Tameness and extending frames

Abstract

We combine two notions in AECs, tameness and good λ-frames, and show that they together give a very well-behaved nonforking notion in all cardinalities. This helps to fill a longstanding gap in classification theory of tame AECs and increases the applicability of frames. Along the way, we prove a complete stability transfer theorem and uniqueness of limit models in these AECs.VanDieren [11], which came from the latter's thesis, and says that two different types over a large model must differ over some smaller model. Tameness has been used successfully in categoricity transfers (see Grossberg and VanDieren [12,13] and Lessman [18]) and stability transfer (see Grossberg and VanDieren [11]; Baldwin, Kueker and VanDieren [3]; and Lieberman [19]). Unfortunately, not all AECs are tame as Baldwin and Shelah [5] have constructed an AEC that is not tame from the exact sequences of an almost free, non-Whitehead group which exists in ZFC at ℵ 1 and consistently exists in all cardinals. On the other hand, the author has shown in [6] that tameness follows for all AECs from large cardinals. Theorem 1.2 ([6]).• If K is an AEC with LS(K) < κ and κ is strongly compact, then K is < κ tame.• Suppose there is a proper class of strongly compact cardinals. Then every AEC is tame.1450007-2 J. Math. Log. 2014.14. Downloaded from www.worldscientific.com by FLINDERS UNIVERSITY LIBRARY on 02/05/15. For personal use only. Tameness and extending framesThese assumptions commonly appear in addition to tameness: amalgamation is used to make types well behaved and joint embedding then follows from λ-joint embedding. However, these global assumptions are in contrast to the project of frames, which aims to inductively build up a structure theory, cardinal by cardinal, and derive these properties along the way with the aid of weak diamond. On the other hand, the existence of frames in the most general setting (see [26, II, Sec. 3]) uses categoricity in two successive cardinals (and more). If we add no maximal models to this hypothesis, this is already enough to apply the full categoricity transfer of [13].On the other hand, the combination of these hypotheses gives much more than just the sum of their parts. Despite the categoricity transfer results under a tameness hypothesis, there is no robust independence notion for these classes. The closest approximation is likely Boney and Grossberg [7], where an independence notion of "< κ satisfiability" is developed. Although this notion is well-behaved, additional methods beyond tameness are needed. Using the method in this paper, we have an independence notion for tame and categorical AECs under some very mild cardinal arithmetic assumptions; see Theorem 8.3. Looking at good λ-frames, the method for building larger frames is a complicated process that changes the Abstract Elementary Class and drops many of the models; see [26], especially II, Sec. 9.1. Although this is fine for the end goal, a process that deals with the whole class would likely have more applications. We provide such a process for tam...

Building independence relations in abstract elementary classes

“…Later, [Bon14a] (with improvements in [BVc]) gave conditions under which all the properties transferred. Similar ideas are used in [Vas16] to directly build a good frame.…”

“…However this is secondary for us (since tameness already implies that a frame extends to larger models, see [Bon14a,BVc]). Really, we want to extend the good frame to longer types.…”

“…Recall that extending a (≤ 1, λ)-frame to bigger models was investigated in [She09, Chapter II] and [Bon14a,BVc]. Here, most of the arguments are similar but the longer types cause some additional difficulties (e.g.…”

“…However a coherent chain of types (see below) always has an upper bound. We adapt Definition 5.1 in [Bon14a] (which is implicit already in [She01, Claim 0.32.2] or [GV06a, Lemma 2.12]) to our purpose. Definition 14.11.…”

“…Examples of the use of tameness and amalgamation include [BKV06] (an upward stability transfer), [Lie11] (showing that tameness is equivalent to a natural topology on Galois types being Hausdorff), [GV06c] (an upward categoricity transfer theorem, which can be combined with Fact 1.1 and the downward transfer of Shelah [She99] to prove that Shelah's eventual categoricity conjecture for a successor follows from the existence of a proper class of strongly compact cardinals) and [Bon14a,BVc,Jar16], showing that good frames behave well in tame classes.…”