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<!DOCTYPE html> <html > <head> <meta charset="utf-8"> <meta rel="search" type="application/opensearchdescription+xml" href="/open_search.xml" title="Academia.edu"> <meta content="width=device-width, initial-scale=1" name="viewport"> <meta name="google-site-verification" content="bKJMBZA7E43xhDOopFZkssMMkBRjvYERV-NaN4R6mrs"> <meta name="csrf-param" content="authenticity_token" /> <meta name="csrf-token" content="0ebklLO_clBSd8D6OWQkHmAdw4DfpAS2Sa-D-sT9shP8wxoMhL9E5LDVOpd6UedYs1n_7cCp3v8xM1dtHMf7BQ" /> <meta name="citation_title" content="Transseries as germs of surreal functions" /> <meta name="citation_publication_date" content="2018/01/01" /> <meta name="citation_journal_title" content="Transactions of the American Mathematical Society" /> <meta name="citation_author" content="Alessandro Berarducci" /> <meta name="twitter:card" content="summary" /> <meta name="twitter:url" content="https://www.academia.edu/80112883/Transseries_as_germs_of_surreal_functions" /> <meta name="twitter:title" content="Transseries as germs of surreal functions" /> <meta name="twitter:description" content="We show that Écalle’s transseries and their variants (LE and EL-series) can be interpreted as functions from positive infinite surreal numbers to surreal numbers. The same holds for a much larger class of formal series, here called omega-series." /> <meta name="twitter:image" content="https://0.academia-photos.com/18337933/22540702/21734456/s200_alessandro.berarducci.jpg" /> <meta property="fb:app_id" content="2369844204" /> <meta property="og:type" content="article" /> <meta property="og:url" content="https://www.academia.edu/80112883/Transseries_as_germs_of_surreal_functions" /> <meta property="og:title" content="Transseries as germs of surreal functions" /> <meta property="og:image" content="http://a.academia-assets.com/images/open-graph-icons/fb-paper.gif" /> <meta property="og:description" content="We show that Écalle’s transseries and their variants (LE and EL-series) can be interpreted as functions from positive infinite surreal numbers to surreal numbers. The same holds for a much larger class of formal series, here called omega-series." /> <meta property="article:author" content="https://independent.academia.edu/AlessandroBerarducci" /> <meta name="description" content="We show that Écalle’s transseries and their variants (LE and EL-series) can be interpreted as functions from positive infinite surreal numbers to surreal numbers. The same holds for a much larger class of formal series, here called omega-series." /> <title>Transseries as germs of surreal functions</title> <link rel="canonical" href="https://www.academia.edu/80112883/Transseries_as_germs_of_surreal_functions" /> <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 = '6b4daa9be5f38c297ec3ee4ed17d8b628bfdfdc7'; 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(1741722983000); window.Aedu.timeDifference = new Date().getTime() - 1741722983000; </script> <script type="application/ld+json">{"@context":"https://schema.org","@type":"ScholarlyArticle","abstract":"We show that Écalle’s transseries and their variants (LE and EL-series) can be interpreted as functions from positive infinite surreal numbers to surreal numbers. The same holds for a much larger class of formal series, here called omega-series. Omega-series are the smallest subfield of the surreal numbers containing the reals, the ordinal omega, and closed under the exp and log functions and all possible infinite sums. They form a proper class, can be composed and differentiated, and are surreal analytic. The surreal numbers themselves can be interpreted as a large field of transseries containing the omega-series, but, unlike omega-series, they lack a composition operator compatible with the derivation introduced by the authors in an earlier paper.","author":[{"@context":"https://schema.org","@type":"Person","name":"Alessandro Berarducci","url":"https://independent.academia.edu/AlessandroBerarducci","image":"https://0.academia-photos.com/18337933/22540702/21734456/s200_alessandro.berarducci.jpg","sameAs":[]}],"contributor":[],"dateCreated":"2022-05-28","dateModified":"2025-02-03","datePublished":"2018-01-01","headline":"Transseries as germs of surreal functions","image":"https://attachments.academia-assets.com/86603419/thumbnails/1.jpg","inLanguage":"en","keywords":["Pure Mathematics"],"publication":"Transactions of the American Mathematical Society","publisher":{"@context":"https://schema.org","@type":"Organization","name":"American Mathematical Society 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The same holds for a much larger class of formal series, here called omega-series. Omega-series are the smallest subfield of the surreal numbers containing the reals, the ordinal omega, and closed under the exp and log functions and all possible infinite sums. They form a proper class, can be composed and differentiated, and are surreal analytic. The surreal numbers themselves can be interpreted as a large field of transseries containing the omega-series, but, unlike omega-series, they lack a composition operator compatible with the derivation introduced by the authors in an earlier paper.","publisher":"American Mathematical Society (AMS)","ai_title_tag":"Transseries and Omega-Series in Surreal Analysis","publication_date":"2018,,","publication_name":"Transactions of the American Mathematical