David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Ezio Di Nucci
Jack Alan Reynolds
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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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Gunter Fuchs (2010). Generic Embeddings Associated to an Indestructibly Weakly Compact Cardinal. Annals of Pure and Applied Logic 162 (1):89-105.
Christoph Weiß (2012). The Combinatorial Essence of Supercompactness. Annals of Pure and Applied Logic 163 (11):1710-1717.
Trevor M. Wilson (2015). The Envelope of a Pointclass Under a Local Determinacy Hypothesis. Annals of Pure and Applied Logic 166 (10):991-1018.
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