Strong unfoldability, shrewdness and combinatorial consequences

Lücke, Philipp

doi:10.1090/proc/15943

Strong unfoldability, shrewdness and combinatorial consequences
HTML articles powered by AMS MathViewer

by Philipp Lücke

Proc. Amer. Math. Soc. 150 (2022), 4005-4020

DOI: https://doi.org/10.1090/proc/15943

Published electronically: April 7, 2022

PDF | Request permission

Abstract:

We show that the notions of strongly unfoldable cardinals, introduced by Villaveces [J. Symbolic Logic 63 (1998), pp. 1116–1136] in his model-theoretic studies of models of set theory, and shrewd cardinals, introduced by Rathjen [Bull. Symbolic Logic 1 (1995), pp. 468–485] in a proof-theoretic context, coincide. We then proceed by using ideas from the proof of this equivalence to establish the existence of ordinal anticipating Laver functions for strong unfoldability. With the help of these functions, we show that the principle $\Diamond _\kappa (\mathrm {Reg})$ holds at every strongly unfoldable cardinal $\kappa$ with the property that there exists a subset $z$ of $\kappa$ such that every subset of $\kappa$ is ordinal definable from $z$. While a result of Džamonja and Hamkins [Ann. Pure Appl. Logic 144 (2006), pp. 83–95] shows that $\Diamond _\kappa (\mathrm {Reg})$ can consistently fail at a strongly unfoldable cardinal $\kappa$, this implication can be used to prove that various canonical extensions of the axioms of $\mathrm {ZFC}$ are either compatible with the assumption that $\Diamond _\kappa (\mathrm {Reg})$ holds at every strongly unfoldable cardinal $\kappa$ or outright imply this statement. Finally, we will also use our methods to contribute to the study of strong chain conditions of partials orders and their productivity.

