Mini-Workshop: Surreal Numbers, Surreal Analysis, Hahn Fields and Derivations

Berarducci, Alessandro; Ehrlich, Philip; Kuhlmann, Salma

doi:10.4171/owr/2016/60

Cited by 11 publications

(13 citation statements)

References 71 publications

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“…At the mini-workshop on surreal numbers, surreal analysis, Hahn fields and derivations held in Oberwolfach in 2016, the following question was raised: “Let . Is there a good way to introduce and on and an exponential map on ?” [9, page 3315]. In this section we make some observations related to this question.…”

Section: Trigonometric Fields and Surcomplex Exponentiationmentioning

confidence: 99%

Surreal Ordered Exponential Fields

Ehrlich¹,

Kaplan²

2021

J. symb. log.

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In 2001, the algebraico-tree-theoretic simplicity hierarchical structure of J. H. Conway’s ordered field of surreal numbers was brought to the fore by the first author and employed to provide necessary and sufficient conditions for an ordered field (ordered -vector space) to be isomorphic to an initial subfield ( -subspace) of , i.e. a subfield ( -subspace) of that is an initial subtree of . In this sequel, analogous results are established for ordered exponential fields, making use of a slight generalization of Schmeling’s conception of a transseries field. It is further shown that a wide range of ordered exponential fields are isomorphic to initial exponential subfields of . These include all models of , where 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 trigonometric-exponential subfields of , which includes itself, extend to canonical exponential functions on their surcomplex counterparts. The image of the canonical map of the ordered exponential field of logarithmic-exponential transseries into is shown to be initial, as are the ordered exponential fields and .

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Section: Trigonometric Fields and Surcomplex Exponentiationmentioning

confidence: 99%

Surreal Ordered Exponential Fields

Ehrlich¹,

Kaplan²

2021

J. symb. log.

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“…These structures are also of great interest on their own, as the study of these has connections to open problems such as the decidability of R exp and consequently Schanuel's Conjecture. An overview of these connections is given in Krapp [14]. We will also point out how the results in this section relate to them.…”

Section: O-minimal Exponential Fieldsmentioning

confidence: 95%

Value groups and residue fields of models of real exponentiation

Krapp

2019

JLA

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Let F be an archimedean field, G a divisible ordered abelian group and h a group exponential on G. A triple (F, G, h) is realised in a non-archimedean exponential field (K, exp) if the residue field of K under the natural valuation is F and the induced exponential group of (K, exp) is (G, h). We give a full characterisation of all triples (F, G, h) which can be realised in a model of real exponentiation in the following two cases: i) G is countable. ii) G is of cardinality κ and κ-saturated for an uncountable regular cardinal κ with κ <κ = κ. Moreover, we show that for any o-minimal exponential field (K, exp) satisfying the differential equation exp ′ = exp, its residue exponential field is a model of real exponentiation.

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“…A linearly ordered structure is called o-minimal if any definable subset is a finite union of intervals and points. The study of o-minimal exponential fields whose exponential satisfies EXP has strong links to the decidability of Th(R exp ), as described in Krapp [11]. 4.13 Let K be a real closed field.…”

Section: Peano Arithmeticmentioning

confidence: 99%

“…Moreover, under the assumption of Schanuel's Conjecture, if this real closed exponential field is model complete, then it satisfies the existential theory of R exp . 2 We do not know where the exact benchmark is.…”

Section: Introductionmentioning

confidence: 99%

Models of true arithmetic are integer parts of models of real exponentation

Carl

Krapp

2021

J. Log. Anal.

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Exploring further the connection between exponentiation on real closed fields and the existence of an integer part modelling strong fragments of arithmetic, we demonstrate that each model of true arithmetic is an integer part of an exponential real closed field that is elementarily equivalent to the real numbers with exponentiation and that each model of Peano arithmetic is an integer part of a real closed field that admits an isomorphism between its ordered additive and its ordered multiplicative group of positive elements. Under the assumption of Schanuel’s Conjecture, we obtain further strengthenings for the last statement.

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Mini-Workshop: Surreal Numbers, Surreal Analysis, Hahn Fields and Derivations

Cited by 11 publications

References 71 publications

Surreal Ordered Exponential Fields

Surreal Ordered Exponential Fields

Value groups and residue fields of models of real exponentiation

Models of true arithmetic are integer parts of models of real exponentation

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