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Large cardinals and gap-1 morasses. (English) Zbl 1165.03033
Summary: 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 ($$1\leq n\leq \omega$$), 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.

##### MSC:
 3e+35 Consistency and independence results 3e+55 Large cardinals
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##### References:
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