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A forcing axiom deciding the generalized Souslin Hypothesis

• Chris Lambie-Hanson,
Department of Mathematics, Bar-Ilan University, Ramat-Gan 5290002, Israel
• Assaf Rinot,
Department of Mathematics, Bar-Ilan University, Ramat-Gan 5290002, Israel
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Abstract

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 $\lambda$, if $\lambda^{++}$ is not a Mahlo cardinal in Gödel's constructible universe, then $2^\lambda = \lambda^+$ entails the existence of a $\lambda^+$-complete $\lambda^{++}$-Souslin tree.
 Keywords: Souslin tree, square, diamond, sharply dense set, forcing axiom, SDFA
 MSC Classifications: 03E05 - Other combinatorial set theory 03E35 - Consistency and independence results 03E57 - Generic absoluteness and forcing axioms [See also 03E50]

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