Non-Absoluteness of Model Existence at $\aleph_ω$

Abstract

In [FHK13], the authors considered the question whether model-existence of $L_{ω_1,ω}$-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, $φ\in V$ and $V\models "φ\text{ is an } L_{ω_1,ω}\text{-sentence}"$, then $V \models "φ\text{ has a model of size } \aleph_α"$ if and only if $W \models "φ\text{ has a model of size } \aleph_α"$. From [FHK13] we know that the answer is positive for $α=0,1$ and under the negation of CH, the answer is negative for all $α>1$. Under GCH, and assuming the consistency of a supercompact cardinal, the answer remains negative for each $α>1$, except the case when $α=ω$ which is an open question in [FHK13]. We answer the open question by providing a negative answer under GCH even for $α=ω$. Our examples are incomplete sentences. In fact, the same sentences can be used to prove a negative answer under GCH for all $α>1$ assuming the consistency of a Mahlo cardinal. Thus, the large cardinal assumption is relaxed from a supercompact in [FHK13] to a Mahlo cardinal. Finally, we consider the absoluteness question for the $\aleph_α$-amalgamation property of $L_{ω_1,ω}$-sentences (under substructure). We prove that assuming GCH, $\aleph_α$-amalgamation is non-absolute for $1<α<ω$. This answers a question from [SS]. The cases $α=1$ and $α$ infinite remain open. As a corollary we get that it is non-absolute that the amalgamation spectrum of an $L_{ω_1,ω}$-sentence is empty.