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

It had happened more than once that combinatorial model theoretic dividing lines introduced by Shelah were invented independently in different fields of mathematics. This curious note by Itai Ben Yaacov gives another example of this phenomenon:

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 “Critères de compacité” from 1952. The familiar forms for the defining formulae then follow using Mazur’s Lemma regarding weak convergence in Banach spaces.

In a meeting in Kolkata in January 2013, the author asked the audience who had first defined the notion
of a stable formula and when, and to the expected answer replied that, no, it had been Grothendieck, in
the fifties.

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One response to “Model theoretic stability and definability of types, after A. Grothendieck

  1. According to Andrés this is kinda old news: Iovino had already pointed out some of those connections when he developed a stability theory for Banach spaces. But I guess making them explicit is always nice.

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