Bernhard Irrgang
We introduce a method of constructing a forcing along a simplified $(κ,1)$-morass such that the forcing satisfies the $κ$-chain condition. Alternatively, this may be seen as a method to thin out a larger forcing to get a chain condition. As an application, we construct a ccc forcing that adds an $ω_2$-Suslin tree. Related methods are Shelah's historic forcing and Todorcevic's $ρ$-functions.
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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 (0<n<omega+1), 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 refer to them as mangroves and prove that their existence is equi…
This is an overview about a method of constructing ccc forcings: Suppose first that a continuous, commutative system of complete embeddings between countable forcings indexed along $ω_1$ is given. Then its direct limit satisfies ccc by a well-known theorem on finite support iterations. However, this limit has size at most $ω_1$. To get larger forcings, we do not consider linear systems but higher-dimensional systems which are indexed along simplified morasses.
Whenever P is a proper definable forcing for adding a real, the countable support iteration of P has all the preservation properties it can possibly have, within a wide syntactically identified class of properties.
We consider here Easton support iterations of Prikry type forcing notions. New ways of constructing normal ultrafilters in extensions are presented. It turns out that, in contrast with other supports, seemingly unrelated measures or extenders can be involved here.
The preservation theorems for semi-properness, hemi-properness, and pseudo-completeness hold for countable support iterations as well as revised countable support iterations, notwithstanding the fact that the "factor lemma" fails for the countable support versions. Example: The countable support iteration of Namba forcing over a ground model of CH does not add reals. Example: A countable support iteration built from Namba forcing and Cohen forcing, over a ground model of CH, does not collapse omega_1
We prove that any countable support iteration formed with posets with $ω_2$-p.i.c.\ has $ω_2$-c.c., assuming CH in the ground model and assuming also that $ω_1$ is not collapsed. This improves earlier results of Shelah by removing the restriction on the length of the iteration. Thus, we solve the problem of obtaining a large continuum via such forcing iterations.
This is an expository paper about several sophisticated forcing techniques closely related to standard finite support iterations of ccc partial orders. We focus on the four topics of ultrapowers of forcing notions, iterations along templates, Boolean ultrapowers of forcing notions, and restrictions of forcing notions to elementary submodels.
This article continues Rosłanowski and Shelah math.LO/9906024, math.LO/0508272, math.LO/0210205, math.LO/0611131 and math.LO/0605067. We introduce here a new property of <lambda-strategically complete forcing notions which implies that their lambda-support iterations do not collapse lambda^+ (for a strongly inaccessible cardinal lambda).
We show that many countable support iterations of proper forcings preserve Souslin trees. We establish sufficient conditions in terms of games and we draw connections to other preservation properties. We present a proof of preservation properties in countable support interations in the so-called Case A that does not need a division into forcings that add reals and those who do not.
One aim of this work is to get a universe in which weak versions of Martin axioms holds for some forcing notions of cardinality aleph_0, aleph_1 and aleph_2 while on aleph_2 club, the ``small'' brother of diamond, holds. As a consequence we get the consistency of ``there is no Gross space''. Another aim is to present a case of ``Historic iteration''.
We deal with (< kappa)-supported iterated forcing notions which are (E_0,E_1)-complete, have in mind problems on Whitehead groups, uniformizations and the general problem. We deal mainly with the successor of a singular case. This continues [Sh:587]. We also deal with complimentary combinatorial results.
We deal with the problem of preserving various versions of completeness in (< kappa) --support iterations of forcing notions, generalizing the case ``S --complete proper is preserved by CS iterations for a stationary co-stationary S subseteq omega_1''. We give applications to Uniformization and the Whitehead problem. In particular, for a strongly inaccessible cardinal kappa and a stationary set S subseteq kappa with fat complement we can have uniformization for < A_delta : delta in S'>, A_delta subseteq delta = sup A_delta, cf(delta)=otp(A_delta) and a stationary non-reflecting set S' subseteq…
Shelah introduced the revised countable support (RCS) iteration to iterate semiproperness. This was an endpoint in the search for an iteration of a weak condition, still implying that aleph1 is preserved. Dieter Donder found a better manageable approach to this iteration, which is presented here.
We introduce a forcing technique to construct three-dimensional arrays of generic extensions through FS (finite support) iterations of ccc posets, which we refer to as 3D-coherent systems. We use them to produce models of new constellations in Cichoń's diagram, in particular, a model where the diagram can be separated into 7 different values. Furthermore, we show that this constellation of 7 values is consistent with the existence of a $Δ^1_3$ well-order of the reals.
We study the nonstationary-support iteration of Prikry forcings below a measurable cardinal κ, characterizing all the normal measures it carries in the generic extension. We then analyze the restriction of ultrapower embeddings, taken with such a normal measure in the generic extension, to the ground model. We prove that every such restriction is an iterated ultrapwer of the ground model, and provide a sufficient condition for its definability there. This is done without core-model theoretic arguments: the assumption that GCH holds in the ground model up to κsuffices.
