## Journal of Symbolic Logic

### Club guessing sequences and filters

Tetsuya Ishiu

#### Abstract

We investigate club guessing sequences and filters. We prove that assuming V=L, there exists a strong club guessing sequence on μ if and only if μ is not ineffable for every uncountable regular cardinal μ. We also prove that for every uncountable regular cardinal μ, relative to the existence of a Woodin cardinal above μ, it is consistent that every tail club guessing ideal on μ is precipitous.

#### Article information

Source
J. Symbolic Logic, Volume 70, Issue 4 (2005), 1037-1071.

Dates
First available in Project Euclid: 18 October 2005

https://projecteuclid.org/euclid.jsl/1129642114

Digital Object Identifier
doi:10.2178/jsl/1129642114

Mathematical Reviews number (MathSciNet)
MR2194236

Zentralblatt MATH identifier
1100.03036

#### Citation

Ishiu, Tetsuya. Club guessing sequences and filters. J. Symbolic Logic 70 (2005), no. 4, 1037--1071. doi:10.2178/jsl/1129642114. https://projecteuclid.org/euclid.jsl/1129642114

#### References

• James E. Baumgartner and Alan D. Taylor Saturation properties of ideals in generic extensions. II, Transactions of the American Mathematical Society, vol. 271 (1982), no. 2, pp. 587--609.
• James E. Baumgartner, Alan D. Taylor, and Stanley Wagon On splitting stationary subsets of large cardinals, Journal of Symbolic Logic, vol. 42 (1977), no. 2, pp. 203--214.
• Keith J. Devlin Constructibility, Springer-Verlag,1984.
• Mirna Džamonja and Saharon Shelah On squares, outside guessing of clubs and $I_<f[\lambda]$, Fundamenta Mathematicae, vol. 148 (1995), pp. 165--198.
• Matthew Foreman Games played on Boolean algebras, Journal of Symbolic Logic, vol. 48 (1983), no. 3, pp. 714--723.
• Matthew Foreman and Péter Komjáth The club guessing ideal (commentary on a theorem of Gitik and Shelah), to appear.
• Matthew Foreman and Menachem Magidor Large cardinals and definable counterexamples to the continuum hypothesis, Annals of Pure and Applied Logic, vol. 76 (1995), no. 1, pp. 47--97.
• Matthew Foreman, Menachem Magidor, and Saharon Shelah Martin's maximum, saturated ideals, and nonregular ultrafilters. I, Annals of Mathematics. 2nd Series, vol. 127 (1988), no. 1, pp. 1--47.
• Moti Gitik and Saharon Shelah Less saturated ideals, Proceedings of the American Mathematical Society, vol. 125 (1997), no. 5, pp. 1523--1530.
• Noa Goldring Woodin cardinals and presaturated ideals, Annals of Pure and Applied Logic, vol. 55 (1992), no. 3, pp. 285--303.
• Thomas Jech A game theoretic property of Boolean algebras, Logic Colloquium '77 (Proceedings of the Conference, Wrocław, 1977), Studies in Logic and the Foundations of Mathematics, vol. 96, North-Holland,1978, pp. 135--144.
• Thomas Jech and Karel Prikry On ideals of sets and the power set operation, Bulletin of the American Mathematical Society, vol. 82 (1976), no. 4, pp. 593--595.
• Ronald B. Jensen Souslin's hypothesis is incompatible with V=L, Notices of the American Mathematical Society, vol. 15 (1968), p. 935.
• Akihiro Kanamori The Higher Inifinite, Springer-Verlag,1997.
• Menachem Kojman and Saharon Shelah The universality spectrum of stable unsuperstable theories, Annals of Pure and Applied Logic, vol. 58 (1992), no. 1, pp. 57--72.
• Adam J. Ostaszewski On countably compact perfectly normal spaces, Journal of the London Mathematical Society, vol. 14 (1976), no. 2, pp. 505--516.
• Andrzej Rosłanowski and Saharon Shelah Iteration of $\lambda$-complete forcing notions not collapsing $\lambda^+$, International Journal of Mathematics and Mathematical Sciences, vol. 28 (2001), pp. 63--82.
• Saharon Shelah On successors of singular cardinals, Logic Colloquium '78 (Mons, 1978), Studies in Logic and the Foundations of Mathematics, vol. 97, North-Holland,1979, pp. 357--380.
• Robert M. Solovay Real-valued Measurable Ccardinals, Axiomatic Set Theory (D. Scott, editor), American Mathematical Society,1971, pp. 397--428.