Towards solid abelian groups: A formal proof of N\"obeling's theorem

TL;DR

This paper formalizes N"obeling’s theorem within condensed mathematics, providing a detailed proof using ordinal induction in the Lean theorem prover, advancing the understanding of solid abelian groups.

Contribution

It offers the first formal proof of N"obeling’s theorem in Lean, employing ordinal induction, and enhances the theoretical foundation of solid abelian groups in condensed mathematics.

Findings

Formal proof of N"obeling’s theorem completed in Lean

Introduces ordinal induction technique in formalized mathematics

Strengthens the theoretical basis of solid abelian groups

Abstract

Condensed mathematics, developed by Clausen and Scholze over the last few years, is a new way of studying the interplay between algebra and geometry. It replaces the concept of a topological space by a more sophisticated but better-behaved idea, namely that of a condensed set. Central to the theory are solid abelian groups and liquid vector spaces, analogues of complete topological groups. N\"obeling's theorem, a surprising result from the 1960s about the structure of the abelian group of continuous maps from a profinite space to the integers, is a crucial ingredient in the theory of solid abelian groups; without it one cannot give any nonzero examples of solid abelian groups. We discuss a recently completed formalisation of this result in the Lean theorem prover, and give a more detailed proof than those previously available in the literature. The proof is somewhat unusual in that it requires induction over ordinals -- a technique which has not previously been used to a great extent in formalised mathematics.