## 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

## Bibliographic Information

**Philipp Lücke**- Affiliation: Institut de Matemàtica, Universitat de Barcelona, Gran via de les Corts Catalanes 585, 08007 Barcelona, Spain
- ORCID: 0000-0001-8746-5887
- Email: philipp.luecke@ub.edu
- Received by editor(s): July 27, 2021
- Received by editor(s) in revised form: December 6, 2021
- Published electronically: April 7, 2022
- Additional Notes: This project had received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 842082 (Project
*SAIFIA: Strong Axioms of Infinity – Frameworks, Interactions and Applications*). - Communicated by: Vera Fischer
- © Copyright 2022 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**150**(2022), 4005-4020 - MSC (2020): Primary 03E55, 03E05, 03E45
- DOI: https://doi.org/10.1090/proc/15943
- MathSciNet review: 4446247