Skip to main content Accessibility help
×
Home

A relative of the approachability ideal, diamond and non-saturation

  • Assaf Rinot (a1)

Abstract

Let λ denote a singular cardinal. Zeman, improving a previous result of Shelah, proved that together with 2λ = λ+ implies ⋄S for every S ⊆ λ+ that reflects stationarily often.

In this paper, for a set S ⊆ λ+, a normal subideal of the weak approachability ideal is introduced, and denoted by I[S; λ]. We say that the ideal is fat if it contains a stationary set. It is proved:

1. if I[S; λ] is fat, then NSλ + ∣ S is non-saturated;

2. if I[S; λ] is fat and 2λ = λ+, then ⋄S holds;

3. implies that I[S; λ] is fat for every Sλ+ that reflects stationarily often;

4. it is relatively consistent with the existence of a supercompact cardinal that fails, while I[S; λ] is fat for every stationary S ⊆ λ+ that reflects stationarily often.

The stronger principle is studied as well.

Copyright

References

Hide All
[1]Cummings, James, Foreman, Matthew, and Magidor, Menachem, Squares, scales and stationary reflection, Journal of Mathematical Logic, vol. 1 (2001), no. 1, pp. 3598.
[2]Devlin, Keith J., Variations on ⋄, this Journal, vol. 44 (1979), no. 1, pp. 5158.
[3]Devlin, Keith J. and Johnsbráten, Hávard, The Souslin problem, Lecture Notes in Mathematics, Vol. 405, Springer-Verlag, Berlin, 1974.
[4]Džamonja, Mirna and Shelah, Saharon, Saturatedfilters at successors of singular, weak reflection and yet another weak club principle. Annals of Pure and Applied Logic, vol. 79 (1996), no. 3, pp. 289316.
[5]Foreman, Matthew and Magidor, Menachem, A very weak square principle, this Journal, vol. 62 (1997), no. 1, pp. 175196.
[6]Gitik, Moti and Rinot, Assaf, The failure of diamond on a reflecting stationary set, Transactions of the American Mathematical Society, To appear.
[7]Gitik, Moti and Shelah, Saharon, Less saturated ideals, Proceedings of the American Mathematical Society, vol. 125 (1997), no. 5, pp. 15231530.
[8]Gregory, John, Higher Souslin trees and the generalized continuum hypothesis, this Journal, vol. 41 (1976), no. 3, pp. 663671.
[9]Harrington, Leo and Shelah, Saharon, Some exact equiconsistency results in set theory, Notre Dame Journal of Formal Logic, vol. 26 (1985), no. 2, pp. 178188.
[10]Jech, Thomas and Shelah, Saharon, Full reflection of stationary sets below ℵω, this Journal, vol. 55 (1990), no. 2, pp. 822830.
[11]Jensen, R. Björn, The fine structure of the constructible hierarchy, Annals of Pure and Applied Logic, vol. 4 (1972), pp. 229308; erratum, R. Björn Jensen, The fine structure of the constructible hierarchy, Annals of Pure and Applied Logic. 4 (1972), 443, With a section by Jack Silver.
[12]Kanamori, A. and Magidor, M., The evolution of large cardinal axioms in set theory, Higher set theory (Mathematisches Forschungsinstitut, Oberwolfach, 1977), Lecture Notes in Mathematics, vol. 669, Springer, Berlin, 1978, pp. 99275.
[13]Krueger, John, Fat sets and saturated ideals, this Journal, vol. 68 (2003), no. 3, pp. 837845.
[14]Kunen, Kenneth, Set theory, Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland, Amsterdam, 1980.
[15]Magidor, Menachem, Reflecting stationary sets, this Journal, vol. 47 (1982), no. 4, pp. 755771 (1983).
[16]Moore, Justin Tatch, The proper forcing axiom, Prikry forcing, and the singular cardinals hypothesis, Annals of Pure and Applied Logic, vol. 140 (2006), no. 1-3, pp. 128132.
[17]Rinot, Assaf, A cofinality-preserving smallforcing may introduce a special aronszajn tree, Archive for Mathematical Logic, vol. 48 (2009), no. 8, pp. 817823.
[18]Shelah, Saharon, On successors of singular cardinals, Logic Colloquium '78 (Möns, 1978), Studies in Logic and the Foundations of Mathematics, vol. 97, North-Holland, Amsterdam, 1979, pp. 357380.
[19]Shelah, Saharon, Diamonds, uniformization, this Journal, vol. 49 (1984), no. 4, pp. 10221033.
[20]Shelah, Saharon, Advances in cardinal arithmetic, Finite and infinite combinatorics in sets and logic (Banff, AB, 1991), NATO Advanced Science Institutes Series C: Mathematics and Physics Sciences, vol. 411, Kluwer, Dordrecht, 1993, pp. 355383.
[21]Shelah, Saharon, Cardinal arithmetic, Oxford Logic Guides, vol. 29, The Clarendon Press Oxford University Press, New York, 1994.
[22]Shelah, Saharon, Diamonds, Proceedings of the American Mathematical Society, vol. 138 (2010), pp. 21512161.
[23]Steinhorn, Charles I. and King, James H., The uniformization property for ℵ2, Israeljournal of Mathematics, vol. 36 (1980), no. 3–4, pp. 248256.
[24]Viale, Matteo and Sharon, Assaf, Some consequences of reflection on the approachability ideal, Transactions of the American Mathematical Society, vol. 362 (2010), no. 8, pp. 42014212.
[25]Zeman, Martin, Diamond, GCH and weak square, Proceedings of the American Mathematical Society, vol. 138 (2010), no. 5, pp. 18531859.

Keywords

Related content

Powered by UNSILO

A relative of the approachability ideal, diamond and non-saturation

  • Assaf Rinot (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.