Development of new iteration techniques for forcing

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We develop new iterated forcing techniques and investigate the interplay between forcing theory and set theory of the reals, cardinal invariants of the continuum as well as descriptive set theory.(1)Shattered iterations.We further develop the technique of shattered iteration which is due to the present researcher. In particular, we prove the simultaneous consistency of cov(Μ)=b=N_2 and non(Μ)=N_3 by replacing Cohen reals by Hechler reals in the iteration framework.(2)Mixed support constructions.We show that in the model obtained by adding Cohen reals with mixed support over a model satisfying ◇,〓CH,〓_s for all stationary S⊆ω_1, and MA for countable forcing notions simultaneously hold, thus answering a question of Fuchino, Shelah, and Soukup.(3)Cardinal invariants related to the rationals.For any cardinal invariant 〓 of P(ω)/fin, let 〓_Q denote the corresponding invariant of Dense(Q) / nwd(Q). We answer questions of Balcar, Hernandez, and Hrusak by proving s_Q【less than or equal】min{add(Μ),s}, as well as the consistency of h_Q < s_Q and of h < h_Q.(4)Cardinal invariants related to partitions of ω.Let (ω) stand for the partitions of natural numbers ordered by almost refinement, and denote by 〓_c the cardinal invariant of (ω) corresponding to the cardinal invariant 〓. We show that 〓= 〓_c for 〓 = p,h,s,τ, that a_c = a_s, and that t_c = p.(5)Forcing indestructibility of mad families.Extending work of Hrusak and Kurilic, we provide a combinatorial characterization of P-indestructibility of mad families for several classical forcing notions P and, assuming a weak fragment of M A, we construct mad families which are P-indestructible yet Q-destructible for several pairs of forcing notions (P, Q).(6)Silver measurability.We investigate the doughnut property, a notion of measurability related to Silver forcing. In particular, we characterize the Δ^1_2 doughnut property and the Σ^1_2 doughnut property as transcendence statements over the constructible universe L..