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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

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