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Stacking mice

Published online by Cambridge University Press:  12 March 2014

Ronald Jensen ,
Ernest Schimmerling ,
Ralf Schindler  and
John Steel
Ronald Jensen
Affiliation:
Institut für Mathematik, Humboldt-Universität Zu Berlin, Rudower Chausee 25, 12489 Berlin, Germany, E-mail: jensen@math.hu-berlin.de
Ernest Schimmerling
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pa 15213, USA, E-mail: eschimme@andrew.cmu.edu
Ralf Schindler
Affiliation:
Institut für Mathematische Logik und Grundlagenforschung, Universität Münster, Einsteinstr. 62, 48149 Münster, Germany, E-mail: rds@uni-muenster.de
John Steel
Affiliation:
Department of Mathematics, 717 Evans Hall, University of California, Berkeley Ca 94720, USA, E-mail: steel@math.berkeley.edu
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Abstract

We show that either of the following hypotheses imply that there is an inner model with a proper class of strong cardinals and a proper class of Woodin cardinals. 1) There is a countably closed cardinal κ ≥ ℵ such that □κ and □(κ) fail. 2) There is a cardinal κ such that κ is weakly compact in the generic extension by Col(κ, κ+). Of special interest is 1) with κ = ℵ3 since it follows from PFA by theorems of Todorcevic and Velickovic. Our main new technical result, which is due to the first author, is a weak covering theorem for the model obtained by stacking mice over Kc∥κ.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 74 , Issue 1 , March 2009 , pp. 315 - 335
Copyright
Copyright © Association for Symbolic Logic 2009

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References

REFERENCES

[1]Andretta, A., Neeman, I., and Steel, J., The domestic levels of Kc are iterable, Israel Journal of Mathematics, vol. 125 (2001), pp. 157–201.CrossRefGoogle Scholar
[2]Bekkali, M., Topics in set theory. Lebesgue measurability, large cardinals, forcing axioms, rho-functions, Lecture Notes in Mathematics, vol. 1476, Springer-Verlag, Berlin, 1991, notes on lectures by Stevo Todorcevic.Google Scholar
[3]Fuchs, G., Neeman, I., and Schindler, R., A criterion for coarse iterability, Archive for Mathematical Logic, submitted.Google Scholar
[4]Jech, T., Set theory, third ed., Springer-Verlag, 2002.Google Scholar
[5]Jensen, R., A new fine structure, handwritten notes, 1997, www.mathematik.hu-berlin.de/~raesch/org/jensen.html.Google Scholar
[6]Jensen, R., Robust extenders, handwritten notes, 2003, www.mathematik.hu-berlin.de/~raesch/org/jensen.html.Google Scholar
[7]Jensen, R. and Steel, J., in preparation.Google Scholar
[8]Mitchell, W. and Schindler, R., A universal extender model without large cardinals in V, this Journal, vol. 69 (2004), pp. 371–386.Google Scholar
[9]Mitchell, W. and Steel, J., Fine structure and iteration trees, Lecture Notes in Logic, vol. 3, Springer-Verlag, 1994.CrossRefGoogle Scholar
[10]Schimmerling, E., A core model toolbox and guide, Handbook of set theory (Foreman, , Kanamori, , and Magidor, , editors), Springer-Verlag, to appear.Google Scholar
[11]Schimmerling, E., Combinatorial principles in the core model for one Woodin cardinal, Annals of Pure and Applied Logic, vol. 74 (1995), pp. 153–201.CrossRefGoogle Scholar
[12]Schimmerling, E., Coherent sequences and threads, Advances in Mathematics, vol. 216 (2007), pp. 89–117.CrossRefGoogle Scholar
[13]Schimmerling, E. and Steel, J., The maximality of the core model, Transactions of the American Mathematical Society, vol. 351 (1999), pp. 3119–3141.CrossRefGoogle Scholar
[14]Schimmerling, E. and Zeman, M., Characterization of □k in core models, Journal of Mathematical Logic, vol. 4 (2004), pp. 1–72.CrossRefGoogle Scholar
[15]Schindler, R. and Steel, J., The core model induction, book in preparation.Google Scholar
[16]Schindler, R. and Zeman, M., Fine structure theory, Handbook of set theory (Foreman, , Kanamori, , and Magidor, , editors), Springer-Verlag, to appear.Google Scholar
[17]Steel, J., The derived model theorem, preprint, math.berkeley.edu/~steel/papers/Publications.html.Google Scholar
[18]Steel, J., An outline of inner model theory, Handbook of set theory (Foreman, , Kanamori, , and Magidor, , editors), Springer-Verlag, to appear.Google Scholar
[19]Steel, J., The core model iterability problem, Lecture Notes in Logic, vol. 8, Springer-Verlag, 1996.CrossRefGoogle Scholar
[20]Steel, J., PFA implies ADL(ℝ), this Journal, vol. 70 (2005), pp. 1255–1296.Google Scholar
[21]Todorcevic, S., A note on the Proper Forcing Axiom, Contemporary Mathematics, vol. 95 (1984), pp. 209–218.CrossRefGoogle Scholar
[22]Velickovic, B., Forcing axioms and stationary sets, Advances in Mathematics, vol. 94 (1992), pp. 256–284.CrossRefGoogle Scholar
[23]Woodin, H., unpublished.Google Scholar
[24]Zeman, M., Inner models and large cardinals, de Gruyter, Berlin, New York, 2002.CrossRefGoogle Scholar