# Non-Absoluteness of Model Existence at $\aleph_\omega$

## TL;DR

This paper demonstrates that the existence of models for certain infinitary sentences is not absolute between models of set theory, even under GCH, by constructing specific examples and analyzing their properties.

## Contribution

It provides the first negative answer under GCH for the model existence question at \u221e, relaxing large cardinal assumptions from supercompact to Mahlo.

## Key findings

- Negative answer for model existence at l_\u221e under GCH
- Construction of incomplete sentences proving non-absoluteness
- Non-absoluteness of l_lpha-amalgamation property for 1<lpha<

## Abstract

In [FHK13], the authors considered the question whether model-existence of $L_{\omega_1,\omega}$-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 $V\models "\varphi \text{ is an } L_{\omega_1,\omega}\text{-sentence}"$, then $V \models "\varphi \text{ has a model of size } \aleph_\alpha"$ if and only if $W \models "\varphi \text{ has a model of size } \aleph_\alpha"$. From [FHK13] we know that the answer is positive for $\alpha=0,1$ and under the negation of CH, the answer is negative for all $\alpha>1$. Under GCH, and assuming the consistency of a supercompact cardinal, the answer remains negative for each $\alpha>1$, except the case when $\alpha=\omega$ which is an open question in [FHK13]. 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 $\alpha>1$ 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 $\aleph_\alpha$-amalgamation property of $L_{\omega_1,\omega}$-sentences (under substructure). We prove that assuming GCH, $\aleph_\alpha$-amalgamation is non-absolute for $1<\alpha<\omega$. This answers a question from [SS]. The cases $\alpha=1$ and $\alpha$ infinite remain open. As a corollary we get that it is non-absolute that the amalgamation spectrum of an $L_{\omega_1,\omega}$-sentence is empty.