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  1.  29
    Large Cardinals and Definable Well-Orders on the Universe.Andrew D. Brooke-Taylor - 2009 - Journal of Symbolic Logic 74 (2):641-654.
    We use a reverse Easton forcing iteration to obtain a universe with a definable well-order, while preserving the GCH and proper classes of a variety of very large cardinals. This is achieved by coding using the principle ◊ $_{k^ - }^* $ at a proper class of cardinals k. By choosing the cardinals at which coding occurs sufficiently sparsely, we are able to lift the embeddings witnessing the large cardinal properties without having to meet any non-trivial master conditions.
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  2.  17
    Indestructibility of Vopěnka's Principle.Andrew D. Brooke-Taylor - 2011 - Archive for Mathematical Logic 50 (5-6):515-529.
    Vopěnka’s Principle is a natural large cardinal axiom that has recently found applications in category theory and algebraic topology. We show that Vopěnka’s Principle and Vopěnka cardinals are relatively consistent with a broad range of other principles known to be independent of standard (ZFC) set theory, such as the Generalised Continuum Hypothesis, and the existence of a definable well-order on the universe of all sets. We achieve this by showing that they are indestructible under a broad class of forcing constructions, (...)
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    Large Cardinals and Gap-1 Morasses.Andrew D. Brooke-Taylor & Sy-David Friedman - 2009 - Annals of Pure and Applied Logic 159 (1-2):71-99.
    We present a new partial order for directly forcing morasses to exist that enjoys a significant homogeneity property. We then use this forcing in a reverse Easton iteration to obtain an extension universe with morasses at every regular uncountable cardinal, while preserving all n-superstrong , hyperstrong and 1-extendible cardinals. In the latter case, a preliminary forcing to make the GCH hold is required. Our forcing yields morasses that satisfy an extra property related to the homogeneity of the partial order; we (...)
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