Non-absoluteness of model existence at $\aleph _\omega $
Volume 243 / 2018
Abstract
Friedman et al. (2013) considered the question whether model existence of $\mathcal{L}_{\omega_1,\omega}$-sentences is absolute for transitive models of ZFC, in the sense that if $V \subseteq W$ are transitive models of ZFC with the same ordinals, $\varphi\!\in\! V$ and $V\!\models\! “\varphi \text{ is an } \mathcal{L}_{\omega_1,\omega}\text{-sentence}”$, then $V \models\Phi$ if and only if $W \models\Phi$ where $\Phi$ is a first-order sentence with parameters $\varphi$ and $\alpha$ asserting that $\varphi$ has a model of size $\aleph_\alpha$.
From Friedman et al. (2013) we know that the answer is positive for $\alpha=0,1$, and under the negation of CH the answer is negative for all $\alpha \gt 1$. Under GCH, and assuming the consistency of a supercompact cardinal, the answer remains negative for each $\alpha \gt 1$, except the case when $\alpha=\omega$ which is an open question in Friedman et al. (2013).
We answer the open question by providing a negative answer under GCH even for $\alpha=\omega$. Our examples are incomplete sentences. In fact, the same sentences can be used to prove a negative answer under GCH for all $\alpha \gt 1$ assuming the consistency of a Mahlo cardinal. Thus, the large cardinal assumption is relaxed from a supercompact in Friedman et al. (2013) to a Mahlo cardinal.
Finally, we consider the absoluteness question for the $\aleph_\alpha$-amalgamation property of $\mathcal{L}_{\omega_1,\omega}$-sentences (under substructure). We prove that assuming GCH, $\aleph_\alpha$-amalgamation is non-absolute for $1 \lt \alpha \lt \omega$. This answers a question of Sinapova and Souldatos (2017). The cases $\alpha=1$ and $\alpha$ infinite remain open. As a corollary we show that it is non-absolute that the amalgamation spectrum of an $\mathcal{L}_{\omega_1,\omega}$-sentence is empty.
