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We derive a forcing axiom from the conjunction of square and diamond, and present a few applications, primary among them being the existence of super-Souslin trees. It follows that for every uncountable cardinal
is not a Mahlo cardinal in Gödel’s constructible universe, then
entails the existence of a
We study countable saturation of metric reduced products and introduce continuous fields of metric structures indexed by locally compact, separable, completely metrizable spaces. Saturation of the reduced product depends both on the underlying index space and the model. By using the Gelfand–Naimark duality we conclude that the assertion that the Stone–Čech remainder of the half-line has only trivial automorphisms is independent from ZFC (Zermelo-Fraenkel axiomatization of set theory with the Axiom of Choice). Consistency of this statement follows from the Proper Forcing Axiom, and this is the first known example of a connected space with this property.
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