We present the classical theory of preservation of $\sqsubset$-unbounded families in generic extensions by ccc posets, where $\sqsubset$ is a definable relation of certain type on spaces of real numbers, typically associated with some classical cardinal invariant. We also prove that, under some conditions, these preservation properties can be preserved in direct limits of an iteration, so applications are extended beyond the context of finite support iterations. Also, we make a breve exposition of Shelah's theory of forcing with an ultrapower of a poset by a measurable cardinal.
Continuing [Fuchino, Ottenbreit and Sakai[9, 10]] and [Fuchino and Ottenbreit[11]], we further study reflection principles in connection with the Löwenheim-Skolem Theorems of stationary logics. In this paper, we mainly analyze the situations in the models obtained by mixed support iteration of a supercompact length and then collapsing another supercompact cardinal to make it $(2^{\aleph_0})^+$. We show, among other things, that the reflection down to $< 2^{\aleph_0}$ of the non-metrizability of topological spaces with small character is independent from the reflection properties studied in [Fu…
Shelah shows that certain revised countable support (RCS) iterations do not add reals. His motivation is to establish the independence (relative to large cardinals) of Avraham's problem on the existence of uncountable non-constuctible sequences all of whose proper initial segments are constructible, Friedman's problem on whether every 2-coloring of $S^2_0=\{α<ω_2\colon\cf(α)=ω\}$ has an uncountable sequentially closed homogeneous subset, and existence of a precipitous normal filter on $ω_2$ with $S^2_0\in{\cal F}$. The posets which Shelah uses in these constructions are Prikry forcing, Namba f…
We use coherent systems of FS iterations on a power set, which can be seen as matrix iteration that allows restriction on arbitrary subsets of the vertical component, to prove general theorems about preservation of certain type of unbounded families on definable structures and of certain mad families (like those added by Hechler's poset for adding an a.d. family) regardless of the cofinality of their size. In particular, we define a class of posets called $σ$-Frechet-linked and show that they work well to preserve mad families, and unbounded families on $ω^ω$. As applications of this method, w…
Property B: A Baumgartner-style Property that Applies to Preservation of $\aleph_1$ and $\aleph_2$ under Iterations with Supports of Size $\aleph_1$
本文研究了在支持大小为$\aleph_1$的迭代中保持$\aleph_1$和$\aleph_2$的性质B，扩展到其他满足$κ^{<κ}=κ$的基数，并引入了修正的完美集forcing。
We deal with an iteration theorem of forcing notion with a kind of countable support of nice enough forcing notion which is proper aleph_2-c.c. forcing notions. We then look at some special cases (Q_D 's preceded by random forcing).
Iteration theorems for subversions of forcing classes
针对Jensen提出的子真和子完全力迫相关类，证明了修正可数支撑迭代下三类子真力迫的可迭代性，还通过宫本良迭代理论去掉子完全/子真性的技术条件，仍保可迭代性等性质并推广类似定理。
Abstract Corrected Iterations
研究 $(<λ)$-支撑迭代可定义力迫概念，证明此类迭代可被修正，从而得出 Judah 和 Shelah 关于 Suslin ccc 力迫有限支撑迭代结果的一个类比。
For $λ$ inaccessible, we may consider $(< λ)$-support iteration of some specific $(<λ)$-complete $λ^+$-c.c. forcing notion. But this fails a "preservation by restricting to a sub-sequence of the forcing, we "correct" the iteration to regain it. This is used in another paper in the consistency of $cov(meagre) < \mathfrak{d}_λ$.
We introduce several properties of forcing notions which imply that their lambda-support iterations are lambda-proper. Our methods and techniques refine those studied in math.LO/9906024, math.LO/0210205, math.LO/0508272 and math.LO/0605067, covering some new forcing notions (though the exact relation of the new properties to the old ones remains undecided).
Symmetric Iterations with Countable and $<κ$-Support: A Framework for Choiceless ZF Extensions
本文提出了一种基于对称迭代的框架，用于扩展选择性ZF，并在不同支持条件下实现了对称构造和DC的保持。
A countable-support symmetric iteration separating PP from AC
本文通过可数支持对称迭代构造一个满足ZF+DC+PP+AC_{wo}+¬AC的对称模型，利用轨道对称化包和局部分裂原理实现PP与AC的分离。
In this paper we isolate the notion of Stratified class forcing and show that Stratification implies cofinality-preservation and is preserved by iterations with the appropriate support. Many familiar class forcings are stratified and therefore can be simultaneously iterated without changing cofinalities, provided the proper support is used. Easton forcing, Backward Easton forcings and some modifications of Jensen coding are stratified. Jensen coding is not stratified but instead obeys a related property, Delta-Stratification, which is also preservedby iteration with an appropriate larger suppo…
Limit Filters and Dependent Choice in Countable-Support Symmetric Iterations
本文研究了可数支持对称迭代中极限滤子的构造及其在ZF和DC中的应用，证明了特定模型满足ZF和DC，并展示了ω₁-完备滤子对控制AC失败程度的重要性。
We investigate the problem of when $\leqλ$--support iterations of $<λ$--complete notions of forcing preserve $λ^+$. We isolate a property -- {\em properness over diamonds} -- that implies $λ^+$ is preserved and show that this property is preserved by $λ$--support iterations. We close with an application of our technology by presenting a consistency result on uniformizing colorings of ladder systems on $\{δ<λ^+:\cf(δ)=λ\}$ that complements a theorem of Shelah.