Society"},"document_type":"paper","pre_hit_view_count_baseline":null,"quality":"high","language":"en","title":"Transseries as germs of surreal functions","broadcastable":false,"draft":null,"has_indexable_attachment":true,"indexable":true,"seo_quality":null}}["work"]; window.loswp.workCoauthors = [18337933]; window.loswp.locale = "en"; window.loswp.countryCode = "SG"; window.loswp.cwvAbTestBucket = ""; window.loswp.designVariant = "safe_v1"; window.loswp.fullPageMobileSutdModalVariant = "control"; window.loswp.useOptimizedScribd4genScript = false; 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The same holds for a much larger class of formal series, here called omega-series. Omega-series are the smallest subfield of the surreal numbers containing the reals, the ordinal omega, and closed under the exp and log functions and all possible infinite sums. They form a proper class, can be composed and differentiated, and are surreal analytic. The surreal numbers themselves can be interpreted as a large field of transseries containing the omega-series, but, unlike omega-series, they lack a composition operator compatible with the derivation introduced by the authors in an earlier paper.</p></div></div></div></div></div></div><div class="ds-top-related-works--grid-container"><div class="ds-related-content--container ds-top-related-works--container"><style type="text/css">.ds-loswp-section--container { display: flex; flex-direction: column; width: 100%; padding-right: 40px; }</style><div class="ds2-5-content-section"><div class="ds2-5-content-section__heading"><h2 class="ds2-5-content-section__heading__text">Related papers</h2></div><div class="ds2-5-content-section__main"><div class="ds2-5-content-section__main__content"><div class="ds-loswp-section--container"><div class="ds-related-content--container"><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="0" data-entity-id="80112891" 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/80112891/Surreal_numbers_derivations_and_transseries">Surreal numbers, derivations and transseries</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 the European Mathematical Society, 2018</p><p class="ds-related-work--abstract ds2-5-body-sm">Several authors have conjectured that Conway's field of surreal numbers, equipped with the exponential function of Kruskal and Gonshor, can be described as a field of transseries and admits a compatible differential structure of Hardy-type. In this paper we give a complete positive solution to both problems. We also show that with this new differential structure, the surreal numbers are Liouville closed, namely the derivation is surjective.</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":"Surreal numbers, derivations and transseries","attachmentId":86603426,"attachmentType":"pdf","work_url":"https://www.academia.edu/80112891/Surreal_numbers_derivations_and_transseries","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/80112891/Surreal_numbers_derivations_and_transseries"><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="118335569" 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/118335569/Surreal_Analysis_An_Analogue_of_Real_Analysis_for_Surreal_Numbers">Surreal Analysis: An Analogue of Real Analysis for Surreal 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="39227294" href="https://independent.academia.edu/AshvinAnandSwaminathan">Ashvin Anand Swaminathan</a></div><p class="ds-related-work--metadata ds2-5-body-xs">arXiv (Cornell University), 2013</p><p class="ds-related-work--abstract ds2-5-body-sm">The class No of surreal numbers, which John Conway discovered while studying combinatorial games, possesses a rich numerical structure and shares many arithmetic and algebraic properties with the real numbers. Some work has also been done to develop analysis on No. In this paper, we extend this work with a treatment of functions, limits, derivatives, power series, and integrals. We propose surreal definitions of the arctangent and logarithm functions using truncations of Maclaurin series. Using a new representation of surreals, we present a formula for the limit of a sequence, and we use this formula to provide a complete characterization of convergent sequences and to evaluate certain series and infinite Riemann sums via extrapolation. A similar formula allows us to evaluates limits (and hence derivatives) of functions. By defining a new topology on No, we obtain the Intermediate Value Theorem even though No is not Cauchy complete, and we prove that the Fundamental Theorem of Calculus would hold for surreals if a consistent definition of integration exists. Extending our study to defining other analytic functions, evaluating power series in generality, finding a consistent definition of integration, proving Stokes' Theorem to generalize surreal integration, and studying differential equations remains open.</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":"Surreal Analysis: An Analogue of Real Analysis for Surreal Numbers","attachmentId":113986889,"attachmentType":"pdf","work_url":"https://www.academia.edu/118335569/Surreal_Analysis_An_Analogue_of_Real_Analysis_for_Surreal_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/118335569/Surreal_Analysis_An_Analogue_of_Real_Analysis_for_Surreal_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="2" data-entity-id="112056738" 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/112056738/Surreal_numbers_exponentiation_and_derivations">Surreal numbers, exponentiation and derivations</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">arXiv (Cornell University), 2020</p><p class="ds-related-work--abstract ds2-5-body-sm">We give a presentation of Conway's surreal numbers focusing on the connections with transseries and Hardy fields and trying to simplify when possible the existing treatments.