The field of Omicran-reals A new approach to nonstandard analysis

Abdeljalil Saghe (1)

1 FST de Beni-Mellal

From the works of Abraham Robinson, we know that the heuristic idea of infinite and infinitesimal numbers has obtained a formal rigor, he proved that the field of real numbers R can be considered as a proper subset of a new field, * R, which is called field of hyperreal [1] numbers and contains the infinite and infinitesimal numbers. From the approach of Robinson we can represent every hyperreal by a sequence o f R N modulo a maximal ideal M, this ideal is defined by using an ultrafilter U. Unfortunately, the Ultrafilter U and the order relation defined on * R are unknown, only the existence can be proved by the axiom of choice. In this paper, we find a new extension of real numbers which contains the infinite and infinitesimal numbers as the field of hyperreals, the new set is called the field of Omicran-reals O. Moreover, the new approach of construction is simple compared to other methods [1, 7, 8]and very precise, and the field O is endowed with a well defined total order relation.

Mots clés: infinitesimal; Internal set theory; Abraham Robinson; hyppereals; Keyword Nonstadard analysis

Domaines: Mathématiques générales [math.GM]

Abdeljalil Saghe. The field of Omicran-reals A new approach to nonstandard analysis. 2015. ⟨hal-01248379⟩