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We call such fields omega-fields and we prove that any omega-field of bounded Hahn series with real coefficients admits an exponential function making it into a model of the theory of the real exponential field. We also consider relative versions with more general coefficient fields.","publication_date":"2023,3,21","publication_name":"Proceedings of the American Mathematical Society","grobid_abstract_attachment_id":"109405621"},"document_type":"paper","pre_hit_view_count_baseline":null,"quality":"high","language":"en","title":"Exponential fields and Conway’s omega-map","broadcastable":false,"draft":null,"has_indexable_attachment":true,"indexable":true}}["work"]; window.loswp.workCoauthors = [18337933]; 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';</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":109405621,"attachmentType":"pdf"}"><img alt="First page of “Exponential fields and Conway’s omega-map”" class="ds-work-cover--cover-thumbnail" src="https://0.academia-photos.com/attachment_thumbnails/109405621/mini_magick20231222-1-eo0jn3.png?1703238873" /><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">Exponential fields and Conway’s omega-map</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="18337933" href="https://independent.academia.edu/AlessandroBerarducci"><img alt="Profile image of Alessandro Berarducci" class="ds-work-card--author-avatar" src="https://0.academia-photos.com/18337933/22540702/21734456/s65_alessandro.berarducci.jpg" />Alessandro Berarducci</a></div><div class="ds-work-card--detail"><p class="ds-work-card--detail ds2-5-body-sm">2023, Proceedings of the American Mathematical Society</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">16 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 = 112056698; 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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">Inspired by Conway's surreal numbers, we study real closed fields whose value group is isomorphic to the additive reduct of the field. We call such fields omega-fields and we prove that any omega-field of bounded Hahn series with real coefficients admits an exponential function making it into a model of the theory of the real exponential field. We also consider relative versions with more general coefficient fields.</p><div class="ds-work-card--button-container"><button class="ds2-5-button js-swp-download-button" data-signup-modal="{"location":"continue-reading-button--work-card","attachmentId":109405621,"attachmentType":"pdf","workUrl":"https://www.academia.edu/112056698/Exponential_fields_and_Conway_s_omega_map"}">See full PDF</button><button class="ds2-5-button ds2-5-button--secondary js-swp-download-button" data-signup-modal="{"location":"download-pdf-button--work-card","attachmentId":109405621,"attachmentType":"pdf","workUrl":"https://www.academia.edu/112056698/Exponential_fields_and_Conway_s_omega_map"}"><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="109405621" data-landing_url="https://www.academia.edu/112056698/Exponential_fields_and_Conway_s_omega_map" 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="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="1" 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="2" 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="3" data-entity-id="88119048" 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/88119048/On_the_Theory_of_Exponential_Fields">On the Theory of 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="32963925" href="https://uni-landau.academia.edu/IDahn">I. Dahn</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Mathematical Logic Quarterly, 1983</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 the Theory of Exponential Fields","attachmentId":92157295,"attachmentType":"pdf","work_url":"https://www.academia.edu/88119048/On_the_Theory_of_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/88119048/On_the_Theory_of_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="4" data-entity-id="74290691" 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/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-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">Erratum to "Fields of surreal numbers and exponentiation" (Fund. Math. 167 (2001), 173-188) by Lou van den Dries (Urbana, IL) and Philip Ehrlich (Athens, OH) Lemma 4.5 in [2] is false. The correct result is the Lemma below. We use the following conventions and notations: Γ is an ordered abelian group, S ⊆ Γ ; we let [S] := {s 1 +. .. + s k : k ∈ N, s 1 ,. .. , s k ∈ S} be the additive monoid generated by S in Γ ; for a ∈ Γ , put S <a := {s ∈ S : s < a} and define S ≤a and S ≥a similarly; if S is well-ordered, we let o(S) be its ordinal. Also α, λ, µ are ordinals, and sums and products of ordinals are their natural sums and natural products. Lemma. Suppose S ⊆ Γ ≥0 is well-ordered with o(S) ≤ µ. Then [S] is well-ordered with o([S]) ≤ ω ωµ .</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":"Erratum to ``Fields of surreal numbers and exponentiation\" (Fund. Math. 167 (2001), 173–188)","attachmentId":82494371,"attachmentType":"pdf","work_url":"https://www.academia.edu/74290691/Erratum_to_Fields_of_surreal_numbers_and_exponentiation_Fund_Math_167_2001_173_188_","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/74290691/Erratum_to_Fields_of_surreal_numbers_and_exponentiation_Fund_Math_167_2001_173_188_"><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="77432936" 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/77432936/Fields_with_pseudo_exponentiation">Fields with pseudo-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="38774345" href="https://independent.academia.edu/BorisZilber">Boris Zilber</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 study the structure of an algebraically closed field with extra function resembling the classical exponentiation on complex 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":"Fields with pseudo-exponentiation","attachmentId":84800841,"attachmentType":"pdf","work_url":"https://www.academia.edu/77432936/Fields_with_pseudo_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/77432936/Fields_with_pseudo_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="93235164" 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/93235164/Fields_with_pseudo_exponentiation_B">Fields with pseudo-exponentiation B</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="38774345" href="https://independent.academia.edu/BorisZilber">Boris Zilber</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2000</p><p class="ds-related-work--abstract ds2-5-body-sm">1 Introduction This research is motivated by the study of model-theoretical properties of classical &#39;analytic&#39; structures, i.e. ones having natural analytic representation (see also [Z]). For example, the structure of complex numbers as a field with exponentiation C exp = (C, +, ·, exp). One of the questions we can ask is whether C exp is quasi-minimal, i.e. any definable subset of C exp is either countable or of power continuum. Another question is about homogeneity of the structure; we do not know any its automorphism except the identity and the complex conjugation. In general we would like to understand the nature of analytic dimension in a context close to model-theoretic stability theory. A slightly weaker analytic structure C (2) exp is a two-sorted structure with both sorts C(1) and C(2) being copies of complex numbers, on both sorts the field structure is given and there is a mapping exp : C(1) → C(2) in the language. The model theory of the both structures, as well ...