Large cardinals and gap-1 morasses

Annals of Pure and Applied Logic 159 (1-2):71-99 (2009)
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Abstract

We present a new partial order for directly forcing morasses to exist that enjoys a significant homogeneity property. We then use this forcing in a reverse Easton iteration to obtain an extension universe with morasses at every regular uncountable cardinal, while preserving all n-superstrong , hyperstrong and 1-extendible cardinals. In the latter case, a preliminary forcing to make the GCH hold is required. Our forcing yields morasses that satisfy an extra property related to the homogeneity of the partial order; we refer to them as mangroves and prove that their existence is equivalent to the existence of morasses. Finally, we exhibit a partial order that forces universal morasses to exist at every regular uncountable cardinal, and use this to show that universal morasses are consistent with n-superstrong, hyperstrong, and 1-extendible cardinals. This all contributes to the second author’s outer model programme, the aim of which is to show that L-like principles can hold in outer models which nevertheless contain large cardinals

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Citations of this work

On extendible cardinals and the GCH.Konstantinos Tsaprounis - 2013 - Archive for Mathematical Logic 52 (5-6):593-602.
On c-extendible cardinals.Konstantinos Tsaprounis - 2018 - Journal of Symbolic Logic 83 (3):1112-1131.
Indestructibility of Vopěnka’s Principle.Andrew D. Brooke-Taylor - 2011 - Archive for Mathematical Logic 50 (5-6):515-529.
Ultrahuge cardinals.Konstantinos Tsaprounis - 2016 - Mathematical Logic Quarterly 62 (1-2):77-87.

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References found in this work

Morasses, diamond, and forcing.Daniel J. Velleman - 1982 - Annals of Mathematical Logic 23 (2):199.
Simplified morasses.Dan Velleman - 1984 - Journal of Symbolic Logic 49 (1):257-271.
The PCF Conjecture and Large Cardinals.Luís Pereira - 2008 - Journal of Symbolic Logic 73 (2):674 - 688.
Morasses, Diamond, and Forcing.Lee Stanley - 1989 - Journal of Symbolic Logic 54 (2):639-646.

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