We show that large fragments of MM, e. g. the tree property and stationary reflection, are preserved by strongly (ω1 + 1)-game-closed forcings. PFA can be destroyed by a strongly (ω1 + 1)-game-closed forcing but not by an ω2-closed.
A technique for approximating the behaviour of graph transformation systems (GTSs) by means of Petri net-like structures has been recently defined in the literature. In this paper we introduce a monadic second-order logic over graphs expressive enough to characterise typical graph properties, and we show how its formulae can be effectively verified. More specifically, we provide an encoding of such graph formulae into quantifier-free formulae over Petri net markings and we characterise, via a type assignment system, a subclass of formulae F such that the validity of F over a GTS G is implied by the validity of the encoding of F over the Petri net approximation of G. This allows us to reuse existing verification techniques, originally developed for Petri nets, to model-check the logic, suitably enriched with temporal operators
Abstract. We present principles for guessing clubs in the generalized club filter on P κ λ. These principles are shown to be weaker than classical diamond principles but often serve as sufficient substitutes. One application is a new construction of a λ + -Suslin-tree using assumptions different from previous constructions. The other application partly solves open problems regarding the cofinality of reflection points for stationary subsets of [λ] ℵ 0 .
Abstract. In [13] it was demonstrated that the Proper Forcing Axiom implies that there is a five element basis for the class of uncountable linear orders. The assumptions needed in the proof have consistency strength of at least infinitely many Woodin cardinals. In this paper we reduce the upper bound on the consistency strength of such a basis to something less than a Mahlo cardinal, a hypothesis which can hold in the constructible universe L.A crucial notion in the proof is the saturation of an Aronszajn tree, a statement which may be of broader interest. We show that if all Aronszajn trees are saturated and PFA(ω 1 ) holds, then there is a five element basis for the uncountable linear orders. We show that PFA(ω 2 ) implies that all Aronszajn trees are saturated and that it is consistent to have PFA(ω 1 ) plus every Aronszajn tree is saturated relative to the consistency of a reflecting Mahlo cardinal. Finally we show that a hypothesis weaker than the existence of a Mahlo cardinal is sufficient to force the existence of a five element basis for the uncountable linear orders.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.