We show how to force, with finite conditions, the forcing axiom PFA(T), a relativization of PFA to proper forcing notions preserving a given Souslin tree T. The proof uses a Neeman style iteration with generalized side conditions consisting of models of two types, and a preservation theorem for such iterations. The consistency of this axiom was previously known by the standard countable support iteration, using a preservation theorem due to Miyamoto.
Iterating Generalised Perfect Set Forcing Along Well-Founded Orders
提出几何力迫迭代技术的推广版本，证明沿任意良基偏序可迭代广义完美集力迫并保持基数至κ⁺，应用于树状力迫保持ℵ₁。
We develop a version of Cichon's diagram for cardinal invariants on the generalized Cantor space 2^kappa or the generalized Baire space kappa^kappa where kappa is an uncountable regular cardinal. For strongly inaccessible kappa, many of the ZFC-results about the order relationship of the cardinal invariants which hold for omega generalize; for example we obtain a natural generalization of the Bartoszynski-Raisonnier-Stern Theorem. We also prove a number of independence results, both with <kappa-support iterations and kappa-support iterations and products, showing that we consistently have stri…
We introduce more properties of forcing notions which imply that their lambda-support iterations are lambda-proper, where lambda is an inaccessible cardinal. This paper is a direct continuation of section A.2 of math.LO/0210205. As an application of our iteration result we show that it is consistent that dominating numbers associated with two normal filters on lambda are distinct.
I prove preservation theorems for countable support iteration of proper forcing concerning certain classes of capacities and submeasures. New examples of forcing notions and connections with measure theory are included.
Comments on Choiceless Chain Conditions
本文针对无选择公理下的可数链条件，证明若其有限支持迭代仍成立则需可数可数集族选择公理，修正相关定义问题，同时研究Knaster原理与该条件的关系。
A note on iterating strongly $(<λ)$-closed stationary $λ^+$-cc forcing
本文探讨了迭代$(<λ)$-闭 stationary $λ^+$-cc forcing 的定理，讨论其与其它迭代定理及集合论公理的关系，特别是Sh80中的结果。
We introduce a simplified framework for ord-transitive models and Shelah's non elementary proper (nep) theory. We also introduce a new construction for the countable support nep iteration.
There is a proper countable support iteration of length $ω$ adding no new reals at finite stages and adding a Sacks real in the limit.
We provide a general preservation theorem for preserving selective independent families along countable support iterations. The theorem gives a general framework for a number of results in the literature concerning models in which the independence number $\mathfrak{i}$ is strictly below $\mathfrak{c}$, including iterations of Sacks forcing, Miller partition forcing, $h$-perfect tree forcings, coding with perfect trees. Moreover, applying the theorem, we show that $\mathfrak{i} = \aleph_1$ in the Miller Lite model. An important aspect of the preservation theorem is the notion of "Cohen preserva…
This article continues Roslanowski and Shelah math.LO/9906024 and 1105.6049 We introduce here yet another property of (<lambda)-strategically complete forcing notions which implies that their lambda-support iterations do not collapse lambda^+.
We prove two general results about the preservation of extendible and $C^{(n)}$-extendible cardinals under a wide class of forcing iterations (Theorems 5.4 and 7.5). As applications we give new proofs of the preservation of Vopěnka's Principle and $C^{(n)}$-extendible cardinals under Jensen's iteration for forcing the GCH, previously obtained by Brooke-Taylor and Tsaprounis, res\-pectively. We prove that $C^{(n)}$-extendible cardinals are preserved by forcing with standard Easton-support iterations for any possible $Δ_2$-definable behaviour of the power-set function on regular cardinals. We sh…
Using countable support iteration of $S$-proper posets, for some appropriate stationary set $S$, we obtain a generic extension of the constructible universe, in which $\mathfrak{b}=\mathfrak{c}=\aleph_2$ and there is a maximal cofinitary group with a $Π^1_2$-definable set of generators.
The splitting number can be singular. The key method is to construct a forcing poset with finite support matrix iterations of ccc posets introduced by Blass and the second author "Ultrafilters with small generating sets", Israel J. Math., 65, (1989)
Towards a theory of symmetric extensions
本文发展了对称扩展的理论，包括不同类型迭代、商、等价物及结构性结果，并研究了Kinna-Wagner原理在对称扩展中的应用。
The Ordering Principle and Dependent Choice
本文通过有限支持对称系统的迭代，证明了依赖选择公理在ZF基础上结合顺序原理且选择公理失效时的独立性。
Using a finite support iteration of ccc forcings, we construct a model of $\aleph_1<\mathrm{add}(\mathcal{N})<\mathrm{cov}(\mathcal{N})<\mathfrak{b}<\mathrm{non}(\mathcal{M})<\mathrm{cov}(\mathcal{M})=\mathfrak{c}$.
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