Kurepa trees and spectra of ${\mathcal{L}}_{{\omega }_1,\omega }$-sentences

Abstract

We use set-theoretic tools to make a model-theoretic contribution. In particular, we construct a \emph{single} ${\mathcal{L}}_{{\omega }_1,\omega }$-sentence $\psi$ that codes Kurepa trees to prove the consistency of the following: (1) The spectrum of $\psi$ is consistently equal to $[{\aleph }_0,{\aleph }_{{\omega }_1}]$ and also consistently equal to $[{\aleph }_0,2^{{\aleph }_1})$, where $2^{{\aleph }_1}$ is weakly inaccessible. (2) The amalgamation spectrum of $\psi$ is consistently equal to $[{\aleph }_1,{\aleph }_{{\omega }_1}]$ and $[{\aleph }_1,2^{{\aleph }_1})$, where again $2^{{\aleph }_1}$ is weakly inaccessible. This is the first example of an ${\mathcal{L}}_{{\omega }_1,\omega }$-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, $\psi$ 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) $2^{{\aleph }_0}<{\aleph }_{{\omega }_1}<2^{{\aleph }_1}$ and there exists an ${\mathcal{L}}_{{\omega }_1,\omega }$-sentence with models in ${\aleph }_{{\omega }_1}$, but no models in $2^{{\aleph }_1}$. This relates to a conjecture by Shelah that if ${\aleph }_{{\omega }_1}<2^{{\aleph }_0}$, then any ${\mathcal{L}}_{{\omega }_1,\omega }$-sentence with a model of size ${\aleph }_{{\omega }_1}$ 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 $2^{{\aleph }_0}<{\aleph }_{{\omega }_1}$.

to appear in the Journal of Mathematical Logic

References in corpus (3)

• Generalized Descriptive Set Theory and Classification Theory
• Notes on cardinals that are characterizable by a complete (Scott) sentence
• Non-Absoluteness of Model Existence at ${\aleph }_{\omega }$

Cited by in corpus (1)

• Characterizing the spectra of cardinalities of branches of Kurepa trees