Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-05T11:29:19.020Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected]
Ernest Schimmerling
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pa 15213, USA, E-mail: [email protected]
Ralf Schindler
Affiliation:
Institut für Mathematische Logik und Grundlagenforschung, Universität Münster, Einsteinstr. 62, 48149 Münster, Germany, E-mail: [email protected]
John Steel
Affiliation:
Department of Mathematics, 717 Evans Hall, University of California, Berkeley Ca 94720, USA, E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

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

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., 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