## Notre Dame Journal of Formal Logic

### Some Problems in Singular Cardinals Combinatorics

Matthew Foreman

#### Abstract

This paper attempts to present and organize several problems in the theory of Singular Cardinals. The most famous problems in the area (bounds for the ℶ-function at singular cardinals) are well known to all mathematicians with even a rudimentary interest in set theory. However, it is less well known that the combinatorics of singular cardinals is a thriving area with results and problems that do not depend on a solution of the Singular Cardinals Hypothesis. We present here an annotated collection of representative problems with some references. Where the problems are novel, attribution is attempted and it is noted where money is attached to particular problems.

Three closely related themes are represented in these problems: stationary sets and stationary set reflection, variations of square and approachability, and the singular cardinals hypothesis. Underlying many of them are ideas from Shelah's PCF theory. Important subthemes were mutual stationarity, Aronszajn trees, and superatomic Boolean Algebras.

The author notes considerable overlap between this paper and the unpublished report submitted to the Banff Center for the Workshop on Singular Cardinals Combinatorics, May 1–5, 2004.

#### Article information

Source
Notre Dame J. Formal Logic, Volume 46, Number 3 (2005), 309-322.

Dates
First available in Project Euclid: 30 August 2005

https://projecteuclid.org/euclid.ndjfl/1125409329

Digital Object Identifier
doi:10.1305/ndjfl/1125409329

Mathematical Reviews number (MathSciNet)
MR2160660

Zentralblatt MATH identifier
1092.03022

Subjects
Primary: 03E35: Consistency and independence results 03E55: Large cardinals
Secondary: 03E05: Other combinatorial set theory

#### Citation

Foreman, Matthew. Some Problems in Singular Cardinals Combinatorics. Notre Dame J. Formal Logic 46 (2005), no. 3, 309--322. doi:10.1305/ndjfl/1125409329. https://projecteuclid.org/euclid.ndjfl/1125409329

#### References

• [1] Baumgartner, J. E., and S. Shelah, "Remarks on superatomic Boolean algebras", Annals of Pure and Applied Logic, vol. 33 (1987), pp. 109--29.
• [2] Bukovský, L., and E. Copláková-Hartová, "Minimal collapsing extensions of models of ZFC", Annals of Pure and Applied Logic, vol. 46 (1990), pp. 265--98.
• [3] Cummings, J., M. Foreman, and M. Magidor, "Canonical structure in the universe of set theory: Part two", forthcoming in Annals of Pure and Applied Logic.
• [4] Cummings, J., M. Foreman, and M. Magidor, "Squares, scales and stationary reflection", Journal of Mathematical Logic, vol. 1 (2001), pp. 35--98.
• [5] Cummings, J., M. Foreman, and M. Magidor, "Canonical structure in the universe of set theory. I", Annals of Pure and Applied Logic, vol. 129 (2004), pp. 211--43.
• [6] Eisworth, T., "On ideals related to $i[\lambda]$", Notre Dame Journal of Formal Logic, vol. 46 (2005), pp. 301--307.
• [7] Foreman, M., and M. Magidor, "A very weak square principle", The Journal of Symbolic Logic, vol. 62 (1997), pp. 175--96.
• [8] Foreman, M., and M. Magidor, "Mutually stationary sequences of sets and the non-saturation of the non-stationary ideal on $P\sb \varkappa(\lambda)$", Acta Mathematica, vol. 186 (2001), pp. 271--300.
• [9] Galvin, F., and A. Hajnal, "Inequalities for cardinal powers", Annals of Mathematics, vol. 101 (1975), pp. 491--98.
• [10] Gitik, M., "Around Silver's theorem", Notre Dame Journal of Formal Logic, vol. 46 (2005), pp. 323--25.
• [11] Jech, T., "Singular cardinals and the PCF theory", The Bulletin of Symbolic Logic, vol. 1 (1995), pp. 408--24.
• [12] Magidor, M., "On the singular cardinals problem. II", Annals of Mathematics (2), vol. 106 (1977), pp. 517--47.
• [13] Magidor, M., and S. Shelah, "The tree property at successors of singular cardinals", Archive for Mathematical Logic, vol. 35 (1996), pp. 385--404.
• [14] Martínez, J. C., "Some open questions for superatomic Boolean algebras", Notre Dame Journal of Formal Logic, vol. 46 (2005), pp. 353--56.
• [15] Mitchell, W. J., "Adding closed unbounded subsets of $\omega_2$ with finite forcing", Notre Dame Journal of Formal Logic, vol. 46 (2005), pp. 357--71.
• [16] Ruyle, J., Cardinal Sequences of PCF Structures, Ph.D. thesis, University of California, Riverside, 1998.
• [17] Schimmerling, E., "A finite family weak square principle", The Journal of Symbolic Logic, vol. 64 (1999), pp. 1087--110.
• [18] Schimmerling, E., "A question about Suslin trees and the weak square hierarchy", Notre Dame Journal of Formal Logic, vol. 46 (2005), pp. 373--74.
• [19] Shelah, S., Cardinal Arithmetic, vol. 29 of Oxford Logic Guides, The Clarendon Press, New York, 1994.
• [20] Silver, J., "On the singular cardinals problem", pp. 265--68 in Proceedings of the International Congress of Mathematicians (Vancouver BC, 1974), Vol. 1, Canadian Mathematical Congress, Montreal, 1975.
• [21] Solovay, R. M., "Strongly compact cardinals and the GCH", pp. 365--72 in Proceedings of the Tarski Symposium (Symposium on Pure Mathematics, Vol. XXV, University of California, Berkeley, 1971), American Mathematical Society, Providence, 1974.
• [22] Welch, P. D., "Some open problems in mutual stationarity involving inner model theory: A commentary", Notre Dame Journal of Formal Logic, vol. 46 (2005), pp. 375--79.