Uniqueness of limit models in categorical abstract elementary classes with no maximal models: removing the tower hypotheses of the Shelah-Villaveces program

The 2006 paper 'Categoricity in abstract elementary classes with no maximal models' proved, inside the Shelah-Villaveces framework - an abstract elementary class with no maximal models that is categorical in some cardinal $\lambda$ at or above its Hanf number, assuming GCH and instances of the weak diamond principle, but without assuming the amalgamation property - that limit models of cardinality $\mu<\lambda$ are unique up to isomorphism over a common base, but only under a working hypothesis: every continuous tower inside the auxiliary monster-substitute $\mathfrak{C}$ has an amalgamable extension. Whether that hypothesis, or the stronger closure statements designed to imply it (amalgamation bases closed under unions of chains; unions of chains of limit models being limit models), follows from categoricity itself has stood open since. The current status in the primary and secondary literature: the analogous uniqueness theorem under full amalgamation is completely settled; the superstability ingredient of the original program was re-proved in ZFC in 2017; the single claimed unconditional proof in the amalgamation-free framework (Boney-VanDieren-Vasey, 2016) was withdrawn owing to a fatal mistake; and the sharpest surviving result equates the uniqueness conclusion, for $\lambda=\mu^{+n}$, with a union-of-chains amalgamation-base statement. The unconditional uniqueness of limit models remains an open problem.

Background

An abstract elementary class (AEC) is a class $K$ of structures in a fixed vocabulary, equipped with a binary relation $\prec_{\mathcal{K}}$ refining the substructure relation, that is closed under isomorphisms and under unions of $\prec_{\mathcal{K}}$-increasing chains (with the union behaving as the direct limit), satisfies a coherence axiom for $\prec_{\mathcal{K}}$, and has a Loewenheim-Skolem number $\operatorname{LS}(\mathcal{K})$: every structure in the class has $\prec_{\mathcal{K}}$-substructures of size at most $\operatorname{LS}(\mathcal{K})+|A|$ containing any prescribed small set $A$. AECs, introduced by Shelah, capture the model classes of many logics beyond first order, such as $L_{\omega_1,\omega}$, where compactness fails. Each AEC has a Hanf number $\operatorname{Hanf}(\mathcal{K})$, a cardinal above which the existence of arbitrarily large models already follows from the existence of one model of that size. A class is categorical in $\lambda$ if it has exactly one model of cardinality $\lambda$ up to isomorphism, and Shelah's eventual categoricity conjecture - the main test question of the field - predicts that an AEC categorical in some sufficiently large cardinal is categorical in all sufficiently large cardinals.

The amalgamation property (AP) - any two extensions of a common model can be jointly embedded over it - is the structural hypothesis that in first-order model theory follows from compactness. Without AP there is no monster model, and the notion of type is replaced by Galois types (orbits of pairs under $\prec_{\mathcal{K}}$-embeddings, well defined through amalgamation diagrams of amalgamation bases). A model $M\in K_{\mu}$ is an amalgamation base if any two $\mu$-sized $\prec_{\mathcal{K}}$-extensions of $M$ can be amalgamated. A limit model is the substitute for a saturated model: $M$ is a $(\mu,\theta)$-limit model, for a limit ordinal $\theta<\mu^{+}$, if $M$ is the union of an increasing continuous chain $\langle M_i\mid i<\theta\rangle$ of amalgamation bases of cardinality $\mu$ with each $M_{i+1}$ universal over $M_i$. Two limit models over the same base with the same cofinality of length are isomorphic by a back-and-forth argument, so the substantive uniqueness question compares different cofinalities; even in first-order logic, uniqueness of limit models across all regular lengths is essentially equivalent to superstability and fails for strictly stable theories.