</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 with pseudo-exponentiation B","attachmentId":96027857,"attachmentType":"pdf","work_url":"https://www.academia.edu/93235164/Fields_with_pseudo_exponentiation_B","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/93235164/Fields_with_pseudo_exponentiation_B"><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="120240219" 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/120240219/Restricted_analytic_valued_fields_with_partial_exponentiation">Restricted analytic valued fields with partial 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="33465440" href="https://independent.academia.edu/LeonardoAngel1">Leonardo Angel</a></div><p class="ds-related-work--metadata ds2-5-body-xs">arXiv (Cornell University), 2023</p><p class="ds-related-work--abstract ds2-5-body-sm">Non-archimedean fields with restricted analytic functions may not support a full exponential function, but they always have partial exponentials defined in convex subrings. On face of this, we study the first order theory of the class of non-archimedean ordered valued fields augmented by all restricted analytic functions and an exponential function defined in the valuation ring, which extends the restricted analytic exponential. We obtain model completeness and other desirable properties for this theory. In particular, any model embeds in a model where the partial exponential extends to a full one.</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":"Restricted analytic valued fields with partial exponentiation","attachmentId":115455604,"attachmentType":"pdf","work_url":"https://www.academia.edu/120240219/Restricted_analytic_valued_fields_with_partial_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/120240219/Restricted_analytic_valued_fields_with_partial_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="8" data-entity-id="69055301" 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/69055301/Finitely_presented_exponential_fields">Finitely presented 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="111521258" href="https://independent.academia.edu/JuliaKirby3">Julia Kirby</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Algebra & Number Theory, 2013</p><p class="ds-related-work--abstract ds2-5-body-sm">We develop the algebra of exponential fields and their extensions. The focus is on ELA-fields, which are algebraically closed with a surjective exponential map. In this context, we define finitely presented extensions, show that finitely generated strong extensions are finitely presented, and classify these extensions. We give an algebraic construction of Zilber's pseudoexponential fields. As applications of the general results and methods of the paper, we show that Zilber's fields are not model-complete, answering a question of Macintyre, and we give a precise statement explaining how Schanuel's conjecture answers all transcendence questions about exponentials and logarithms. We discuss connections with the Kontsevich-Zagier, Grothendieck, and André transcendence conjectures on periods, and suggest open problems.</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":"Finitely presented exponential fields","attachmentId":79300183,"attachmentType":"pdf","work_url":"https://www.academia.edu/69055301/Finitely_presented_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/69055301/Finitely_presented_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="9" data-entity-id="20792399" 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/20792399/ORDERED_FIELDS_WITH_SEVERAL_EXPONENTIAL_FUNCTIONS">ORDERED FIELDS WITH SEVERAL EXPONENTIAL FUNCTIONS</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="42077770" href="https://independent.academia.edu/BDahn">B. Dahn</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 1984</p><p class="ds-related-work--abstract ds2-5-body-sm">In the present paper we regard ordered fields with several exponential functions and investigate the theory of such exponential fields. For this let F be an ordered field and E a unary function from F into F. Then ( F , E ) is said to be an ordered exponential field and E an exponential function on F if for all x, y E F E(x + y) = = E ( x ) . E(y) and E ( x ) 2 ek(x) = Cfor all odd natural numbers k. The corresponding theory of ordered exponential fields is denoted by OEF(e) where e is a function symbol and E the interpretation of e in ( F , E ) . Similarly we write ( F , E , E') for ordered fields with two exponential functions and OEF(e, e') = OEF(e) w OEF(e') for the corresponding theory. I n [l] it is proved that OEF(e) is stronger than the axiom system prop&ed by TARSKI in [2].</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":41922479,"attachmentType":"pdf","work_url":"https://www.academia.edu/20792399/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-wsj-grid-card-view-pdf" href="https://www.academia.edu/20792399/ORDERED_FIELDS_WITH_SEVERAL_EXPONENTIAL_FUNCTIONS"><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":109405621,"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":109405621,"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_109405621" 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/109202282/Rings_of_real_valued_continuous_functions_II">Rings of real-valued continuous functions. II</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="39341047" href="https://independent.academia.edu/GregoryChudnovsky">Gregory Chudnovsky</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Mathematische Zeitschrift, 1981</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":"Rings of real-valued continuous functions. 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href="https://www.academia.edu/71339447/On_definable_germs_of_functions_in_expansions_of_the_real_field_by_power_functions">On definable germs of functions in expansions of the real field by power functions</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="47557596" href="https://independent.academia.edu/mouradberraho">mourad berraho</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2020</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"On definable germs of functions in expansions of the real field by power functions","attachmentId":80724968,"attachmentType":"pdf","work_url":"https://www.academia.edu/71339447/On_definable_germs_of_functions_in_expansions_of_the_real_field_by_power_functions","alternativeTracking":true}"><span 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data-entity-id="78157444" 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/78157444/Truncation_in_Differential_Hahn_Fields">Truncation in Differential Hahn Fields</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="108592212" href="https://independent.academia.edu/SantiagoCamacho25">Santiago Camacho</a></div><p class="ds-related-work--metadata ds2-5-body-xs">arXiv: 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":"Truncation in Differential Hahn Fields","attachmentId":85303346,"attachmentType":"pdf","work_url":"https://www.academia.edu/78157444/Truncation_in_Differential_Hahn_Fields","alternativeTracking":true}"><span 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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><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-related-work-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 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