Real closed exponential fields

D'Aquino, Paola; Knight, Julia F.; Kuhlmann, Salma; Lange, Karen

doi:10.4064/fm219-2-6

Cited by 5 publications

(4 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Ressayre ([40]; also see [15]) showed if A is a real closed exponential field with residue class field and value group , then A is isomorphic to a truncation closed, cross sectional subfield K of a Hahn field , where the logarithm of is purely infinite for each . It is an open question whether every model of the theory of real numbers with exponentiation is isomorphic to a truncation closed, cross sectional exponential subfield of a transserial Hahn field (or even of a logarithmic Hahn field).…”

Section: Initial Embeddings Of Ordered Exponential Fieldsmentioning

confidence: 99%

“…The definition of a development triple is inspired by [15]. Our development triples are slightly different from the ones in [15], as we insist that our substructures A be -closed. Let be a development triple.…”

Section: Exponential Fields Which Define Convergent Weierstrass Systemsmentioning

confidence: 99%

“…Some of the methods employed in Sections 6–10 are adaptations or expansions of methods developed by Ressayre [40], van den Dries, Macintyre and Marker [47, 48], and D’Aquino, Knight, Kuhlmann and Lange [15] in their treatments of truncation closed embeddings into Hahn fields of ordered exponential fields and ordered fields with additional structure more generally. However, as is evident from Proposition 1.4, even in the case of an ordered field K the existence of a truncation closed embedding into a Hahn field does not suffice to establish the existence of an initial embedding into .…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Surreal Ordered Exponential Fields

Ehrlich¹,

Kaplan²

2021

J. symb. log.

View full text Add to dashboard Cite

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 .

show abstract

Section: Initial Embeddings Of Ordered Exponential Fieldsmentioning

confidence: 99%

Section: Exponential Fields Which Define Convergent Weierstrass Systemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Surreal Ordered Exponential Fields

Ehrlich¹,

Kaplan²

2021

J. symb. log.

View full text Add to dashboard Cite

show abstract

“…An exponential integer part of an exponential field

\langle R,\exp \rangle

is an IP

Z\subseteq R

such that

Z_{&gt;0}

is closed under exp . Ressayre shows that every real‐closed exponential field has an exponential IP (this is further elaborated in [8]).…”

Section: Preliminariesmentioning

confidence: 99%

Models of VTC0$\mathsf {VTC^0}$ as exponential integer parts

Jeřábek

2023

Mathematical Logic Qtrly

View full text Add to dashboard Cite

We prove that (additive) ordered group reducts of nonstandard models of the bounded arithmetical theory are recursively saturated in a rich language with predicates expressing the integers, rationals, and logarithmically bounded numbers. Combined with our previous results on the construction of the real exponential function on completions of models of , we show that every countable model of is an exponential integer part of a real‐closed exponential field.

show abstract

Real closures of models of weak arithmetic

Jeřábek

Kołodziejczyk

2012

Arch. Math. Logic

View full text Add to dashboard Cite

(2010)) have recently shown that every real-closed field with an integer part satisfying the arithmetic theory IΣ 4 is recursively saturated, and that this theorem fails if IΣ 4 is replaced by I∆ 0 . We prove that the theorem holds if IΣ 4 is replaced by weak subtheories of Buss' bounded arithmetic:It also holds for I∆ 0 (and even its subtheory IE 2 ) under a rather mild assumption on cofinality. On the other hand, it fails for the extension of IOpen by an axiom expressing the Bézout property, even under the same assumption on cofinality.A discretely ordered subring A of a real-closed field (henceforth often: rcf) R is an integer part of R if for every r ∈ R there exists a ∈ A such that a ≤ r < a + 1. It is well-known that every rcf has an integer part [MR93], which is then a model of the weak arithmetic theory IOpen (induction for quantifier-free formulas in the language of ordered rings). On the other hand, every model of IOpen is an integer part of its real closure (or, more precisely, the real closure of its fraction field), as shown by Shepherdson [She64].Recently, d'Aquino et al.[DKS10] studied the question which rcfs have integer parts satisfying more arithmetic, e.g. Peano Arithmetic. It turns out *

show abstract

Real closed exponential fields

Cited by 5 publications

References 11 publications

Surreal Ordered Exponential Fields

Surreal Ordered Exponential Fields

Models of VTC0$\mathsf {VTC^0}$ as exponential integer parts

Real closures of models of weak arithmetic

Contact Info

Product

Resources

About