A clean way to separate sets of surreals

TL;DR

This paper presents a simplified proof that for any two sets of surreal numbers with one set less than the other, there exists a surreal number strictly between them, using sign sequences.

Contribution

The paper introduces a streamlined proof for the separation of sets of surreals, leveraging properties of sign sequences to simplify classical arguments.

Findings

Existence of a surreal number between two sets when one is less than the other.

Simplification of classical proofs using sign sequence properties.

Compatibility conditions for separating sets of surreals.

Abstract

Let surreal numbers be defined by means of sign sequences. We give a proof that if $S < T$ are sets of surreals, then there is some surreal $w$ such that $S < w < T$. The classical proof is simplified by observing that, for every set $S$ of surreals, there exists a surreal $s$ such that, for every surreal $w$, we have $S<w$ if and only if the restriction of $w$ to the length of $s$ is $ \geq s$. Hence $S < w < T$ if and only if $w$ satisfies the above condition, as well as its symmetrical version with respect to $T$. It is now enough to check that if $S < T$, then the two conditions are compatible.