Time: 9h00, Wednesday, March 15, 2017
Location: Room 4, Building A14, Institute of Mathematics, 18 Hoang Quoc Viet, Cau Giay, Hanoi
Abstract: In the framework of Nonstandard Analysis the existence of infinitesimals is admitted, making it possible to have many different convex subgroups of the real numbers. Neutrices are additive convex subgroups of a nonstandard model for the real numbers. An external number is the algebraic sum of a (hyper)real number and a neutrix. Due to the stability by some shifts, external numbers may be seen as mathematical models for orders of magnitude. It was shown in [3] that the class of external numbers equipped with addition and the class of external numbers which are not neutrices equipped with multiplication form commutative regular semigroups. Unlike real numbers, external numbers have individualized neutral and inverse elements for both addition and multiplication. It was also shown that the distributive law is valid under some restrictions that can be completely characterized. Moreover, the external numbers are totally ordered, even allowing for a sort of generalized completeness property [2] [1] [5]. Hence external numbers have to a large extent algebraic properties similar to those of real numbers. This justifies the introduction of common algebraic structures defined by axiomatic rules, the so-called solids. Solids are a sort of ``mellowed" version of ordered fields, having a restricted distributivity law. However, necessary and sufficient conditions can be given for distributivity to hold. We present an axiomatics for the external numbers given in [4]. 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 all these axioms is called a complete arithmetical solid. It was shown that the external numbers form a complete arithmetical solid indeed, and therefore the axioms presented are consistent. Also, up to isomorphism, the precise elements (elements with minimal magnitude) of a model are situated between the nonstandard rationals and the nonstandard reals.