The categoricity spectrum of large abstract elementary classes

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Abstract

The categoricity spectrum of a class of structures is the collection of cardinals in which the class has a single model up to isomorphism. Assuming that cardinal exponentiation is injective (a weakening of the generalized continuum hypothesis, GCH), we give a complete list of the possible categoricity spectrums of an abstract elementary class with amalgamation and arbitrarily large models. Specifically, the categoricity spectrum is either empty, an end segment starting below the Hanf number, or a closed interval consisting of finite successors of the Löwenheim–Skolem–Tarski number (there are examples of each type). We also prove (assuming a strengthening of the GCH) that the categoricity spectrum of an abstract elementary class with no maximal models is either bounded or contains an end segment. This answers several longstanding questions around Shelah’s categoricity conjecture.

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Notes

1. That is, sentences like $$\forall x \exists y: x \cdot y = 1 \wedge y \cdot x = 1$$: quantification is over elements and only finite conjunctions and disjunctions are allowed.

2. The conjecture also appears as open problem D.3(a) in [3].

3. For example, Theorem 5.33 gives new conditions for forking symmetry of independent sequences, a key difficulty in [36].

4. Although this will not be used, it is known [39] that taking $$\theta = \text {cf} (\lambda )$$ suffices: the conjunction of $$2^\lambda = \lambda ^+$$ with the principle $$\Phi _{\lambda ^+} (\{\delta < \lambda ^+ \mid \text {cf} (\delta ) = \text {cf} (\lambda )\})$$ is equivalent to $${\text {GCHWD}}(\lambda )$$.

5. If $${\mathbf {K}}$$ is an AEC, $${\mathbf {S}}(M)$$ will of course be a set.

6. In Shelah’s original definition, only the set of basic types is required to be stable. However full stability follows, see [12, II.4.2].

7. This is a special case of the definition of a skeleton, see [44, 5.3] but since we have no use for skeletons in this paper, we chose to only study the simpler case.

8. Recall that by definition of $${\text {Cat}}({\mathbf {K}})$$, this implies that $$\mu _1 \ge \text {LS}({\mathbf {K}})$$.

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Acknowledgements

We thank John T. Baldwin, Will Boney, and Marcos Mazari-Armida for comments that helped improve the presentation of this paper. We also thank the referee for multiple thorough reports.

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Correspondence to Sebastien Vasey.

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Vasey, S. The categoricity spectrum of large abstract elementary classes. Sel. Math. New Ser. 25, 65 (2019). https://doi.org/10.1007/s00029-019-0511-x