Uniqueness of limit models from categoricity in abstract elementary classes without full amalgamation: deriving Assumption 1 from categoricity (the Shelah-Villaveces problem)

An abstract elementary class (AEC) with no maximal models that is categorical in a cardinal $\lambda>\mu$ is conjectured (Shelah-Villaveces, 1999, Claim/Theorem 3.37) to have unique limit models of cardinality $\mu$ over any common base, assuming only GCH and weak-diamond principles and, crucially, not assuming the amalgamation property. VanDieren (2016) proved, for $\lambda=\mu^{+n}$, that this uniqueness is equivalent to 'Assumption 1' — that the union of every increasing continuous chain of limit models of length $<\mu^{+}$ is an amalgamation base — but left Assumption 1 itself underived from categoricity, deferred the extension to arbitrary $\mu<\lambda$, and flagged the set-theoretic hypotheses for elimination. The follow-up preprint that VanDieren (2016) cites as answering the Assumption 1 question affirmatively (Boney-VanDieren-Vasey, arXiv:1611.05292) was withdrawn in March 2017 'due to a fatal mistake' and no corrected proof has appeared; the superstability step was later made ZFC by Boney-Grossberg-VanDieren-Vasey (APAL 2017) but only given diamond-derived density hypotheses; and the limit-model literature since 2018 assumes amalgamation or tameness. The problem — deriving uniqueness of limit models (equivalently Assumption 1) from categoricity in the no-full-amalgamation setting, across the full cardinal range and with minimal set-theoretic hypotheses — remains open.

Background

An abstract elementary class (AEC) is a pair $\mathbf{K}=(K,\prec_{\mathbf{K}})$ consisting of a class of structures in a fixed vocabulary and a strong substructure relation satisfying category-theoretic axioms abstracting the class of models of a first-order theory with elementary embeddings: closure under isomorphisms and under unions of $\prec_{\mathbf{K}}$-increasing chains, coherence, and a downward Löwenheim-Skolem property, which yields the Löwenheim-Skolem number $LS(\mathbf{K})$. Unlike elementary classes, AECs need satisfy neither compactness nor the amalgamation property (AP): given $M_{0}\prec_{\mathbf{K}}M_{1},M_{2}$ there need not be a common $\prec_{\mathbf{K}}$-extension amalgamating them. $\mathbf{K}$ is categorical in $\lambda$ if all its models of cardinality $\lambda$ are isomorphic, and has no maximal models if every model has a proper $\prec_{\mathbf{K}}$-extension. Since there is in general no syntactic notion of type, one uses Galois (orbital) types: the orbit $\mathrm{gtp}(a/M)$ of an element under automorphisms of a sufficiently large ambient model fixing $M$.

The classification theory of AECs is guided by Shelah's eventual categoricity conjecture (an AEC categorical in one sufficiently large cardinal is categorical in all sufficiently large cardinals) and by Grossberg's 1986 conjecture that amalgamation follows from categoricity. In the strategy of Kolman and Shelah (as recounted in VanDieren (2016)), the uniqueness of limit models is the key intermediate step from categoricity to amalgamation. Here $M\in K_{\mu}$ is an amalgamation base if any two of its extensions of cardinality $\mu$ can be amalgamated, and $M$ is a $(\mu,\theta)$-limit model over $M_{0}$ if $M=\bigcup_{i<\theta}M_{i}$ for an increasing continuous chain of amalgamation bases with $M_{i+1}$ universal over $M_{i}$ (every extension of $M_{i}$ of cardinality $\mu$ embeds into $M_{i+1}$ over $M_{i}$). Limit models are the AEC surrogate for saturated models; uniqueness of limit models says that any two limit models of cardinality $\mu$ over the same base $M_{0}$ are isomorphic over $M_{0}$.

The technical engine is $\mu$-splitting: a Galois type $p$ over $M$ $\mu$-splits over $N\prec_{\mathbf{K}}M$ if there are $N\prec_{\mathbf{K}}N_{1},N_{2}\prec_{\mathbf{K}}M$ and an isomorphism $h:N_{1}\cong N_{2}$ fixing $N$ with $h(p\upharpoonright N_{1})\neq p\upharpoonright N_{2}$. A class is $\mu$-superstable (roughly) if every type over the union of a chain of limit models does not $\mu$-split over some early stage (no long splitting chains), and has $\mu$-symmetry if non-splitting independence is symmetric in the two variables of a type. The main combinatorial tools are towers — increasing sequences of amalgamation bases with distinguished elements tracking non-splitting — whose reducedness and continuity properties encode whether limit models built along different chains coalesce. GCH is the generalized continuum hypothesis ($2^{\kappa}=\kappa^{+}$ for all infinite $\kappa$), and $\Phi_{\chi^{+}}(S^{\chi^{+}}_{\mathrm{cf}(\chi)})$ is a weak form of Jensen's diamond principle predicting colorings along the stationary set of points of cofinality $\mathrm{cf}(\chi)$ in $\chi^{+}$; together with categoricity it yields the density of amalgamation bases.

