Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Parametrized $\diamondsuit$ principles


Authors: Justin Tatch Moore, Michael Hrusák and Mirna Dzamonja
Journal: Trans. Amer. Math. Soc. 356 (2004), 2281-2306
MSC (2000): Primary 03E17, 03E65
DOI: https://doi.org/10.1090/S0002-9947-03-03446-9
Published electronically: October 8, 2003
MathSciNet review: 2048518
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We will present a collection of guessing principles which have a similar relationship to $\diamondsuit$ as cardinal invariants of the continuum have to ${CH}$. The purpose is to provide a means for systematically analyzing $\diamondsuit$ and its consequences. It also provides for a unified approach for understanding the status of a number of consequences of ${CH}$ and $\diamondsuit$in models such as those of Laver, Miller, and Sacks.


References [Enhancements On Off] (What's this?)

  • 1. B. Balcar and M. Hrusák.
    Combinatorics of dense subsets of the rationals.
    Preprint, 2003.
  • 2. T. Bartoszynski and H. Judah.
    Set theory. On the structure of the real line.
    A. K. Peters, Ltd., 1995. MR 96k:03002
  • 3. T. Baumgartner.
    Personal communication, January 2003.
  • 4. A. Blass.
    Reductions between cardinal characteristics of the continuum.
    In Set Theory (Boise, ID, 1992-1994), 31-49. Contemp. Math., 192, AMS 1996. MR 97b:03059
  • 5. Jörg Brendle.
    Mad families and iteration theory.
    In Y. Zhang, editor, Logic and Algebra, 1-31. Contemp. Math., 302, AMS 2002.
  • 6. Jörg Brendle.
    Mob families and mad families.
    Arch. Math. Logic, 37(3):183-197, 1997. MR 99m:03098
  • 7. K. Devlin and S. Shelah.
    A weak version of $\diamondsuit$ which follows from $2^{\aleph_0} < 2^{\aleph_1}$.
    Israel Journal of Math, 29(2-3):239-247, 1978. MR 57:9537
  • 8. E. K. van Douwen.
    The integers and topology.
    In K. Kunen and J. Vaughan, editors, Handbook of Set-Theoretic Topology, pages 111-167. North-Holland, 1984. MR 87f:54008
  • 9. A. Dow.
    More set-theory for topologists.
    Topology Appl., 64(3):243-300, 1995. MR 97a:54005
  • 10. T. Eisworth.
    All ladder systems can be anti-Dowker.
    Preprint.
  • 11. T. Eisworth and J. Roitman.
    CH with no Ostaszewski spaces.
    Trans. Amer. Math. Soc., 351(7):2675-2693, 1999. MR 2000b:03182
  • 12. M. Goldstern and S. Shelah.
    Ramsey ultrafilters and the reaping number-- ${\rm {c}on}(\mathfrak{r}<\mathfrak{u})$.
    Ann. Pure Appl. Logic, 49(2):121-142, 1990. MR 91m:03050
  • 13. J. Hirschorn.
    Cohen and random reals.
    Ph.D. thesis, University of Toronto, 2000.
  • 14. M. Hrusák.
    Life in the Sacks model.
    Acta Univ. Carolin. Math. Phys., 42(2):43-58, 2001. MR 2003b:03069
  • 15. M. Hrusák.
    Rendezvous with madness.
    Ph.D. thesis, York University, 1999.
  • 16. M. Hrusák.
    Another $\diamondsuit$-like principle.
    Fund. Math., 167(3):277-289, 2001. MR 2002e:03075
  • 17. R. B. Jensen.
    Souslin's Hypothesis is incompatible with $V=L$.
    Notices Amer. Math. Soc., 15:935, 1968.
  • 18. Judah-Shelah.
    Killing Luzin and Sierpinski sets.
    Proc. Amer. Math. Soc., 120(3):917-920, 1994. MR 94e:03046
  • 19. K. Kunen.
    An introduction to independence proofs, volume 102 of Studies in Logic and the Foundations of Mathematics.
    North-Holland, 1983. MR 85e:03003
  • 20. P. Larson and S. Todorcevic.
    Katetov's problem.
    Trans. Amer. Math. Soc., 354:1783-1792, 2002. MR 2003b:54033
  • 21. P. Larson and S. Todorcevic.
    Chain conditions in maximal models.