Shelah and Villaveces (1999) initiated the study of AECs that have no maximal models (every model has a proper $\prec_{\mathcal{K}}$-extension) without assuming AP, under GCH (the generalized continuum hypothesis, $2^{\mu}=\mu^{+}$ for every infinite cardinal $\mu$) together with instances of the weak diamond principle $\Phi_{\mu^{+}}(S^{\operatorname{cf}(\mu)}_{\mu^{+}})$ (a weak form of the diamond principle, following from $2^{\mu}<2^{\mu^{+}}$, that supplies predictions on stationary sets and enables diagonalization constructions; it is weaker than full diamond and consistent with failures of GCH, but the source framework simply assumes GCH plus these instances). In this framework they proved from categoricity the density of amalgamation bases and a superstability-like property for $\mu$-splitting (a dependence notion for Galois types), and asserted - with a flawed proof - that limit models below the categoricity cardinal are unique. VanDieren identified the gaps (after two years of correspondence the original authors acknowledged they could not repair them), and the 2006 paper proved the uniqueness theorem under the working Hypothesis 1: every continuous tower inside the fixed $(\mu,\mu^{+})$-limit model $\mathfrak{C}$ (which serves as a weak substitute for the monster model) has an amalgamable $<^{c}_{\mu,\alpha}$-extension inside $\mathfrak{C}$. Here a tower is an increasing sequence of amalgamation bases together with distinguished elements and base models recording amalgamation data, ordered by $<^{c}_{\mu,\alpha}$ (a strengthening of extension requiring continuity and element-avoidance); the uniqueness proof builds a long $<^{c}$-increasing chain of towers whose direct limits must stay inside $\mathfrak{C}$, and it is exactly at the limit stages of these constructions that an extra closure property of amalgamation bases is needed. The paper shows that Hypothesis 1 would follow from Hypothesis 2 (the class $\mathcal{K}^{am}_{\mu}$ of amalgamation bases of cardinality $\mu$ is closed under unions of $\prec_{\mathcal{K}}$-increasing chains of length $<\mu^{+}$, a weakening of Grossberg's intermediate categoricity conjecture, stated as Conjecture 0.12 there) or from Hypothesis 3 (the union of such a chain of limit models is again a limit model), proves the $<^{c}_{\mu,\alpha}$-extension property only for the class of 'nice' towers - towers whose limit-stage initial unions $\bigcup_{j<i}M_{j}$ are amalgamation bases - (noting explicitly that the original Shelah-Villaveces argument for all towers 'does not converge'), and proves that reduced towers (towers that are minimal with respect to a certain $<^{c}$-type extension ordering) are continuous only under Hypothesis 3. A 2013 erratum corrected a further error, found by Tapani Hyttinen, in the surrounding arguments.

The uniqueness of limit models is positioned as the first step toward deriving the amalgamation property itself from categoricity in this framework, and ultimately toward categoricity transfer without amalgamation. The subsequent literature fully resolves the analogous uniqueness question under AP, re-proves the superstability ingredient in ZFC, and contains one claimed unconditional proof in the AP-free framework that was withdrawn; the unconditional uniqueness question is therefore the surviving open core of the program.

