Abstract
We develop a notion of forking for Galois-types in the context of Abstract Elementary Classes (AECs). Under the hypotheses that an AEC K is tame, type-short, and failure of an order-property, we considerDefinition 1Let M0≺N be models from K and A be a set. We say that the Galois-type of A over N does not fork overM0, written A⫝M0N, iff for all small a∈A and all small N−≺N, we have that Galois-type of a over N− is realized in M0.Assuming property (E) (Existence and Extension, see Definition 3.3) we show that this non-forking is a well behaved notion of independence, in particular satisfies symmetry and uniqueness and has a corresponding U-rank. We find conditions for a universal local character, in particular derive superstability-like property from little more than categoricity in a “big cardinal”. Finally, we show that under large cardinal axioms the proofs are simpler and the non-forking is more powerful.In [10], it is established that, if this notion is an independence notion, then it is the only one.
Highlights
Much of first order model theory has focused on Shelah’s forking
Already in 1970, Shelah [Sh3] introduced splitting as a weak independence notion for a nonelementary context that is known as homogeneous model theory
For the more general cases of classes axiomatizeable by an Lλ+,ω sentence or Abstract Elementary Classes (AECs), very little is known in this direction, there have been several attempts
Summary
Much of first order model theory has focused on Shelah’s forking. In the last fifteen years, significant progress has been made towards understanding of unstable theories, especially simple theories (Kim [Ki98] and Kim and Pillay [KP97]), N IP theories (see surveys by Adler [Ad09] and Simon [Si]), and, most recently, N T P2 (Ben-Yaacov and Chernikov [BYCh] and Chenikov, Kaplan and Shelah [CKS1007]). Makkai and Shelah [MaSh285] studied the case when a class is axiomatized by an Lκ,ω theory and κ is strongly compact They managed to obtain an eventual categoricity theorem by introducing a forking-like relation on types. One of the more important notions is that of good λ-frame This is a forking-like relation defined using Galoistypes over models of cardinality λ. Our approach is orthogonal to Shelah’s recent work on good λ-frames and we manage to obtain a forking notion on the class of all models above a natural threshold size (instead of models of a single cardinality). Boney [Bonc] shows that the hypotheses of the above theorem hold for any AEC with LS(K) < κ We improve these papers by using purely model theoretic properties: tameness and type shortness. We would like to thank John Baldwin, Adi Jarden, Sebastien Vasey, and Andres Villaveces for comments on an early drafts of this paper
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