Development of new iteration techniques for forcing

我们开发了新的迭代强迫技术，并调查强迫理论和集合论的实数，基数不变量的连续体以及描述集理论之间的相互作用。（1）破碎迭代：我们进一步发展了本研究者提出的破碎迭代技术。特别地，在迭代框架下，用Hechler实数代替Cohen实数，证明了cov（M）=B= N2和non（M）= N3的同时相容性.（2）混合支撑结构：我们证明了在满足条件的模型上添加混合支撑的Cohen实所得到的模型中，对于所有平稳的S <$ω_1和可数强迫概念的MA同时成立，从而回答了Fuchino，Shelah和Soukup的一个问题。（3）与有理数有关的基数不变量：对于P（ω）/fin的任何基数不变量，设φ_Q表示Dense（Q）/ nwd（Q）的相应不变量。我们通过证明s_Q[小于或等于]min{add（M），s}以及h_Q < s_Q和h < h_Q的一致性来回答Balcar，埃尔南德斯和Hrusak的问题。（4）与ω的分拆有关的基数不变量：设（ω）表示自然数的几乎加细分拆，用ω_c表示（ω）的基数不变量。我们证明，对于= p、h、s、τ，=_c，a_c = a_s，t_c = p。（5）mad族的强迫不可破坏性推广了Hrusak和Kurilic的工作，我们给出了几个经典强迫概念P的mad族的P-不可破坏性的组合刻画，并假设MA的一个弱片段，我们构造了几对强迫概念（P，Q）的P-不可破坏而Q-可破坏的mad族。（6）银的可测性：我们研究了与银强迫有关的可测性概念--甜甜圈性质。特别地，我们将Δ^1_2甜甜圈性质和Δ ^1_2甜甜圈性质刻画为可构造论域L上的超越陈述。

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..