References

Brent Cody, Characterizations of the weakly compact ideal on $P_\kappa \lambda$, Ann. Pure Appl. Logic 171 (2020), no. 6, 102791, 23. MR 4082998, DOI 10.1016/j.apal.2020.102791
Brent Cody, Moti Gitik, Joel David Hamkins, and Jason A. Schanker, The least weakly compact cardinal can be unfoldable, weakly measurable and nearly $\theta$-supercompact, Arch. Math. Logic 54 (2015), no. 5-6, 491–510. MR 3372604, DOI 10.1007/s00153-015-0423-1
Sean Cox, Layered posets and Kunen’s universal collapse, Notre Dame J. Form. Log. 60 (2019), no. 1, 27–60. MR 3911105, DOI 10.1215/00294527-2018-0022
Sean Cox and Philipp Lücke, Characterizing large cardinals in terms of layered posets, Ann. Pure Appl. Logic 168 (2017), no. 5, 1112–1131. MR 3620068, DOI 10.1016/j.apal.2016.11.008
Keith J. Devlin, Constructibility, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1984. MR 750828, DOI 10.1007/978-3-662-21723-8
Frank R. Drake, Set theory—an introduction to large cardinals, Studies in Logic and the Foundations of Mathematics, vol. 76, North-Holland Publishing Co., Amsterdam, 1974. MR 3408725
Mirna D amonja and Joel David Hamkins, Diamond (on the regulars) can fail at any strongly unfoldable cardinal, Ann. Pure Appl. Logic 144 (2006), no. 1-3, 83–95. MR 2279655, DOI 10.1016/j.apal.2006.05.001
Joel David Hamkins, A class of strong diamond principles, arXiv:math/0211419, 2002.
Joel David Hamkins and Thomas A. Johnstone, The proper and semi-proper forcing axioms for forcing notions that preserve $\aleph _2$ or $\aleph _3$, Proc. Amer. Math. Soc. 137 (2009), no. 5, 1823–1833. MR 2470843, DOI 10.1090/S0002-9939-08-09727-X
Kai Hauser, Indescribable cardinals and elementary embeddings, J. Symbolic Logic 56 (1991), no. 2, 439–457. MR 1133077, DOI 10.2307/2274692
Kai Hauser, Indescribable cardinals without diamonds, Arch. Math. Logic 31 (1992), no. 5, 373–383. MR 1164732, DOI 10.1007/BF01627508
Thomas Jech, Set theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. The third millennium edition, revised and expanded. MR 1940513
Thomas Jech, Stationary sets, Handbook of set theory. Vols. 1, 2, 3, Springer, Dordrecht, 2010, pp. 93–128. MR 2768680, DOI 10.1007/978-1-4020-5764-9_{2}
Ronald B. Jensen and Kenneth Kunen. Some combinatorial properties of $L$ and $V$. Handwritten notes, 1969.
Ronald Jensen, Ernest Schimmerling, Ralf Schindler, and John Steel, Stacking mice, J. Symbolic Logic 74 (2009), no. 1, 315–335. MR 2499432, DOI 10.2178/jsl/1231082314
Thomas A. Johnstone, Strongly unfoldable cardinals made indestructible, J. Symbolic Logic 73 (2008), no. 4, 1215–1248. MR 2467213, DOI 10.2178/jsl/1230396915
Akihiro Kanamori, The higher infinite, 2nd ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. Large cardinals in set theory from their beginnings. MR 1994835
Richard Laver, Making the supercompactness of $\kappa$ indestructible under $\kappa$-directed closed forcing, Israel J. Math. 29 (1978), no. 4, 385–388. MR 472529, DOI 10.1007/BF02761175
Azriel Lévy, A hierarchy of formulas in set theory, Mem. Amer. Math. Soc. 57 (1965), 76. MR 189983
Azriel Lévy, The sizes of the indescribable cardinals, Axiomatic Set Theory (Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, Calif., 1967) Amer. Math. Soc., Providence, R.I., 1971, pp. 205–218. MR 0281606
Philipp Lücke, Structural reflection, shrewd cardinals and the size of the continuum, J. Math. Logic (2022), DOI 10.1142/S0219061322500076.
M. Magidor, On the role of supercompact and extendible cardinals in logic, Israel J. Math. 10 (1971), 147–157. MR 295904, DOI 10.1007/BF02771565
Tadatoshi Miyamoto, A note on weak segments of PFA, Proceedings of the Sixth Asian Logic Conference (Beijing, 1996) World Sci. Publ., River Edge, NJ, 1998, pp. 175–197. MR 1789737
Tadatoshi Miyamoto, A note on fast function. Unpublished notes, 2001.
Michael Rathjen, Recent advances in ordinal analysis: $\Pi ^1_2$-$\textrm {CA}$ and related systems, Bull. Symbolic Logic 1 (1995), no. 4, 468–485. MR 1369172, DOI 10.2307/421132
Michael Rathjen, An ordinal analysis of parameter free $\Pi ^1_2$-comprehension, Arch. Math. Logic 44 (2005), no. 3, 263–362. MR 2140614, DOI 10.1007/s00153-004-0232-4
Assaf Rinot, Chain conditions of products, and weakly compact cardinals, Bull. Symb. Log. 20 (2014), no. 3, 293–314. MR 3271280, DOI 10.1017/bsl.2014.24
Jason Aaron Schanker, Partial near supercompactness, Ann. Pure Appl. Logic 164 (2013), no. 2, 67–85. MR 2989393, DOI 10.1016/j.apal.2012.08.001
Stevo Todorcevic, Walks on ordinals and their characteristics, Progress in Mathematics, vol. 263, Birkhäuser Verlag, Basel, 2007. MR 2355670, DOI 10.1007/978-3-7643-8529-3
Andrés Villaveces, Chains of end elementary extensions of models of set theory, J. Symbolic Logic 63 (1998), no. 3, 1116–1136. MR 1649079, DOI 10.2307/2586730