Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-22T20:43:42.727Z Has data issue: false hasContentIssue false
  • English
  • Français

GAMES AND RAMSEY-LIKE CARDINALS

Published online by Cambridge University Press:  30 January 2019

DAN SAATTRUP NIELSEN  and
PHILIP WELCH
DAN SAATTRUP NIELSEN
Affiliation:
SCHOOL OF MATHEMATICSUNIVERSITY OF BRISTOL TYNDALL AVE BRISTOL BS8 1TH, UKE-mail: [email protected]
PHILIP WELCH
Affiliation:
SCHOOL OF MATHEMATICSUNIVERSITY OF BRISTOL TYNDALL AVE BRISTOL BS8 1TH, UKE-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We generalise the α-Ramsey cardinals introduced in Holy and Schlicht (2018) for cardinals α to arbitrary ordinals α, and answer several questions posed in that paper. In particular, we show that α-Ramseys are downwards absolute to the core model K for all α of uncountable cofinality, that strategic ω-Ramsey cardinals are equiconsistent with remarkable cardinals and that strategic α-Ramsey cardinals are equiconsistent with measurable cardinals for all α > ω. We also show that the n-Ramseys satisfy indescribability properties and use them to provide a game-theoretic characterisation of completely ineffable cardinals, as well as establishing further connections between the α-Ramsey cardinals and the Ramsey-like cardinals introduced in Gitman (2011), Feng (1990), and Sharpe and Welch (2011).

Keywords

03E3503E4503E55Ramsey-like cardinalslarge cardinalsgamesweakly compact cardinalsineffable cardinalscompletely ineffable cardinalsremarkable cardinalsvirtually measurable cardinalsmeasurable cardinalscore model
Type
Articles
Information
The Journal of Symbolic Logic , Volume 84 , Issue 1 , March 2019 , pp. 408 - 437
Copyright
Copyright © The Association for Symbolic Logic 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Abramson, F. G., Harrington, L. A., Kleinberg, E. M., and Zwicker, W. S., Flipping properties: A unifying thread in the theory of large cardinals. Annals of Mathematical Logic, vol. 12 (1977), pp. 25–58.10.1016/0003-4843(77)90005-5CrossRefGoogle Scholar
Dodd, A. J., The Core Model, London Mathematical Society Lecture Note Series, vol. 61, Cambridge University Press, New York, 1982.10.1017/CBO9780511600586CrossRefGoogle Scholar
Donder, D., Jensen, R. B., and Koppelberg, B. J., Some applications of the core model, Set Theory and Model Theory (Jensen, R. and Prestel, A., editors), Springer, Berlin, 1981, pp. 55–97.10.1007/BFb0098619CrossRefGoogle Scholar
Feng, Q., A hierarchy of Ramsey cardinals. Annals of Pure and Applied Logic, vol. 49 (1990), no. 2, pp. 257–277.10.1016/0168-0072(90)90028-ZCrossRefGoogle Scholar
Gitman, V., Ramsey-like cardinals, this Journal, vol. 76 (2011), no. 2, pp. 519–540.Google Scholar
Gitman, V. and Schindler, R., Virtual large cardinals. Annals of Pure and Applied Logic, vol. 169 (2018), no. 12, pp. 1317–1334.10.1016/j.apal.2018.08.005CrossRefGoogle Scholar
Gitman, V. and Welch, P., Ramsey-like cardinals II, this Journal, vol. 76 (2011), no. 2, pp. 541–560.Google Scholar
Holy, P. and Schlicht, P., A hierarchy of Ramsey-like cardinals. Fundamenta Mathematicae, vol. 242 (2018), pp. 49–74.10.4064/fm396-9-2017CrossRefGoogle Scholar
Jensen, R. and Steel, J., K without the measurable, this Journal, vol. 78 (2013), no. 3, pp. 708–734.Google Scholar
Kanamori, A., The Higher Infinite: Large Cardinals in Set Theory from their Beginnings, second edition, Springer, Berlin, 2009.Google Scholar
Kanamori, A. and Magidor, M., The evolution of large cardinal axioms in set theory, Higher Set Theory (Müller, G. H. and Scott, D. S., editors), Springer, Berlin, 1978, pp. 99–275.10.1007/BFb0103104CrossRefGoogle Scholar
Kellner, J. and Shelah, S., More on the pressing down game. Archive for Mathematical Logic, vol. 50 (2011), no. 3–4, pp. 477–501.10.1007/s00153-011-0227-xCrossRefGoogle Scholar
Mitchell, W. J., Ramsey cardinals and constructibility, this Journal, vol. 44 (1979), no. 2, pp. 260–266.Google Scholar
Schindler, R., Weak covering at large cardinals. Mathematical Logic Quarterly, vol. 43 (1997), pp. 22–28.10.1002/malq.19970430103CrossRefGoogle Scholar
Schindler, R., Iterates of the core model, this Journal, vol. 71 (2006), no. 1, pp. 241–251.Google Scholar
Schindler, R.-D., Proper forcing and remarkable cardinals. The Bulletin of SymbolicLogic, vol. 6 (2000), no. 2, pp. 176–184.Google Scholar
Sharpe, I. and Welch, P. D., Greatly Erdös cardinals with some generalizations to the Chang and Ramsey properties. Annals of Pure and Applied Logic, vol. 162 (2011), pp. 863–902.10.1016/j.apal.2011.04.002CrossRefGoogle Scholar
Zeman, M., Inner Models and Large Cardinals, Series in Logic and its Applications, Vol. 5, de Gruyter, Berlin, New York, 2002.10.1515/9783110857818CrossRefGoogle Scholar