The continuum of the surreal numbers revisited.The surreal numbers defined through transfinite Cauchy fundamental sequences

TL;DR

This paper proves that three different methods of constructing the continuum of surreal numbers—via transfinite power series, Dedekind cuts, and Cauchy sequences—are equivalent, unifying various approaches to transfinite real numbers.

Contribution

It establishes the equivalence of three hierarchies of transfinite real numbers, showing they define the same continuum of surreal numbers through inductive limits.

Findings

All three techniques produce the same class of infinite numbers.

The continuum of surreal numbers can be characterized by multiple equivalent constructions.

This unification parallels the classical constructions of real numbers.

Abstract

In this treatise on the theory of the continuum of the surreal numbers of J.H. Conway, is proved ,that the three different techniques and hierarchies of the continuums of the transfinite real numbers of Glayzal A. (1937) defined through transfinite power series , of the surreal numbers of J.H. Conway (1976) defined by Dedekind cuts ,and of the ordinal real numbers of K. E. Kyritsis (1992) defined by fundamental Cauchy transfinite sequences, give by inductive limit or union the same class and continuum of infinite numbers. This is quite remarkable and is the analogue in the transfinite numbers, of that the real numbers can be constructed either as decimal power series, or by Dedekind cuts, or by Cauchy fundamental sequences.