Model theoretic stability and definability of types, after A. Grothendieck

TL;DR

This paper reveals that Grothendieck's 1952 criteria imply Shelah's 1970s fundamental theorem of stability theory, connecting model theoretic stability with functional analysis concepts.

Contribution

It demonstrates that the core stability theorem is a direct consequence of Grothendieck's compactness criteria, unifying model theory and Banach space theory.

Findings

The fundamental theorem of stability theory follows from Grothendieck's criteria.

Definability of types is linked to weak convergence in Banach spaces.

Familiar formulas for defining types are derived via Mazur's Lemma.

Abstract

We point out how the "Fundamental Theorem of Stability Theory", namely the equivalence between the "non order property" and definability of types, proved by Shelah in the 1970s, is in fact an immediate consequence of Grothendieck's "Crit{\`e}res de compacit{\'e}" from 1952. The familiar forms for the defining formulae then follow using Mazur's Lemma regarding weak convergence in Banach spaces.