Journal of Symbolic Logic 74 (1):315-335 (2009)
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 k ≥ N₃ such that □k and □(k) fail. 2) There is a cardinal k such that k is weakly compact in the generic extension by Col(k, k⁺). Of special interest is 1) with k = N₃ 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 $K^c ||k.$
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References found in this work BETA
Combinatorial Principles in the Core Model for One Woodin Cardinal.Ernest Schimmerling - 1995 - Annals of Pure and Applied Logic 74 (2):153-201.
Characterization of □Κin Core Models.Ernest Schimmerling & Martin Zeman - 2004 - Journal of Mathematical Logic 4 (01):1-72.
Citations of this work BETA
The Combinatorial Essence of Supercompactness.Christoph Weiß - 2012 - Annals of Pure and Applied Logic 163 (11):1710-1717.
Generic Embeddings Associated to an Indestructibly Weakly Compact Cardinal.Gunter Fuchs - 2010 - Annals of Pure and Applied Logic 162 (1):89-105.
The Envelope of a Pointclass Under a Local Determinacy Hypothesis.Trevor M. Wilson - 2015 - Annals of Pure and Applied Logic 166 (10):991-1018.
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