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The modal logic of inner models

Tanmay Inamdar & Benedikt Löwe

Journal of Symbolic Logic 81 (1):225-236 (2016)

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The Consistency Strength of M P C C C.George Leibman - 2010 - Notre Dame Journal of Formal Logic 51 (2):181-193.details The Maximality Principle MPCCC is a scheme which states that if a sentence of the language of ZFC is true in some CCC forcing extension VP, and remains true in any further CCC-forcing extension of VP, then it is true in all CCC-forcing extensions of V, including V itself. A parameterized form of this principle, MPCCC, makes this assertion for formulas taking real parameters. In this paper, we show that MPCCC has the same consistency strength as ZFC, solving an open (...) Download Export citation Bookmark 2 citations
Fatal Heyting Algebras and Forcing Persistent Sentences.Leo Esakia & Benedikt Löwe - 2012 - Studia Logica 100 (1-2):163-173.details Hamkins and Löwe proved that the modal logic of forcing is S4.2 . In this paper, we consider its modal companion, the intermediate logic KC and relate it to the fatal Heyting algebra H ZFC of forcing persistent sentences. This Heyting algebra is equationally generic for the class of fatal Heyting algebras. Motivated by these results, we further analyse the class of fatal Heyting algebras. Download Export citation Bookmark 2 citations
The Necessary Maximality Principle for c. c. c. forcing is equiconsistent with a weakly compact cardinal.Joel D. Hamkins & W. Hugh Woodin - 2005 - Mathematical Logic Quarterly 51 (5):493-498.details The Necessary Maximality Principle for c. c. c. forcing with real parameters is equiconsistent with the existence of a weakly compact cardinal. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim). Download Export citation Bookmark 8 citations
Consistency results about ordinal definability.Kenneth McAloon - 1971 - Annals of Mathematical Logic 2 (4):449.details Download Export citation Bookmark 33 citations
Certain very large cardinals are not created in small forcing extensions.Richard Laver - 2007 - Annals of Pure and Applied Logic 149 (1-3):1-6.details The large cardinal axioms of the title assert, respectively, the existence of a nontrivial elementary embedding j:Vλ→Vλ, the existence of such a j which is moreover , and the existence of such a j which extends to an elementary j:Vλ+1→Vλ+1. It is known that these axioms are preserved in passing from a ground model to a small forcing extension. In this paper the reverse directions of these preservations are proved. Also the following is shown : if V is a model (...) Download Export citation Bookmark 26 citations

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