Categoricity, amalgamation, and tameness

Abstract

ABSTRACT. Theorem. For each 2 ≤ k < ω there is an Lω 1 ,ω -sentence φ k such that:(1) φ k is categorical in µ if µ ≤ ℵ k−2 ; (2) φ k is not ℵ k−2 -Galois stable; (3) φ k is not categorical in any µ with µ > ℵ k−2 ; (4) φ k has the disjoint amalgamation property;indeed, syntactic first-order types determine Galois types over models of cardinality at mostWe adapt an example of [9]. The amalgamation, tameness, stability results, and the contrast between syntactic and Galois types are new; the categoricity results refine the earlier work of Hart and Shelah and answer a question posed by Shelah in [17].Considerable work (e.g. [14,15,16,7,8,6,18,12,11]) has explored the extension of Morley's categoricity theorem to infinitary contexts. While the analysis in [14,15] applies only to L ω 1 ,ω , it can be generalized and in some ways strengthened in the context of abstract elementary classes.Various locality properties of syntactic types do not generalize in general to Galois types (defined as orbits under an automorphism group) in an AEC [5]; much of the difficulty of the work stems from this difference. One such locality properties is called tameness. Roughly speaking, K is (µ, κ)-tame if distinct Galois types over models of size κ have distinct restrictions to some submodel of size µ. For classes with arbitrarily large models, that satisfy amalgamation and tameness, strong categoricity transfer theorems have been proved [7,8,6,13,4,10]. In particular these results yield categoricity in every uncountable power for a tame AEC in a countable language (with arbitrarily large models satisfying amalgamation and the joint embedding property) that is categorical in any single cardinal above ℵ 2 ([6]) or even above ℵ 1 ([13]).In contrast, Shelah's original work [14,15] showed (under weak GCH) that categoricity up to ℵ ω of a sentence in L ω 1 ,ω implies categoricity in all uncountable cardinalities. Hart and Shelah [9] showed the necessity of the assumption by constructing sentences φ k which were categorical up to some ℵ n but not eventually

Tameness From Large Cardinal Axioms

“…In AECs, types as sets of formulas do not behave as nicely as they do in firstorder model theory; any of the examples of non-tameness is an example of this and it is made explicit in [BK09]. However, Shelah isolated a semantic notion of type in [Sh300] that Grossberg named Galois type in [Gro02] this can replace the first-order notion.…”

A topology for galois types in abstract elementary classes

“…A few words about the significance of the assumption of tameness: although it holds automatically in elementary classes (as well as in homogeneous classes, among other settings), it fails in some contexts- [4] and [5], for example, construct examples of non-tame AECs from Abelian groups. Moreover, it is independent from the other properties one associates with well-behaved AECs.…”

Good frames in the Hart–Shelah example

“…For a fixed 1 n ∈ [3, ω), the Hart-Shelah example is an AEC K n that is categorical exactly in the interval [ℵ 0 , ℵ n−2 ]. It was investigated in details by Baldwin and Kolesnikov [BK09] who proved that K n has (disjoint) amalgamation, is (Galois) stable exactly in the infinite cardinals less than or equal to ℵ n−3 , and is (< ℵ 0 , ≤ ℵ n−3 )-tame (i.e. Galois types over models of size at most ℵ n−3 are determined by their restrictions to finite sets, see Definition 2.1).…”