Surreal limits

TL;DR

This paper explores how surreal number sequences converge through their sign expansion strings, establishing a notion of string convergence that aligns with limits and sums in surreal number theory.

Contribution

It introduces a natural notion of string convergence for surreal numbers and connects it with transfinite sums and the supremum of increasing ordinal sequences.

Findings

String convergence of surreal sequences is well-defined and consistent with limits.

Transfinite sums of surreal numbers can be characterized via string limits.

The approach aligns with Conway normal form for surreal numbers.

Abstract

We note that if a sequence of real numbers converges to some limit, then the sequence of the corresponding strings in the surreal $+,-$ sign expansion representation converges, for a natural notion of string convergence, to the string corresponding to the limit, modulo an infinitesimal. The corresponding statement would be obviously false if we were considering, as strings, decimal or binary representations, instead. The string limit of a possibly transfinite sequence of surreal numbers is always defined and, when considering increasing sequences of ordinals, corresponds to taking the supremum. A transfinite sum can be defined using the string limit and this sum agrees with the representation of a surreal number in Conway normal form.