</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":"Surreal numbers, exponentiation and derivations","attachmentId":109405667,"attachmentType":"pdf","work_url":"https://www.academia.edu/112056738/Surreal_numbers_exponentiation_and_derivations","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/112056738/Surreal_numbers_exponentiation_and_derivations"><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="37537717" 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/37537717/Elementary_Theory_of_Surreal_Numbers">Elementary Theory of Surreal 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="23514036" href="https://manchester.academia.edu/TeeJayDack">Tee-Jay Dack</a></div><p class="ds-related-work--abstract ds2-5-body-sm">The aim of this paper is to introduce the theory of surreal numbers in a manner that is accessible to anyone with an undergraduate degree in mathematics. We shall briefly outline some prerequisite predicate logic, with emphasis on the study of ordinal numbers. We then move swiftly on to discuss the basic definition of what a surreal number is and what orderings we may attach to these numbers to induce interesting structures. Both ring and field operations can then be endowed to this ordered structure, showing that No is indeed a totally ordered field. The last few words of this paper are dedicated to the representation of more 'standard' numbers in this new field No, with particular emphasis on Conway's adaption of Cantor's so-called normal form for the ordinals in the context of surreal numbers.</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":"Elementary Theory of Surreal Numbers","attachmentId":57511935,"attachmentType":"pdf","work_url":"https://www.academia.edu/37537717/Elementary_Theory_of_Surreal_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/37537717/Elementary_Theory_of_Surreal_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="4" data-entity-id="112056748" 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/112056748/Mini_Workshop_Surreal_Numbers_Surreal_Analysis_Hahn_Fields_and_Derivations">Mini-Workshop: Surreal Numbers, Surreal Analysis, Hahn Fields and Derivations</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">Oberwolfach Reports, 2017</p><p class="ds-related-work--abstract ds2-5-body-sm">New striking analogies between H. Hahn's fields of generalised series with real coefficients, G. H. Hardy's field of germs of real valued functions, and J. H. Conway's field No of surreal numbers, have been lately discovered and exploited. The aim of the workshop was to bring quickly together experts and young researchers, to articulate and investigate current key questions and conjectures regarding these fields, and to explore emerging applications of this recent discovery.</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":"Mini-Workshop: Surreal Numbers, Surreal Analysis, Hahn Fields and Derivations","attachmentId":109405722,"attachmentType":"pdf","work_url":"https://www.academia.edu/112056748/Mini_Workshop_Surreal_Numbers_Surreal_Analysis_Hahn_Fields_and_Derivations","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/112056748/Mini_Workshop_Surreal_Numbers_Surreal_Analysis_Hahn_Fields_and_Derivations"><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="74290546" 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/74290546/Fields_of_surreal_numbers_and_exponentiation">Fields of surreal numbers and exponentiation</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="125581634" href="https://independent.academia.edu/PhilEhrlich">Philip L . Ehrlich</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Fundamenta Mathematicae, 2001</p><p class="ds-related-work--abstract ds2-5-body-sm">We show that Conway's field of surreal numbers with its natural exponential function has the same elementary properties as the exponential field of real numbers. We obtain ordinal bounds on the length of products, reciprocals, exponentials and logarithms of surreal numbers in terms of the lengths of their inputs. It follows that the set of surreal numbers of length less than a given ordinal is a subfield of the field of all surreal numbers if and only if this ordinal is an ε-number. In that case, this field is even closed under surreal exponentiation, and is an elementary extension of the real exponential field.