281 research outputs found

    Combinatorial Properties and Dependent choice in symmetric extensions based on L\'{e}vy Collapse

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    We work with symmetric extensions based on L\'{e}vy Collapse and extend a few results of Arthur Apter. We prove a conjecture of Ioanna Dimitriou from her P.h.d. thesis. We also observe that if VV is a model of ZFC, then DC<κDC_{<\kappa} can be preserved in the symmetric extension of VV in terms of symmetric system ⟨P,G,F⟩\langle \mathbb{P},\mathcal{G},\mathcal{F}\rangle, if P\mathbb{P} is κ\kappa-distributive and F\mathcal{F} is κ\kappa-complete. Further we observe that if VV is a model of ZF + DCκDC_{\kappa}, then DC<κDC_{<\kappa} can be preserved in the symmetric extension of VV in terms of symmetric system ⟨P,G,F⟩\langle \mathbb{P},\mathcal{G},\mathcal{F}\rangle, if P\mathbb{P} is κ\kappa-strategically closed and F\mathcal{F} is κ\kappa-complete.Comment: Revised versio

    Laver and set theory

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    In this commemorative article, the work of Richard Laver is surveyed in its full range and extent.Accepted manuscrip

    Stationary reflection principles and two cardinal tree properties

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    We study consequences of stationary and semi-stationary set reflection. We show that the semi stationary reflection principle implies the Singular Cardinal Hypothesis, the failure of weak square principle, etc. We also consider two cardinal tree properties introduced recently by Weiss and prove that they follow from stationary and semi stationary set reflection augmented with a weak form of Martin's Axiom. We also show that there are some differences between the two reflection principles which suggest that stationary set reflection is analogous to supercompactness whereas semi-stationary set reflection is analogous to strong compactness.Comment: 19 page
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