46Z"}