</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":"Fields of surreal numbers and exponentiation","attachmentId":82494265,"attachmentType":"pdf","work_url":"https://www.academia.edu/74290546/Fields_of_surreal_numbers_and_exponentiation","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/74290546/Fields_of_surreal_numbers_and_exponentiation"><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="79025328" 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/79025328/An_algebraic_set_theory_of_surreal_numbers_I">An algebraic (set) theory of surreal numbers, I</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="63139934" href="https://independent.academia.edu/Hugoluizmariano">Hugo luiz mariano</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2019</p><p class="ds-related-work--abstract ds2-5-body-sm">The notion of surreal number was introduced by J.H. Conway in the mid 1970&#39;s: the surreal numbers constitute a linearly ordered (proper) class $No$ containing the class of all ordinal numbers ($On$) that, working within the background set theory NBG, can be defined by a recursion on the class $On$. Since then, have appeared many constructions of this class and was isolated a full axiomatization of this notion that been subject of interest due to large number of interesting properties they have, including model-theoretic ones. Such constructions suggests strong connections between the class $No$ of surreal numbers and the classes of all sets and all ordinal numbers. In an attempt to codify the universe of sets directly within the surreal number class, we have founded some clues that suggest that this class is not suitable for this purpose. The present work, that expounds parts of the PhD thesis of the first author (\cite{Ran18}), establishes a basis to obtain an &quot;algebraic (...</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":"An algebraic (set) theory of surreal numbers, I","attachmentId":85885838,"attachmentType":"pdf","work_url":"https://www.academia.edu/79025328/An_algebraic_set_theory_of_surreal_numbers_I","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/79025328/An_algebraic_set_theory_of_surreal_numbers_I"><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="74290556" 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/74290556/Surreal_Ordered_Exponential_Fields">Surreal Ordered Exponential Fields</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="125581634" href="https://independent.academia.edu/PhilEhrlich">Philip L . Ehrlich</a></div><p class="ds-related-work--metadata ds2-5-body-xs">The Journal of Symbolic Logic, 2021</p><p class="ds-related-work--abstract ds2-5-body-sm">In [15], the algebraico-tree-theoretic simplicity hierarchical structure of J. H. Conway&#39;s ordered field $\mathbf{No}$ of surreal numbers was brought to the fore and employed to provide necessary and sufficient conditions for an ordered field (ordered $K$-vector space) to be isomorphic to an initial subfield ($K$-subspace) of $\mathbf{No}$, i.e. a subfield ($K$-subspace) of $\mathbf{No}$ that is an initial subtree of $\mathbf{No}$. In this sequel to [15], piggybacking on the just-said results, analogous results are established for ordered exponential fields. It is further shown that a wide range of ordered exponential fields are isomorphic to initial exponential subfields of $(\mathbf{No}, \exp)$. These include all models of $T(\mathbb{R}_W, e^x)$, where $\mathbb{R}_W$ is the reals expanded by a convergent Weierstrass system $W$. Of these, those we call trigonometric-exponential fields are given particular attention. It is shown that the exponential functions on the initial trig...</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":"Surreal Ordered Exponential Fields","attachmentId":82494274,"attachmentType":"pdf","work_url":"https://www.academia.edu/74290556/Surreal_Ordered_Exponential_Fields","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/74290556/Surreal_Ordered_Exponential_Fields"><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="60047973" 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/60047973/Integration_on_the_Surreals_a_Conjecture_of_Conway_Kruskal_and_Norton">Integration on the Surreals: a Conjecture of Conway, Kruskal and Norton</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="47375209" href="https://independent.academia.edu/FriedmanHarvey">Harvey Friedman</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2015</p><p class="ds-related-work--abstract ds2-5-body-sm">In 1976 Conway introduced the surreal number system No, containing the reals, the ordinals, and numbers such as −ω, 1/ω, √ ω and ln ω. No is a real closed ordered field, with much additional structure. Surreal theory is conveniently developed within the class theory NBG, a conservative extension of ZFC. A longstanding aim has been to develop analysis on No as a powerful extension of ordinary analysis on R. This entails finding a natural way of extending important functions f : R → R to functions f * : No → No, and naturally defining integration on the f *. The usual square root, log : R → R, and exp : R → R were naturally extended to No by Bach, Conway, Kruskal, and Norton, retaining their usual properties. Later Norton also proposed a treatment of integration, but Kruskal discovered flaws. The search for natural extensions from R to No, and natural integration on No continues. This paper addresses this and related unresolved issues with positive and negative results. In the positive direction, we show thatÉcalle-Borel transseriable functions extend naturally to No, and an integral with good properties exists on them. Transseriable functions include semi-algebraic, semi-analytic, analytic, and meromorphic ones as well as solutions of systems of linear and nonlinear systems of ODEs with possible irregular singularities as in [12]. In particular, most classical special functions (such as Airy, Bessel, Ei, erf, Gamma, Painlevé and so on) extend naturally (and are integrable) from finite to infinite values of the variable. In the negative direction, we show there is a fundamental obstruction to naturally extending many larger families of functions to No and to defining integration on surreal functions. We show that there are no descriptions of operators which can be proved within NBG to have the basic properties of integration, even on highly restricted families of real-valued entire functions.