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  1. Combinatorial properties of classical forcing notions.Jörg Brendle - 1995 - Annals of Pure and Applied Logic 73 (2):143-170.
    We investigate the effect of adding a single real on cardinal invariants associated with the continuum. We show:1. adding an eventually different or a localization real adjoins a Luzin set of size continuum and a mad family of size ω1;2. Laver and Mathias forcing collapse the dominating number to ω1, and thus two Laver or Mathias reals added iteratively always force CH;3. Miller's rational perfect set forcing preserves the axiom MA.
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  • Projective absoluteness for Sacks forcing.Daisuke Ikegami - 2009 - Archive for Mathematical Logic 48 (7):679-690.
    We show that ${{\bf \Sigma}^1_3}$ -absoluteness for Sacks forcing is equivalent to the non-existence of a ${{\bf \Delta}^1_2}$ Bernstein set. We also show that Sacks forcing is the weakest forcing notion among all of the preorders that add a new real with respect to ${{\bf \Sigma}^1_3}$ forcing absoluteness.
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  • Forcing absoluteness and regularity properties.Daisuke Ikegami - 2010 - Annals of Pure and Applied Logic 161 (7):879-894.
    For a large natural class of forcing notions, we prove general equivalence theorems between forcing absoluteness statements, regularity properties, and transcendence properties over and the core model . We use our results to answer open questions from set theory of the reals.
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