On the quotient class of non-archimedean fields

Dinis, Bruno; Berg, Imme van den

doi:10.1016/j.indag.2017.05.001

Cited by 6 publications

(6 citation statements)

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“…The axioms for a solid, i.e. Axioms 2.1-2.29, extend the axioms originally presented in [10] and were shown to be consistent in [11] by the construction of a direct model in the language of ZF C. This was given in the form of a set of cosets of a non-Archimedean field. Allowing for definable classes, it was shown in [10] that the external numbers of [19] and [20] satisfy the axioms for addition and for multiplication, together with a modified form of the distributivity axiom; this modified form was shown to be equivalent to Axiom 2.22 in [12].…”

Section: On Consistencymentioning

confidence: 76%

“…The set E is not characterized by Axioms 2.1-2.29 for as showed in [11] the set of all cosets with respect to all convex subgroups for addition of a non-Archimedean field is a model for these axioms. As we will see in the next section all the algebraic axioms, i.e.…”

Section: 2mentioning

confidence: 99%

“…An assembly is a completely regular semigroup (union of groups), in which the magnitude operation is linear. A structure satisfying Axioms 2.1 -2.29 was called a solid in [11].…”

Section: The Axiomsmentioning

confidence: 99%

“…Respecting total order, they are based on semigroup operations more than group operations, a sort of "mellowed" version of the common rules of calculation of real numbers. In [11] it was shown that the set of cosets of non-Archimedean ordered fields with respect to all possible convex subgroups for addition has a similar algebraic structure. Such structures were called solids.…”

Section: Introductionmentioning

confidence: 99%

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Axiomatics for the external numbers of nonstandard analysis

Dinis¹,

Berg

2017

JLA

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Neutrices are additive subgroups of a nonstandard model of the real numbers. An external number is the algebraic sum of a nonstandard real number and a neutrix. Due to the stability by some shifts, external numbers may be seen as mathematical models for orders of magnitude. The algebraic properties of external numbers gave rise to the so-called solids, which are extensions of ordered fields, having a restricted distributivity law. However, necessary and sufficient conditions can be given for distributivity to hold. In this article we develop an axiomatics for the external numbers. The axioms are similar to, but mostly somewhat weaker than the axioms for the real numbers and deal with algebraic rules, Dedekind completeness and the Archimedean property. A structure satisfying these axioms is called a complete arithmetical solid. We show that the external numbers form a complete arithmetical solid, implying the consistency of the axioms presented. We also show that the set of precise elements (elements with minimal magnitude) has a built-in nonstandard model of the rationals. Indeed the set of precise elements is situated between the nonstandard rationals and the nonstandard reals whereas the set of non-precise numbers is completely determined.2010 Mathematics Subject Classification. 03H05, 03C65, 26E35.

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Section: On Consistencymentioning

confidence: 76%

Section: 2mentioning

confidence: 99%

“…An assembly is a completely regular semigroup (union of groups), in which the magnitude operation is linear. A structure satisfying Axioms 2.1 -2.29 was called a solid in [11].…”

Section: The Axiomsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Axiomatics for the external numbers of nonstandard analysis

Dinis¹,

Berg

2017

JLA

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“…External numbers satisfy, to a large extent, the algebraic and analytic properties of the real numbers [17,18,9], including a Generalized Dedekind completeness property (see [2]). Structures with these properties were called complete arithmetical solids in [10](see also [11,12]). This means in particular that we may use the algebraic properties of neutrices to study properties of convergence.…”

Section: Introductionmentioning

confidence: 99%

On Flexible Sequences

2019

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In the setting of nonstandard analysis we introduce the notion of flexible sequence. The terms of flexible sequences are external numbers. These are a sort of analogue for the classical O(·) and o(·) notation for functions, and have algebraic properties similar to those of real numbers. The flexibility originates from the fact that external numbers are stable under some shifts, additions and multiplications. We introduce two forms of convergence, and study their relation. We show that the usual properties of convergence of sequences hold or can be adapted to these new notions of convergence and give some applications.2010 Mathematics Subject Classification. 03H05, 40A05.

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