Proper Forcing and L$$

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Journal of Symbolic Logic 66 (2):801-810 (2001)
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Abstract

We present two ways in which the model L is canonical assuming the existence of large cardinals. We show that the theory of this model, with ordinal parameters, cannot be changed by small forcing; we show further that a set of ordinals in V cannot be added to L by small forcing. The large cardinal needed corresponds to the consistency strength of AD$^L$; roughly $\omega$ Woodin cardinals.

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Generalizations of the Kunen inconsistency.Joel David Hamkins, Greg Kirmayer & Norman Lewis Perlmutter - 2012 - Annals of Pure and Applied Logic 163 (12):1872-1890.
Thin equivalence relations and inner models.Philipp Schlicht - 2014 - Annals of Pure and Applied Logic 165 (10):1577-1625.

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