5,508 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

    Games and Ramsey-like cardinals

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    We generalise the α\alpha-Ramsey cardinals introduced in Holy and Schlicht (2018) for cardinals α\alpha to arbitrary ordinals α\alpha, and answer several questions posed in that paper. In particular, we show that α\alpha-Ramseys are downwards absolute to the core model KK for all α\alpha of uncountable cofinality, that strategic ω\omega-Ramsey cardinals are equiconsistent with remarkable cardinals and that strategic α\alpha-Ramsey cardinals are equiconsistent with measurable cardinals for all α>ω\alpha>\omega. We also show that the nn-Ramseys satisfy indescribability properties and use them to provide a game-theoretic characterisation of completely ineffable cardinals, as well as establishing further connections between the α\alpha-Ramsey cardinals and the Ramsey-like cardinals introduced in Gitman (2011), Feng (1990) and Sharpe and Welch (2011).Comment: 33 pages, 2 figures. Added Theorem 4.20 saying that strategic (ω+1)(\omega{+}1)-Ramsey cardinals are equiconsistent with measurables, and fixed many typos. This version is forthcoming in the JS

    Virtual Set Theory:Taking the Blue Pill

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