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The amalgamation spectrum. (English) Zbl 1177.03039
Summary: We study when classes can have the disjoint amalgamation property for a proper initial segment of cardinals.
Theorem A. For every natural number \(k\), there is a class \({\mathbf K}_k\) defined by a sentence in \(L_{\omega_1,\omega}\) that has no models of cardinality greater than \(\beth_{k+1}\), but \({\mathbf K}_k\) has the disjoint amalgamation property on models of cardinality less than or equal to \(\aleph_{k-3}\) and has models of cardinality \(\aleph_{k-1}\).
More strongly, we can have disjoint amalgamation up to \(\aleph_\alpha\) for \(\alpha<\omega_1\), but have a bound on the size of models.
Theorem B. For every countable ordinal \(\alpha\), there is a class \({\mathbf K_\alpha}\) defined by a sentence in \(L_{\omega_1,\omega}\) that has no models of cardinality greater than \(\beth_{\omega_1}\), but \(\mathbf K\) does have the disjoint amalgamation property on models of cardinality less than or equal to \(\aleph_\alpha\).
Finally we show that we can extend the \(\aleph_\alpha\) to \(\beth_\alpha\) in the second theorem consistently with ZFC and while having \(\aleph_i\ll\beth_i\) for \(0 < i\leq \alpha\).
Similar results hold for arbitrary ordinals \(\alpha\) with \(|\alpha|=\kappa\) and \(L_{\kappa^+,\omega}\).

MSC:
03C45 Classification theory, stability, and related concepts in model theory
03C48 Abstract elementary classes and related topics
03C75 Other infinitary logic
03E35 Consistency and independence results
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References:
[1] Notre Dame Journal of Formal Logic (2007)
[2] Shelah’s categoricity conjecture from a successor for tame abstract elementary classes 71 pp 553– (2006) · Zbl 1100.03023
[3] DOI: 10.1090/conm/302/05080
[4] DOI: 10.1016/S0168-0072(98)00016-5 · Zbl 0945.03049
[5] DOI: 10.1007/BF02762887
[6] DOI: 10.1007/BF02761954 · Zbl 0552.03019
[7] A complete l {\(\omega\)}1, {\(\omega\)}-sentence characterizing 1 42 pp 151– (1977)
[8] Classification theory and the number of nonisomorphic models (1978)
[9] DOI: 10.1007/BF02761994 · Zbl 0384.03032
[10] The theory of models pp 265– (1965)
[11] Upward categoricity from a successor cardinal for an abstract elementary class with amalgamation 70 pp 639– (2005)
[12] On the existence of atomic models 58 pp 1189– (1993)
[13] DOI: 10.1016/0003-4843(80)90009-1 · Zbl 0489.03008
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