Development of new iteration techniques for forcing

我们开发了新的迭代强迫技术，并研究了强迫理论与实数集理论、连续统的基本不变量以及描述集理论之间的相互作用。(1)破碎的迭代。我们进一步发展了破碎迭代技术，这是由于目前的研究人员。特别地，我们在迭代框架中用Hechler实数代替Cohen实数，证明了cov（Μ）=b=N_2和non（Μ）=N_3的同时一致性。(2)混合支护结构。我们表明,在模型中通过添加科恩实数与混合支持模型满足◇〓CH,〓_所有静止的年代⊆ω_1,和马可数强迫观念同时举行,因此Fuchino回答问题,沙拉,Soukup。(3)与有理数相关的基数不变量。对于P(ω)/fin的任意基数不变量〓，设〓_Q表示Dense(Q) / nwd(Q)的对应不变量。我们通过证明s_Q【小于或等于】min{add（Μ），s}，以及h_Q < s_Q和h < h_Q的一致性，回答了Balcar、Hernandez和Hrusak的问题。(4)与ω分区相关的基数不变量。设（ω）表示按几乎精化排序的自然数分割，用〓_c表示（ω）的基数不变量对应于基数不变量〓。我们证明了〓= =〓_c对于〓= p,h,s,τ, a_c = a_s, t_c = p。(5)强迫疯狂家庭的不可毁灭性。推广Hrusak和Kurilic的工作，给出了几个经典强迫概念P的疯狂族的P-不可摧毁性的组合表征，并假设M a的一个弱片段，构造了几个强迫概念（P， Q）对P-不可摧毁但Q-可摧毁的疯狂族。我们研究了甜甜圈性质，一个与银强迫相关的可测量性概念。特别地，我们将Δ^1_2 doughnut性质和Σ^1_2 doughnut性质描述为可构造宇宙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..