Continuous tree-like scales

• Autorzy: James Cummings
• Języki publikacji: EN

Abstrakty

EN

Answering a question raised by Luis Pereira, we show that a continuous tree-like scale can exist above a supercompact cardinal. We also show that the existence of a continuous tree-like scale at ℵω is consistent with Martin’s Maximum.

Słowa kluczowe

EN

PCF theory; Large cardinals; Forcing

Kategorie tematyczne

• 03E04: Ordered sets and their cofinalities; pcf theory
• 03E55: Large cardinals

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2010
• Tom: 8
• Numer: 2
• Strony: 314-318

Daty

wydano

2010-04-01

online

2010-04-14

Twórcy

autor

James Cummings

• Carnegie Mellon University

Bibliografia

• [1] Brooke-Taylor A., Friedman S.D., Large cardinals and gap-1 morasses, Ann. Pure Appl. Logic, 2009, 159, 71–99 http://dx.doi.org/10.1016/j.apal.2008.10.007
• [2] Cummings J., Notes on singular cardinal combinatorics, Notre Dame J. Formal Logic, 2005, 46, 251–282 http://dx.doi.org/10.1305/ndjfl/1125409326
• [3] Cummings J., Foreman M., Magidor M., Squares, scales and stationary reflection, J. Math. Log., 2001, 1, 35–98 http://dx.doi.org/10.1142/S021906130100003X
• [4] Cummings J., Foreman M., Magidor M., The non-compactness of square, J. Symbolic Logic, 2003, 68, 637–643 http://dx.doi.org/10.2178/jsl/1052669068
• [5] Cummings J., Foreman M., Magidor M., Canonical structure in the universe of set theory I, Ann. Pure Appl. Logic, 2004, 129, 211–243 http://dx.doi.org/10.1016/j.apal.2004.04.002
• [6] Cummings J., Foreman M., Magidor M., Canonical structure in the universe of set theory II, Ann. Pure Appl. Logic, 2006, 142, 55–75 http://dx.doi.org/10.1016/j.apal.2005.11.007
• [7] Foreman M., Magidor M., A very weak square principle, J. Symbolic Logic, 1997, 62, 175–196 http://dx.doi.org/10.2307/2275738
• [8] Foreman M., Magidor M., Shelah S., Martin’s maximum, saturated ideals, and nonregular ultrafilters, I. Ann. of Math., 1988, 127, 1–47 http://dx.doi.org/10.2307/1971415
• [9] Gitik M., On a question of Pereira, Arch. Math. Logic, 2008, 47, 53–64 http://dx.doi.org/10.1007/s00153-008-0070-x
• [10] Laver R., Making the supercompactness of κ indestructible under κ-directed closed forcing, Israel J. Math., 1978, 29, 385–388 http://dx.doi.org/10.1007/BF02761175
• [11] Magidor M., Shelah S., When does almost free imply free? (for groups, transversals etc.), J. Amer. Math. Soc., 1994, 7, 769–830 http://dx.doi.org/10.2307/2152733
• [12] Pereira L., Combinatoire des cardinaux singuliers et structures PCF, Thesis, University of Paris VII, 2007
• [13] Shelah S., Semiproper forcing axiom implies Martin maximum but not PFA+, J. Symbolic Logic, 1987, 52, 360–367 http://dx.doi.org/10.2307/2274385
• [14] Shelah S., PCF and infinite free subsets in an algebra, Arch. Math. Logic, 2002, 41, 321–359 http://dx.doi.org/10.1007/s001530100101

Identyfikatory

DOI

10.2478/s11533-010-0001-z