Problem Statement

Work in the Shelah-Villaveces framework: let $\mathcal{K}$ be an abstract elementary class with Loewenheim-Skolem number $\operatorname{LS}(\mathcal{K})$ and Hanf number $\operatorname{Hanf}(\mathcal{K})$ satisfying Assumption 0.7 of VanDieren (2006): (1) $\mathcal{K}$ has no maximal models with respect to $\prec_{\mathcal{K}}$; (2) $\mathcal{K}$ is categorical in a fixed cardinal $\lambda\geq\operatorname{Hanf}(\mathcal{K})$; (3) GCH (the generalized continuum hypothesis, $2^{\mu}=\mu^{+}$ for every infinite cardinal $\mu$) holds; and (4) the weak-diamond principle $\Phi_{\mu^{+}}(S^{\operatorname{cf}(\mu)}_{\mu^{+}})$ holds for every $\mu<\lambda$. The amalgamation property is not assumed. Here a model $M\in\mathcal{K}$ of cardinality $\mu$ is an amalgamation base if any two $\mu$-sized $\prec_{\mathcal{K}}$-extensions of $M$ can be jointly embedded over $M$ by $\prec_{\mathcal{K}}$-embeddings fixing $M$ pointwise, and a model is a $(\mu,\theta)$-limit model over a base $M_{0}$, for a limit ordinal $\theta<\mu^{+}$, if it is the union of an increasing continuous chain $\langle M_{i}\mid i<\theta\rangle$ of amalgamation bases of cardinality $\mu$ with $M_{0}$ as its first element and each $M_{i+1}$ universal over $M_{i}$ (every model of cardinality $\mu$ extending $M_{i}$ $\prec_{\mathcal{K}}$-embeds into $M_{i+1}$ over $M_{i}$); two such models are isomorphic over the common base if there is a $\prec_{\mathcal{K}}$-isomorphism between them fixing the base pointwise. The problem is the unconditional uniqueness of limit models: for every cardinal $\mu$ with $\operatorname{LS}(\mathcal{K})\leq\mu<\lambda$ and all limit ordinals $\theta_1,\theta_2<\mu^{+}$, if $M_1$ and $M_2$ are $(\mu,\theta_1)$- and $(\mu,\theta_2)$-limit models over a common base $M$, must $M_1$ and $M_2$ be isomorphic over $M$? Prove this from Assumption 0.7 alone - that is, remove Hypothesis 1 from the proof of Theorem II.9.1 of VanDieren (2006) - or refute it by an explicit example. For the formulations below: a tower is an increasing sequence of amalgamation bases indexed by an ordinal $\alpha\leq\mu^{+}$, each carrying distinguished elements and base models that record amalgamation data, and $<^{c}_{\mu,\alpha}$ is the associated extension ordering on towers, strengthening the extension relation with continuity and element-avoidance requirements; $\mathfrak{C}$ denotes a fixed $(\mu,\mu^{+})$-limit model - the union of a $\prec_{\mathcal{K}}$-increasing continuous chain $\langle M_i\in\mathcal{K}^{am}_{\mu}\mid i<\mu^{+}\rangle$ of amalgamation bases of cardinality $\mu$ with each $M_{i+1}$ universal over $M_i$ - the framework's substitute for a monster model, inside which the tower constructions of the source paper take place. Related questions posed by the source, where Hypothesis 1 would follow from Hypothesis 2 or from Hypothesis 3: (a) does Hypothesis 1 (every continuous tower inside the fixed $(\mu,\mu^{+})$-limit model $\mathfrak{C}$ has an amalgamable $<^{c}_{\mu,\alpha}$-extension inside $\mathfrak{C}$) follow from Assumption 0.7? (b) does Hypothesis 2 (the class $\mathcal{K}^{am}_{\mu}$ of amalgamation bases of cardinality $\mu$ is closed under unions of $\prec_{\mathcal{K}}$-increasing chains of length $<\mu^{+}$) follow from Assumption 0.7? (c) does Hypothesis 3, or the weaker statement that the union of an increasing continuous chain of limit models is an amalgamation base, follow from Assumption 0.7 (for $\lambda=\mu^{+n}$ this weakened form is equivalent to the uniqueness conclusion, by the characterization in arXiv:1511.09112)? The source paper also poses Question III.11.6: can Hypothesis 3 be removed from Theorem III.11.4 (reduced towers are continuous), or, alternatively, are nice reduced towers dense (a tower is nice if every limit-stage initial union $\bigcup_{j<i}M_{j}$ is an amalgamation base)? The paper attributes to a positive answer only the weaker consequence that every nice tower has a continuous extension without any extra hypothesis; this is a partial step toward Hypothesis 1 (Part III of the paper is described there as 'a partial proof of Hypothesis 1'), not a form that by itself settles the uniqueness question. A refutation requires an explicit abstract elementary class satisfying Assumption 0.7 (in a universe where the stated GCH and weak-diamond instances hold) together with a cardinal $\mu<\lambda$ and two limit models over a common base of cardinality $\mu$ that are not isomorphic over that base. The downstream goals of the program - deriving the amalgamation property, and downward categoricity transfer, from categoricity with no maximal models alone - are the motivation for this question and remain open, but the completion boundary of the present problem is the uniqueness statement (or its refutation).

