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تسجيل مستخدم جديدNon-Absoluteness of Model Existence at $aleph_omega$
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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, $varphiin V$ and $Vmodels varphi text{ is an } L_{omega_1,omega}text{-sentence}$, 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.
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