267 research outputs found
Non-Absoluteness of Model Existence at
In [FHK13], the authors considered the question whether model-existence of
-sentences is absolute for transitive models of ZFC, in
the sense that if are transitive models of ZFC with the same
ordinals, and , then if and only if .
From [FHK13] we know that the answer is positive for and under
the negation of CH, the answer is negative for all . Under GCH, and
assuming the consistency of a supercompact cardinal, the answer remains
negative for each , except the case when which is an
open question in [FHK13].
We answer the open question by providing a negative answer under GCH even for
. Our examples are incomplete sentences. In fact, the same
sentences can be used to prove a negative answer under GCH for all
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
-amalgamation property of -sentences (under
substructure). We prove that assuming GCH, -amalgamation is
non-absolute for . This answers a question from [SS]. The
cases and infinite remain open. As a corollary we get that
it is non-absolute that the amalgamation spectrum of an
-sentence is empty
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