References

1. Monica VanDieren (2006). Categoricity in abstract elementary classes with no maximal models. Annals of Pure and Applied Logic 141, 108-147. DOI: 10.1016/j.apal.2005.10.006. https://doi.org/10.1016/j.apal.2005.10.006 (arXiv: math/0510579)
2. Monica M. VanDieren (2013). Erratum to 'Categoricity in abstract elementary classes with no maximal models' [Ann. Pure Appl. Logic 141 (2006) 108-147]. Annals of Pure and Applied Logic 164(2), 131-133. DOI: 10.1016/j.apal.2012.09.003. https://doi.org/10.1016/j.apal.2012.09.003
3. Saharon Shelah and Andres Villaveces (1999). Toward categoricity for classes with no maximal models. Annals of Pure and Applied Logic 97, 1-25. DOI: 10.1016/S0168-0072(98)00015-3. https://doi.org/10.1016/S0168-0072(98)00015-3 (arXiv: math/9707227)
4. Will Boney, Rami Grossberg, Monica M. VanDieren, and Sebastien Vasey (2017). Superstability from categoricity in abstract elementary classes. Annals of Pure and Applied Logic 168, 1383-1395. DOI: 10.1016/j.apal.2017.01.005. https://doi.org/10.1016/j.apal.2017.01.005 (arXiv: 1609.07101)
5. Rami Grossberg, Monica VanDieren, and Andres Villaveces (2016). Uniqueness of limit models in classes with amalgamation. Mathematical Logic Quarterly 62, 367-382. DOI: 10.1002/malq.201500033. https://doi.org/10.1002/malq.201500033 (arXiv: 1507.02118)
6. Sebastien Vasey (2017). Saturation and solvability in abstract elementary classes with amalgamation. Archive for Mathematical Logic 56, 671-690. DOI: 10.1007/s00153-017-0561-8. https://doi.org/10.1007/s00153-017-0561-8 (arXiv: 1604.07743)
7. Sebastien Vasey (2019). The categoricity spectrum of large abstract elementary classes. Selecta Mathematica 25(5), article 65. DOI: 10.1007/s00029-019-0511-x. https://doi.org/10.1007/s00029-019-0511-x (arXiv: 1805.04068)
8. Will Boney, Monica M. VanDieren, and Sebastien Vasey (2016). Shelah-Villaveces revisited. arXiv:1611.05292. https://arxiv.org/abs/1611.05292 (preprint v1, November 2016; withdrawn by the authors in March 2017 with the note 'Paper withdrawn due to a fatal mistake')
9. Monica M. VanDieren (2016). A Characterization of Uniqueness of Limit Models in Categorical Abstract Elementary Classes. arXiv:1511.09112. https://arxiv.org/abs/1511.09112 (unpublished preprint, v3)
10. Will Boney and Sebastien Vasey (2017). Chains of saturated models in AECs. Archive for Mathematical Logic 56, 187-213. DOI: 10.1007/s00153-017-0532-0. https://doi.org/10.1007/s00153-017-0532-0 (arXiv: 1503.08781)
11. Jeremy Beard and Marcos Mazari-Armida (2025). On the spectrum of limit models. Annals of Pure and Applied Logic 176, 103647. DOI: 10.1016/j.apal.2025.103647. https://doi.org/10.1016/j.apal.2025.103647 (arXiv: 2503.11605)
12. Will Boney and Monica VanDieren (2024). Limit models in strictly stable abstract elementary classes. Mathematical Logic Quarterly 70, 438-453. DOI: 10.1002/malq.202200075. https://doi.org/10.1002/malq.202200075
13. Monica M. VanDieren (2016). Superstability and symmetry. Annals of Pure and Applied Logic 167, 1171-1183. DOI: 10.1016/j.apal.2016.05.004. https://doi.org/10.1016/j.apal.2016.05.004
14. Monica M. VanDieren (2016). Symmetry and the union of saturated models in superstable abstract elementary classes. Annals of Pure and Applied Logic 167, 395-407. DOI: 10.1016/j.apal.2015.12.007. https://doi.org/10.1016/j.apal.2015.12.007
15. Rami Grossberg and Monica VanDieren (2006). Galois-stability for tame abstract elementary classes. Journal of Mathematical Logic 6(1), 25-48. DOI: 10.1142/S0219061306000487. https://doi.org/10.1142/S0219061306000487 (arXiv: math/0509535)