</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":"Integration on the Surreals: a Conjecture of Conway, Kruskal and Norton","attachmentId":73663803,"attachmentType":"pdf","work_url":"https://www.academia.edu/60047973/Integration_on_the_Surreals_a_Conjecture_of_Conway_Kruskal_and_Norton","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/60047973/Integration_on_the_Surreals_a_Conjecture_of_Conway_Kruskal_and_Norton"><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="74290693" 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/74290693/Number_systems_with_simplicity_hierarchies_a_generalization_of_Conways_theory_of_surreal_numbers">Number systems with simplicity hierarchies: a generalization of Conway's theory of surreal 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="125581634" href="https://independent.academia.edu/PhilEhrlich">Philip L . Ehrlich</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Symbolic Logic, 2001</p><p class="ds-related-work--abstract ds2-5-body-sm">Introduction. In his monograph On Numbers and Games [7], J. H. Conway introduced a real-closed field containing the reals and the ordinals as well as a great many other numbers including ω, ω, /2, 1/ω, and ω − π to name only a few. Indeed, this particular real-closed field, which Conway calls No, is so remarkably inclusive that, subject to the proviso that numbers—construed here as members of ordered “number” fields—be individually definable in terms of sets of von Neumann-Bernays-Gödel set theory with Global Choice, henceforth NBG [cf. 21, Ch. 4], it may be said to contain “All Numbers Great and Small.” In this respect, No bears much the same relation to ordered fields that the system of real numbers bears to Archimedean ordered fields. This can be made precise by saying that whereas the ordered field of reals is (up to isomorphism) the unique homogeneous universal Archimedean ordered field, No is (up to isomorphism) the unique homogeneous universal orderedfield [14]; also see [10]...</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":"Number systems with simplicity hierarchies: a generalization of Conway's theory of surreal numbers","attachmentId":82494372,"attachmentType":"pdf","work_url":"https://www.academia.edu/74290693/Number_systems_with_simplicity_hierarchies_a_generalization_of_Conways_theory_of_surreal_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/74290693/Number_systems_with_simplicity_hierarchies_a_generalization_of_Conways_theory_of_surreal_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></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":86603419,"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":86603419,"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_86603419" 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. You can download the paper by clicking the button above.</p></div></div></div></div><div class="ds-sidebar--container js-work-sidebar"><div class="ds-related-content--container"><h2 class="ds-related-content--heading">Related papers</h2><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="0" data-entity-id="74290691" 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/74290691/Erratum_to_Fields_of_surreal_numbers_and_exponentiation_Fund_Math_167_2001_173_188_">Erratum to ``Fields of surreal numbers and exponentiation" (Fund. Math. 167 (2001), 173–188)</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="125581634" href="https://independent.academia.edu/PhilEhrlich">Philip L . 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Ehrlich</a></div><p class="ds-related-work--metadata ds2-5-body-xs">The Journal of Symbolic Logic, 2018</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":"Number Systems with Simplicity Hierarchies: A Generalization of Conway’s Theory of Surreal Numbers II","attachmentId":82494288,"attachmentType":"pdf","work_url":"https://www.academia.edu/74290696/Number_Systems_with_Simplicity_Hierarchies_A_Generalization_of_Conway_s_Theory_of_Surreal_Numbers_II","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/74290696/Number_Systems_with_Simplicity_Hierarchies_A_Generalization_of_Conway_s_Theory_of_Surreal_Numbers_II"><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="74290551" 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/74290551/Conway_names_the_simplicity_hierarchy_and_the_surreal_number_tree">Conway names, the simplicity hierarchy and the surreal number tree</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="125581634" href="https://independent.academia.edu/PhilEhrlich">Philip L . 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Dahn</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 1984</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":"Ordered Fields with Several Exponential Functions","attachmentId":92157296,"attachmentType":"pdf","work_url":"https://www.academia.edu/88119051/Ordered_Fields_with_Several_Exponential_Functions","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/88119051/Ordered_Fields_with_Several_Exponential_Functions"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" 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