×

zbMATH — the first resource for mathematics

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:
03E35 Consistency and independence results
03E55 Large cardinals
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Baumgartner, James E., Iterated forcing, (), 1-59 · Zbl 0524.03040
[2] James Cummings, Iterated forcing and elementary embeddings, in: The Handbook of Set Theory (Volume II), Springer, ISBN: 978-1-4020-4843-2, 2009 (Chapter 12)
[3] Cummings, James; Schimmerling, Ernest, Indexed squares, Israel journal of mathematics, 131, 61-99, (2002) · Zbl 1025.03040
[4] Devlin, Keith J., Aspects of constructibility, Lecture notes in mathematics, vol. 354, (1973), Springer Berlin · Zbl 0312.02054
[5] Devlin, Keith J., Constructibility, (1984), Springer Berlin · Zbl 0542.03029
[6] Dobrinen, Natasha; Friedman, Sy D., Homogeneous iteration and measure one covering relative to HOD, Archive for mathematical logic, 47, 7-8, 711-718, (2008) · Zbl 1153.03034
[7] Friedman, Sy D., Fine structure and class forcing, De gruyter series in logic and its applications, vol. 3, (2000), de Gruyter Berlin · Zbl 0954.03045
[8] Friedman, Sy D., Large cardinals and \(L\)-like universes, (), 93-110 · Zbl 1136.03034
[9] Jech, Thomas, Set theory, (2003), Springer · Zbl 1007.03002
[10] Kanamori, Akihiro, The higher infinite, (2003), Springer · Zbl 1022.03033
[11] Kunen, Kenneth, Set theory, (1980), North-Holland · Zbl 0443.03021
[12] Pereira, Luís, The PCF conjecture and large cardinals, Journal of symbolic logic, 73, 2, 674-688, (2008) · Zbl 1153.03024
[13] Stanley, Lee, Review of papers by dan velleman, Journal of symbolic logic, 54, 2, 639-646, (1989)
[14] Velleman, Daniel J., Morasses, diamond, and forcing, Annals of mathematical logic, 23, 2-3, 199-281, (1982) · Zbl 0521.03034
[15] Velleman, Daniel J., Simplified morasses, The journal of symbolic logic, 49, 1, 257-271, (1984) · Zbl 0575.03035
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.