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Monthly Archives: July 2013
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.
Posted in Classification theory, Model Theory
Tagged definability of types, Grothendieck, Shelah, stability