Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-22T05:30:32.293Z Has data issue: false hasContentIssue false
  • English
  • Français

Square in Core Models

Published online by Cambridge University Press:  15 January 2014

Ernest Schimmerling  and
Martin Zeman
Ernest Schimmerling
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213-3890, USA. E-mail: [email protected]
Martin Zeman
Affiliation:
Institut Für Formale Logik, Währinger Straße 25, A-1090 Wien, Austria. Mathematical Institute Sav, Štefánikova 49, 814 73 Bratislava, Slovakia. E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that in all Mitchell-Steel core models, □k holds for all k. (See Theorem 2.) From this we obtain new consistency strength lower bounds for the failure of □k if k is either singular and countably closed, weakly compact, or measurable. (Corollaries 5, 8, and 9.) Jensen introduced a large cardinal property that we call subcompactness; it lies between superstrength and supercompactness in the large cardinal hierarchy. We prove that in all Jensen core models, □k holds iff k is not subcompact. (See Theorem 15; the only if direction is essentially due to Jensen.)

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 7 , Issue 3 , September 2001 , pp. 305 - 314
Copyright
Copyright © Association for Symbolic Logic 2001

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

[1] Andretta, A., Neeman, I., and Steel, J. R., The domestic levels of Kc are iterable, to appear in Israel Journal of Mathematics.Google Scholar
[2] Burke, D., Generic embeddings and the failure of box, Proceedings of the American Mathematical Society, vol. 123 (1995), no. 9, pp. 2867–2871.Google Scholar
[3] Cummings, J., Foreman, M., and Magidor, M., Squares, scales and stationary reflection, to appear in Journal of Mathematical Logic.Google Scholar
[4] Cummings, J. and Schimmerling, E., Indexed squares, preprint.Google Scholar
[5] Jensen, R. B., Some remarks on □ below 0 , circulated notes.Google Scholar
[6] Jensen, R. B., The fine structure of the constructible hierarchy, Annals of Mathematical Logic, vol. 4 (1972), pp. 229–308.Google Scholar
[7] Jensen, R. B., A new fine structure for higher core models, circulated notes, 1997.Google Scholar
[8] Jensen, R. B., Corrections and remarks, circulated notes, 1998.Google Scholar
[9] Jensen, R. B. and Zeman, M., Smooth categories and global □, Annals of Pure and Applied Logic, vol. 102 (2000), pp. 101–138.Google Scholar
[10] Martin, D. A. and Steel, J. R., Projective determinacy, Proceedings of the National Academy of Sciences of the United States of America, vol. 85 (1988), no. 18, pp. 6582–6586.Google Scholar
[11] Mitchell, W. J. and Schimmerling, E., Weak covering without countable closure, Mathematical Research Letters, vol. 2 (1995), no. 5, pp. 595–609.Google Scholar
[12] Mitchell, W. J., Schimmerling, E., and STEEL, J.R., The covering lemma up to a Woodin cardinal, Annals of Pure and AppliedLogic, vol. 84 (1997), no. 2, pp. 219–255.CrossRefGoogle Scholar
[13] Mitchell, W. J. and Steel, J.R., Fine structure and iteration trees, Lecture notes in logic, vol. 3, Springer-Verlag, Berlin, 1994.Google Scholar
[14] Neeman, I., Inner models in the region of a Woodin limit of Woodin cardinals, preprint.Google Scholar
[15] Neeman, I. and Steel, J. R., A weak Dodd-Jensen lemma, to appear in The Journal of Symbolic Logic.Google Scholar
[16] Schimmerling, E., Combinatorial principles in the core model for one Woodin cardinal, Annals of Pure and Applied Logic, vol. 74 (1995), no. 2, pp. 153–201.Google Scholar
[17] Schimmerling, E., Combinatorial set theory and inner models, Set theory, techniques and applications, Curacao 1995 and Barcelona 1996 conferences (Dordrecht) (Di Prisco, C. A., Larson, J. A., Bagaria, J., and Mathias, A. R. D., editors), Kluwer Academic Publishers, 1998, pp. 207–212.Google Scholar
[18] Schimmerling, E., Covering properties of core models, Sets and proofs (Cambridge) (Cooper, S. B. and Truss, J. K., editors), London Mathematical Society Lecture Note Series 258, Cambridge University Press, 1999, pp. 281–299.Google Scholar
[19] Schimmerling, E., A finite family weak square principle, The Journal of Symbolic Logic, vol. 64 (1999), no. 3, pp. 1087–1110.Google Scholar
[20] Schimmerling, E. and Steel, J. R., The maximality of the core model, Transactions of the American Mathematical Society, vol. 351 (1999), no. 8, pp. 3119–3141.CrossRefGoogle Scholar
[21] Schimmerling, E. and Woodin, W. H., The Jensen covering property, to appear in The Journal of Symbolic Logic.Google Scholar
[22] Schimmerling, E. and Zeman, M., A characterization of □ k in core models, preprint.Google Scholar
[23] Schindler, R. D., Steel, J. R., and Zeman, M., Deconstructing inner model theory, preprint.Google Scholar
[24] Solovay, R. M., Reinhardt, W. N., and Kanamori, A., Strong axioms of infinity and elementary embeddings, Annals of Mathematical Logic, vol. 13 (1978), no. 1, pp. 73–116.Google Scholar
[25] Steel, J. R., An outline of inner model theory, to appear in the Handbook of set theory.Google Scholar
[26] Steel, J. R., The core model iterability problem, Lecture notes in logic (Berlin), vol. 8, 1996, pp. Springer-Verlag.Google Scholar
[27] Welch, P. D., D. Phil. dissertation, Ph.D. thesis , Oxford University, 1979.Google Scholar
[28] Woodin, W. H., The axiom of determinacy, forcing axioms, and the nonstationary ideal, Logic and its applications, vol. 1, de Gruyter, Berlin, 1999.Google Scholar
[29] Wylie, D. J., Ph. D. dissertation, Ph.D. thesis , Massachusetts Institute of Technology, 1990.Google Scholar
[30] Zeman, M., Ph. D. dissertation, Ph.D. thesis , Humboldt-Universität zu Berlin, 1997.Google Scholar
[31] Zeman, M., Towards □ k in higher core models, circulated notes, 1998.Google Scholar