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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
##### Keywords:
disjoint amalgamation property; size of models
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##### References:
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