Friedman et al. (2013) 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 subset of W are transitive models of ZFC with the same ordinals, phi is an element of V and V satisfies"phi is an L-omega 1,(omega)-sentence", then V satisfies Phi if and only if W satisfies Phi where Phi is a first-order sentence with parameters phi and alpha asserting that phi 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 > 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 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 > 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 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 of Sinapova and Souldatos (2017). The cases alpha = 1 and a infinite remain open. As a corollary we show that it is non-absolute that the amalgamation spectrum of an L-omega 1,(omega)-sentence is empty.
