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A Forcing Axiom Deciding the Generalized Souslin Hypothesis

Part of: Set theory

Published online by Cambridge University Press:  07 January 2019

Chris Lambie-Hanson  and
Assaf Rinot
Chris Lambie-Hanson
Affiliation:
Department of Mathematics, Bar-Ilan University, Ramat-Gan 5290002, Israel Email: lambiec@macs.biu.ac.ilrinotas@math.biu.ac.il
Assaf Rinot
Affiliation:
Department of Mathematics, Bar-Ilan University, Ramat-Gan 5290002, Israel Email: lambiec@macs.biu.ac.ilrinotas@math.biu.ac.il
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Abstract

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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 $\unicode[STIX]{x1D706}$, if $\unicode[STIX]{x1D706}^{++}$ is not a Mahlo cardinal in Gödel’s constructible universe, then $2^{\unicode[STIX]{x1D706}}=\unicode[STIX]{x1D706}^{+}$ entails the existence of a $\unicode[STIX]{x1D706}^{+}$-complete $\unicode[STIX]{x1D706}^{++}$-Souslin tree.

Keywords

Souslin treesquarediamondsharply dense setforcing axiomSDFA

MSC classification

Primary: 03E05: Other combinatorial set theory
Secondary: 03E35: Consistency and independence results 03E57: Generic absoluteness and forcing axioms
Type
Article
Information
Canadian Journal of Mathematics , Volume 71 , Issue 2 , April 2019 , pp. 437 - 470
Copyright
© Canadian Mathematical Society 2018 

Footnotes

This research was partially supported by the Israel Science Foundation (grant #1630/14).

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