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K Without the Measurable

Published online by Cambridge University Press:  12 August 2016

Ronald Jensen  and
John Steel
Ronald Jensen
Affiliation:
Humboldt Universität Zu Berlin, Institut für Mathematik, Rudower Chaussee 25, 12489 Berlin, Germany, E-mail: [email protected]
John Steel
Affiliation:
Humboldt Universität Zu Berlin, Institut für Mathematik, Rudower Chaussee 25, 12489 Berlin, Germany, E-mail: [email protected]
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Abstract

We show in ZFC that if there is no proper class inner model with a Woodin cardinal, then there is an absolutely definable core model that is close to V in various ways.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 78 , Issue 3 , September 2013 , pp. 708 - 734
Copyright
Copyright © Association for Symbolic Logic 2013

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References

REFERENCES

[1] Devlin, K. J. and Jensen, R. B., Marginalia to a theorem of Silver, Proceedings of the ISILC logic conference (MÜLler, G. H., Oberschelp, A., and Potthoff, K., editors), Springer Lecture Notes in Mathematics, vol. 499, Springer, Berlin, 1975, pp. 115–142.Google Scholar
[2] Dodd, A. J. and Jensen, R. B., The core model, Annals of Mathematical Logic, vol. 20 (1981), pp. 43–75.Google Scholar
[3] Dodd, A. J. and Jensen, R. B., The covering lemma for K, Annals of Mathematical Logic, vol. 22 (1982), pp. 1–30.Google Scholar
[4] Dodd, A. J. and Jensen, R. B., The covering lemma for L[U], Annals of Mathematical Logic, vol. 22 (1982), pp. 127–135.CrossRefGoogle Scholar
[5] Fuchs, Gunter, λ-structures and s -structures: translating the models, Annals of Pure and Applied Logic, vol. 162 (2011), pp. 257–317.Google Scholar
[6] Gitik, M., Schindler, R., and Shelah, S., Pcf theory and Woodin cardinals, Logic Colloquium '02, Lecture Notes in Logic, vol. 27, Association for Symbolic Logic, Urbana IL, 2006, pp. 172–205.Google Scholar
[7] Jensen, R. B., The fine structure of the constructible hierarchy, Annals of Mathematical Logic, vol. 4 (1972), pp. 229–308.Google Scholar
[8] Jensen, R. B., Robust extenders, handwritten notes, 2003, www.mathematik.hu-berlin.de/~raesch/org/jensen.html. Google Scholar
[9] Jensen, R. B., A new fine structure, handwritten notes, www.mathematik.hu-berlin.de/~raesch/org/jensen.html. Google Scholar
[10] Jensen, R. B., Schimmerling, E., Schindler, R. D., and Steel, J. R., Stacking mice, this Journal, vol. 74 (2009), pp. 315–335.Google Scholar
[11] Mitchell, W. J., The core model for sequences of measures II, unpublished preliminary draft, 1982.Google Scholar
[12] Mitchell, W. J., The core model for sequences of measures I, Mathematical Proceedings of the Cambridge Philosophical Society, vol. 95 (1984), pp. 229–260.CrossRefGoogle Scholar
[13] Mitchell, W. J. and Schimmerling, E., Weak covering without countable closure, Mathematical Research Letters, vol. 2 (1995), no. 5, pp. 595–609.CrossRefGoogle Scholar
[14] Mitchell, W. J., Schimmerling, E., and Steel, J. R., The covering lemma up to a Woodin cardinal, Annals of Pure and Applied Logic, vol. 84 (1997), pp. 219–255.CrossRefGoogle Scholar
[15] Mitchell, W. J. and Schindler, R. D., A universal extender model without large cardinals in V, this Journal, vol. 69 (2004), pp. 371–386.Google Scholar
[16] Mitchell, W. J. and Steel, J. R., Fine structure and iteration trees, Lecture Notes in Logic, vol. 3, Springer-Verlag, 1994.Google Scholar
[17] 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.Google Scholar
[18] Schimmerling, E. and Zeman, M., Square in core models, The Bulletin of Symbolic Logic, vol. 7 (2001), pp. 305–314.Google Scholar
[19] Schimmerling, E. and Zeman, M., Characterization of ☕K in core models, Journal of Mathematical Logic, vol. 4 (2004), pp. 1–72.Google Scholar
[20] Schindler, R. D., The core model for almost linear iterations, Annals of Pure and Applied Logic, vol. 116 (2002), pp. 207–274.Google Scholar
[21] Schlutzenberg, F., Measures in mice, Ph.D. thesis, University of California at Berkeley, 2007.Google Scholar
[22] Steel, J. R., The core model iterability problem, Lecture Notes in Logic, vol. 8, Springer-Verlag, 1996.Google Scholar
[23] Steel, J. R., PFA implies AD in L(ℝ), this Journal, vol. 70 (2005), no. 4, pp. 1255–1296.Google Scholar
[24] Steel, J. R., Derived models associated to mice, Computational prospects of infinity (Chong, C. T., Feng, Q., Slaman, T. A., and Woodin, W. H., editors), World Scientific, 2008, pp. 105–193.CrossRefGoogle Scholar
[25] Steel, J. R., An outline of inner model theory, Handbook of set theory (Foreman, M. and Kanamori, A., editors), vol. 3, Springer, 2010, pp. 1595–1684.CrossRefGoogle Scholar