Continuous tree-like scales

James Cummings

Open Mathematics (2010)

  • Volume: 8, Issue: 2, page 314-318
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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James Cummings. "Continuous tree-like scales." Open Mathematics 8.2 (2010): 314-318. <http://eudml.org/doc/269205>.

@article{JamesCummings2010,
abstract = {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.},
author = {James Cummings},
journal = {Open Mathematics},
keywords = {PCF theory; Large cardinals; Forcing; large cardinals; forcing},
language = {eng},
number = {2},
pages = {314-318},
title = {Continuous tree-like scales},
url = {http://eudml.org/doc/269205},
volume = {8},
year = {2010},
}

TY - JOUR
AU - James Cummings
TI - Continuous tree-like scales
JO - Open Mathematics
PY - 2010
VL - 8
IS - 2
SP - 314
EP - 318
AB - 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.
LA - eng
KW - PCF theory; Large cardinals; Forcing; large cardinals; forcing
UR - http://eudml.org/doc/269205
ER -

References

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  1. [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 Zbl1165.03033
  2. [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 Zbl1121.03053
  3. [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 Zbl0988.03075
  4. [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 Zbl1069.03032
  5. [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 Zbl1058.03051
  6. [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 Zbl1096.03060
  7. [7] Foreman M., Magidor M., A very weak square principle, J. Symbolic Logic, 1997, 62, 175–196 http://dx.doi.org/10.2307/2275738 Zbl0880.03022
  8. [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 Zbl0645.03028
  9. [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 Zbl1145.03024
  10. [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 Zbl0381.03039
  11. [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 Zbl0819.20059
  12. [12] Pereira L., Combinatoire des cardinaux singuliers et structures PCF, Thesis, University of Paris VII, 2007 
  13. [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 Zbl0625.03035
  14. [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 Zbl1049.03034

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