An algebraic (set) theory of surreal numbers, I

TL;DR

This paper develops an algebraic set theory framework for surreal numbers, aiming to relate them to a universe of surreal sets and explore their foundational properties within a categorical set theory context.

Contribution

It introduces an algebraic (set) theory for surreal numbers, establishing a foundational link between surreal numbers and a universe of surreal sets inspired by algebraic set theory.

Findings

Proposes a basis for an algebraic set theory for surreal numbers

Establishes connections between surreal numbers and surreal sets

Explores the relation between surreal numbers and ordinal classes

Abstract

The notion of surreal number was introduced by J.H. Conway in the mid 1970's: the surreal numbers constitute a linearly ordered (proper) class $No$ containing the class of all ordinal numbers ($On$) that, working within the background set theory NBG, can be defined by a recursion on the class $On$. Since then, have appeared many constructions of this class and was isolated a full axiomatization of this notion that been subject of interest due to large number of interesting properties they have, including model-theoretic ones. Such constructions suggests strong connections between the class $No$ of surreal numbers and the classes of all sets and all ordinal numbers. In an attempt to codify the universe of sets directly within the surreal number class, we have founded some clues that suggest that this class is not suitable for this purpose. The present work, that expounds parts of the PhD thesis of the first author (\cite{Ran18}), establishes a basis to obtain an "algebraic (set) theory for surreal numbers" along the lines of the Algebraic Set Theory - a categorial set theory introduced in the 1990's based on the concept of ZF-algebra: to establish abstract and general links between the class of all surreal numbers and a universe of "surreal sets" similar to the relations between the class of all ordinals ($On$) and the class of all sets ($V$), that also respects and expands the links between the linearly ordered class of all ordinals and of all surreal numbers.