Shelah and Villaveces (1999) set out to prove uniqueness of limit models from categoricity with no maximal models, without assuming AP; their argument derived superstability and the density of amalgamation bases from GCH and weak diamond, then ran a tower argument. VanDieren found substantial gaps in this proof (her 2002 thesis and 2006 APAL paper, with a 2013 erratum), which had repaired Shelah-Villaveces by adding the assumption that unions of chains of limit models are limit models; with Grossberg and Villaveces (MLQ 2016) she later proved uniqueness under full amalgamation with superstability-like hypotheses ($\mu$-stability, no $\mu$-Vaughtian pairs, and locality of splitting), and her APAL 2016 papers established the superstability-plus-symmetry route under AP. VanDieren (2016) returns to the original no-AP setting and replaces the strong assumption by the weaker Assumption 1: the union of an increasing and continuous chain of limit models $\langle M_{i}\in K_{\mu}\mid i<\alpha<\mu^{+}\rangle$ is an amalgamation base. Its Theorem 1 states that for $\mu=\kappa^{+}$ with $LS(\mathbf{K})\le\kappa$, $\lambda=\mu^{+n}$ with $0<n<\omega$, under GCH and $\Phi_{\chi^{+}}(S^{\chi^{+}}_{\mathrm{cf}(\chi)})$ for $\kappa\le\chi<\lambda$, in a $\lambda$-categorical AEC with no maximal models, Assumption 1, uniqueness of limit models over a base, uniqueness of limit models, and saturation of unions of chains of saturated models dense with $\kappa$-amalgamation bases are all equivalent. Corollary 1 concludes that Assumption 1 is necessary and sufficient for the Shelah-Villaveces claim when $\lambda=\mu^{+n}$. For contrast: in the presence of full amalgamation the problem is settled — Vasey (AML 2017) proved that categoricity above implies $\lambda$-superstability and $\lambda$-symmetry in 'nice' AECs, which combines with the amalgamation-setting papers above to give uniqueness of limit models, and Grossberg-Vasey (JSL 2017) showed uniqueness of limit models is equivalent to superstability in tame AECs. The open residue is exactly the no-full-amalgamation, no-tameness setting of the original Shelah-Villaveces program.

Problem Statement

Settle the Shelah-Villaveces uniqueness-of-limit-models problem in abstract elementary classes without full amalgamation. Let $\mathbf{K}$ be an AEC with no maximal models that is categorical in some $\lambda>\mu\ge LS(\mathbf{K})$ (with $\mu=\kappa^{+}$, $LS(\mathbf{K})\le\kappa$), under GCH and the weak-diamond instances $\Phi_{\chi^{+}}(S^{\chi^{+}}_{\mathrm{cf}(\chi)})$ for $\kappa\le\chi<\lambda$. The overall objective is to prove, without assuming the amalgamation property (or tameness), that limit models of cardinality $\mu$ over a common base $M_{0}$ are isomorphic over $M_{0}$ (Shelah-Villaveces' Claim 1), which by Theorem 1 of VanDieren (2016) is equivalent, for $\lambda=\mu^{+n}$, to deriving:

1. Assumption 1 from categoricity: prove that the union of every increasing and continuous chain of limit models $\langle M_{i}\in K_{\mu}\mid i<\alpha<\mu^{+}\rangle$ is an amalgamation base, using only categoricity, no maximal models, and the standing set-theoretic hypotheses — the closing question of VanDieren (2016), whose cited affirmative answer was withdrawn in 2017. Alternatively, refute Claim 1 by exhibiting such a categorical AEC in which two limit models of cardinality $\mu$ over a common base are not isomorphic over that base (equivalently, in which Assumption 1 fails).

2. Cardinal range: extend the equivalence and the uniqueness conclusion beyond the successor regime $\lambda=\mu^{+n}$ ($0<n<\omega$) to arbitrary $\mu<\lambda$, which requires transferring $\mu$-symmetry down past limit cardinals without amalgamation — the step VanDieren (2016) deferred and whose attempted adaptation was the withdrawn preprint.

3. Set-theoretic hypotheses: reduce or eliminate GCH and the $\Phi_{\chi^{+}}(S^{\chi^{+}}_{\mathrm{cf}(\chi)})$ instances; Remark 3 of VanDieren (2016) pinpoints the three proof steps that use them (superstability — since made ZFC given the density hypotheses; existence of limit models in the needed cardinalities; limit models of intermediate cardinalities being amalgamation bases).

The three parts above are facets of a single objective — deriving superstability-like structure from categoricity without amalgamation — in the no-full-amalgamation setting, for arbitrary $\mu<\lambda$, and under minimal hypotheses; the counterexample of (1) is its negative direction, and a result confined to the $\lambda=\mu^{+n}$ regime or to GCH is the corresponding restriction of that objective.

References

1. Monica M. VanDieren, 'A Characterization of Uniqueness of Limit Models in Categorical Abstract Elementary Classes', arXiv preprint arXiv:1511.09112 (v1 November 2015, v3 December 2016; unpublished), DOI 10.48550/arXiv.1511.09112, https://arxiv.org/abs/1511.09112
2. Saharon Shelah and Andrés Villaveces, 'Toward categoricity for classes with no maximal models', Annals of Pure and Applied Logic 97 (1999) 1-25, DOI 10.1016/S0168-0072(98)00015-3, https://doi.org/10.1016/S0168-0072(98)00015-3
3. Will Boney, Monica M. VanDieren, and Sebastien Vasey, 'Shelah-Villaveces revisited', arXiv:1611.05292 (v1, 16 November 2016, circulated under the title 'Some instances of uniqueness of limit models in abstract elementary classes with no maximal models'; v2, 5 March 2017, withdrawn with the arXiv comment 'Paper withdrawn due to a fatal mistake'), https://arxiv.org/abs/1611.05292
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