Les nombres de Cuesta-Conway comme extension des ordinaux de Cantor: une courte introduction aux nombres surr\'eels

TL;DR

This paper introduces Cuesta-Conway numbers, also known as surreal numbers, as an accessible extension of Cantor's ordinals, highlighting their construction, algebraic properties, and their role as a totally ordered, real-closed field.

Contribution

It provides a simple construction of surreal numbers from normal forms, demonstrating their extension of reals and ordinals with fundamental algebraic operations.

Findings

Surreal numbers form a totally ordered, commutative, real-closed field.

Construction yields an increasing sequence of real-closed fields.

Complete proof of surreal numbers' algebraic and order properties.

Abstract

On Cuesta-Conway numbers as an extension of Cantor's ordinals: A short introduction to surreal numbers. The class of Cuesta-Conway numbers, the surreal numbers, can be defined simply, starting from their normal forms (families of exponentials indexed by ordinals), as an extension of the reals and ordinals from which easily follow addition, multiplication, and total order relation. A construction of this class yields an increasing sequence of real-closed fields, preliminary, so to say. A complete proof is also given the well-known result that the class of surreal numbers is a totally ordered, commutative, real-closed field!