This is the title of a talk given at the CMU logic seminar on April 28, 2015. The talk presents the corresponding paper.
Forking is one of the central notion of modern model theory. Roughly speaking, it is a notion of independence generalizing both linear independence in vector spaces and algebraic independence in fields. In the first-order framework, it was introduced by Shelah and is one of the main device of his book. One can ask whether there is such a notion for classes that are not first-order axiomatizable, such as classes of models of a sentence in infinitary logic. We will focus on abstract elementary classes (AECs), a very general axiomatic framework introduced by Shelah in 1985. In Shelah's book on AECs, the central concept is again a local generalization of forking: good frames.
We will attempt to explain why good frames are so useful by surveying recent applications to problems like existence of saturated models or stability transfer. Time permitting, we will sketch a proof of the following approximation to a conjecture of Shelah:
Theorem (modulo a claim of Shelah whose proof has yet to appear): Assume there are unboundedly many strongly compact cardinals and the weak generalized continuum hypothesis holds. Then an AEC which for a high-enough cardinal λ has a single model of size λ will have a single model of size μ for every high-enough μ.
Sebastien Vasey, Independence in abstract elementary classes, Preprint: pdf arXiv.
Will Boney, Rami Grossberg, Alexei Kolesnikov, and Sebastien Vasey, Canonical forking in AECs, Preprint: pdf arXiv.
Saharon Shelah, Classification Theory for Abstract Elementary Classes, Studies in Logic: Mathematical Logic and foundations, vol. 18, College Publications, 2009.