The amalgamation spectrum.

*(English)*Zbl 1177.03039Summary: 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 \(L_{\kappa^+,\omega}\).

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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\textit{J. T. Baldwin} et al., J. Symb. Log. 74, No. 3, 914--928 (2009; Zbl 1177.03039)

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