    Fund. Math., 168(1):77-104, 2001. MR 2002e:03067
  • 22. J. T. Moore.
    Random forcing and (S) and (L).
    Submitted to Top. Appl.
  • 23. J. T. Moore.
    Ramsey theory on sets of real numbers.
    Ph.D. thesis, University of Toronto, 2000.
  • 24. A. Ostaszewski.
    On countably compact, perfectly normal spaces.
    J. London Math. Soc., 14(3):505-516, 1976. MR 55:11210
  • 25. A. Ros\lanowski and S. Shelah.
    Norms on possibilities I. Forcing with trees and creatures.
    Mem. Amer. Math. Soc., 141(671), 1999. MR 2000c:03036
  • 26. Saharon Shelah.
    Covering of the null ideal may have countable cofinality.
    Fund. Math., 166(1-2):109-136, 2000.
    Saharon Shelah's anniversary issue. MR 2001m:03101
  • 27. S. Shelah.
    Proper and improper forcing.
    Springer-Verlag, Berlin, second edition, 1998. MR 98m:03002
  • 28. Saharon Shelah.
    On cardinal invariants of the continuum.
    In Axiomatic set theory (Boulder, Colo., 1983), pages 183-207. Amer. Math. Soc., Providence, RI, 1984. MR 86b:03064
  • 29. Otmar Spinas.
    Partition numbers.
    Ann. Pure Appl. Logic, 90(1-3):243-262, 1997. MR 99c:03067
  • 30. S. Todorcevic.
    Coherent sequences (preprint 2002).
    In Handbook of Set Theory. North-Holland.
  • 31. S. Todorcevic.
    Partitioning pairs of countable ordinals.
    Acta Math., 159(3-4):261-294, 1987. MR 88i:04002
  • 32. S. Todorcevic.
    Partition Problems In Topology.
    Amer. Math. Soc., 1989. MR 90d:04001
  • 33. S. Todorcevic.
    Random set mappings and separability of compacta.
    Topology and its Applications, 74:265-274, 1996. MR 97j:03099
  • 34. Peter Vojtás.
    Generalized Galois-Tukey-connections between explicit relations on classical objects of real analysis.
    In Set theory of the reals (Ramat Gan, 1991), pages 619-643. Bar-Ilan Univ., Ramat Gan, 1993. MR 95e:03139
  • 35. Nancy M. Warren.
    Properties of Stone-Cech compactifications of discrete spaces.
    Proc. Amer. Math. Soc., 33:599-606, 1972. MR 45:1123
  • 36. W. H. Woodin.
    The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal.
    Logic and its Applications. de Gruyter, 1999. MR 2001e:03001
  • 37. O. Yiparaki.
    On Some Partition Trees.
    Ph.D. thesis, University of Michigan, 1994.
  • 38. Y. Zhang.
    On a class of mad families.
    J. Symbolic Logic, 64:737-746, 1999. MR 2001m:03104

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 03E17, 03E65

Retrieve articles in all journals with MSC (2000): 03E17, 03E65


Additional Information

Justin Tatch Moore
Affiliation: Department of Mathematics, Boise State University, Boise, Idaho 83725
Email: justin@math.boisestate.edu

Michael Hrusák
Affiliation: Institute of Mathematics, University Nacional Autonoma de Mexico, Apartado Postal 27-3, 58089 Morelia, Mexico
Email: michael@matmor.unam.mx

Mirna Dzamonja
Affiliation: School of Mathematics, University of East Anglia, Norwich, England NR4 7TJ
Email: m.dzamonja@uea.ac.uk

DOI: https://doi.org/10.1090/S0002-9947-03-03446-9
Keywords: Diamond, weak diamond, cardinal invariant, guessing principle
Received by editor(s): September 12, 2002
Published electronically: October 8, 2003
Additional Notes: The first and third authors received support from EPSRC grant GR/M71121 for the research of this paper. The research of the second author was supported in part by the Netherlands Organization for Scientific Research (NWO) – Grant 613.007.039, and in part by the Grant Agency of the Czech Republic – Grant GAČR 201/00/1466.
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society