Kurepa trees and spectra of $\mathcal{L}_{ω_1,ω}$-sentences

We use set-theoretic tools to make a model-theoretic contribution. In particular, we construct a \emph{single} $\mathcal{L}_{ω_1,ω}$-sentence $ψ$ that codes Kurepa trees to prove the consistency of the following: (1) The spectrum of $ψ$ is consistently equal to $[\aleph_0,\aleph_{ω_1}]$ and also consistently equal to $[\aleph_0,2^{\aleph_1})$, where $2^{\aleph_1}$ is weakly inaccessible. (2) The amalgamation spectrum of $ψ$ is consistently equal to $[\aleph_1,\aleph_{ω_1}]$ and $[\aleph_1,2^{\aleph_1})$, where again $2^{\aleph_1}$ is weakly inaccessible. This is the first example of an $\mathcal{L}_{ω_1,ω}$-sentence whose spectrum and amalgamation spectrum are consistently both right-open and right-closed. It also provides a positive answer to a question in [18]. (3) Consistently, $ψ$ has maximal models in finite, countable, and uncountable many cardinalities. This complements the examples given in [1] and [2] of sentences with maximal models in countably many cardinalities. (4) $2^{\aleph_0}<\aleph_{ω_1}<2^{\aleph_1}$ and there exists an $\mathcal{L}_{ω_1,ω}$-sentence with models in $\aleph_{ω_1}$, but no models in $2^{\aleph_1}$. This relates to a conjecture by Shelah that if $\aleph_{ω_1}<2^{\aleph_0}$, then any $\mathcal{L}_{ω_1,ω}$-sentence with a model of size $\aleph_{ω_1}$ also has a model of size $2^{\aleph_0}$. Our result proves that $2^{\aleph_0}$ can not be replaced by $2^{\aleph_1}$, even if $2^{\aleph_0}<\aleph_{ω_1}$.

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