Andrzej Roslanowski, Saharon Shelah
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.
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In the present paper we are interested in simple forcing notions and Forcing Axioms. A starting point for our investigations was the article [JR1] in which several problems were posed. We answer some of those problems here.
We look for a parallel to the notion of ``proper forcing'' among lambda-complete forcing notions not collapsing lambda^+ . We suggest such a definition and prove that it is preserved by suitable iterations.
An $\aleph_1$-Souslin tree is a complicated combinatorial object whose existence cannot be decided on the grounds of ZFC alone. But 15 years after Tennenbaum and independently Jech devised notions of forcing for introducing such a tree, Shelah proved that already the simplest forcing notion --- Cohen forcing --- adds an $\aleph_1$-Souslin tree. In this paper, we identify a rather large class of notions of forcing that, assuming a GCH-type assumption, add a $λ^+$-Souslin tree. This class includes Prikry, Magidor and Radin forcing.
The main result of this paper is a partial answer to [math.LO/9909115, Problem 5.5]: a finite iteration of Universal Meager forcing notions adds generic filters for many forcing notions determined by universality parameters. We also give some results concerning cardinal characteristics of the sigma-ideals determined by those universality parameters.
The present paper has three themes. First, we continue the investigations started in Judah, Roslanowski and Shelah \math.LO/9310224 and Roslanowski and Shelah math.LO/9807172, math.LO/9703222, and we investigate the method of norms on possibilities in the context of ccc forcing notions, getting a number of constructions of nicely definable ccc forcings. The second theme of the paper is a part of the general program ``how special are random and Cohen forcing notions (or: the respective ideals)''. Shelah math.LO/9303208 shows that the two forcing notions may occupy special positions in the realm…
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.
We prove that if Q is a nw-nep forcing then it cannot add a dominating real. We also prove that Amoeba forcing cannot be P(X)/I if I is an aleph_1-complete ideal.
Boban Velickovic asked the following question: Is there a nontrivial forcing notion with the Sacks property which is also ccc? A ``definable'' variant of this question has been answered in [Sh:480] (math.LO/9303208): Every nontrivial Souslin forcing notion which has the Sacks property has an uncountable antichain. Here we show that it is consistent that every nontrivial forcing notion which has the Sacks property has an uncountable antichain. Independently, Velickovic has also proved the consistency of this statement.
We prove that the Sacks forcing collapses the continuum onto the dominating number d, answering the question of Carlson and Laver. Next we prove that if a proper forcing of the size at most continuum collapses omega_2 then it forces diamond_{omega_1} .
We show that for a Suslin ccc forcing notion $\mathbb Q$ adding a Hechler real, ``$\text{ZF}+\text{DC}_{ω_1}+$all sets of reals are $I_{\mathbb Q,\aleph_0}$-measurable'' implies the existence of an inner model with a measurable cardinal. We also introduce a wide class of Suslin ccc forcing notions which add a Hechler real, so that the above result applies to them.
We present reasons for developing a theory of forcing notions which satisfy the properness demand for countable models which are not necessarily elementary submodels of some (H(chi), in). This leads to forcing notions which are ``reasonably'' definable. We present two specific properties materializing this intuition: nep (non-elementary properness) and snep (Souslin non-elementary properness). For this we consider candidates (countable models to which the definition applies), and the older Souslin proper. A major theme here is ``preservation by iteration'', but we also show a dichotomy: if suc…
We describe a method of building ``nice'' sigma-ideals from Souslin ccc forcing notions. [These notes were written down in 1992, but were not submitted to any journal. In a slightly modified form, they were incorporated to: T. Bartoszynski and H. Judah, Set Theory: on the structure of the real line, A K Peters, Wellesley, MA, 1995; pages 193-203.]
We present a version with non-definable forcing notions of Shelah's theory of iterated forcing along a template. Our main result, as an application, is that, if $κ$ is a measurable cardinal and $θ<κ<μ<λ$ are uncountable regular cardinals, then there is a ccc poset forcing $\mathfrak{s}=θ<\mathfrak{b}=μ<\mathfrak{a}=λ$. Another application is to get models with large continuum where the groupwise-density number $\mathfrak{g}$ assumes an arbitrary regular value.
In this paper we analyse some notions of amoeba for tree forcings. In particular we introduce an amoeba-Silver and prove that it satisfies quasi pure decision but not pure decision. Further we define an amoeba-Sacks and prove that it satisfies the Laver property. We also show some application to regularity properties. We finally present a generalized version of amoeba and discuss some interesting associated questions.
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).
Generalizing the proof for Sacks forcing, we show that the $h$-perfect tree forcing notions introduced by Goldstern, Judah and Shelah preserve selective independent families even when iterated. As a result we obtain new proofs of the consistency of $\mathfrak{i} = \mathfrak{u} < \mathrm{non} (\mathcal N) = \mathrm{cof} (\mathcal N)$ and $\mathfrak{i} < \mathfrak{u} = \mathrm{non} (\mathcal N) = \mathrm{cof}( \mathcal N)$ as well as some related results.
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.
We study which $κ$-distributive forcing notions of size $κ$ can be embedded into tree Prikry forcing notions with $κ$-complete ultrafilters under various large cardinal assumptions. An alternative formulation -- can the filter of dense open subsets of a $κ$-distributive forcing notion of size $κ$ be extended to a $κ$-complete ultrafilter.
On fresh sets in iterations of Prikry type forcing notions
该研究考察 Prikry 型力迫迭代中 fresh sets 的存在与不存在性，推广已有结果并回答相关审稿问题，同时讨论驻集保持。
The intersection number for forcing notions
本文定义力迫概念的相交数，证明其基本性质，并引入 μ-相交链这一新链式性质，为迭代力迫理论提供组合工具。
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).
We discuss the effect of adding a single real (for various forcing notions adding reals) on cardinal invariants associated with the continuum (like the unbounding or the dominating number or the cardinals related to measure and category on the real line). For random and Cohen forcing, this question was investigated by Cicho'n and Pawlikowski; for Hechler forcing, by Judah, Shelah and myself. We show here: (1) adding an eventually different or a localization real adjoins a Luzin set of size continuum and a mad family of size omega_1; (2) Laver and Mathias forcing collapse the dominating number …
Starting from large cardinals we construct a pair $V_1\subseteq V_2$ of models of $ZFC$ with the same cardinals and cofinalities such that $GCH$ holds in $V_1$ and fails everywhere in $V_2$.
A central theme in set theory is to find universes with extreme, well-understood behaviour. The case we are interested in is assuming GCH and has a strong forcing axiom of higher order than usual. Instead of "for every suitable forcing notion for~$λ$" we shall say "for every such family of forcing notions, depending on stationary $S\subseteq λ$, for some such stationary set we have\dots". Such notions of forcing are important for Abelian group theory, but this application is delayed for a sequel.
In the paper we probe the possibilities of creating a Kurepa tree in a generic extension of a model of CH plus no Kurepa trees by an omega_1-preserving forcing notion of size at most omega_1. In the first section we show that in the Levy model obtained by collapsing all cardinals between omega_1 and a strongly inaccessible cardinal by forcing with a countable support Levy collapsing order many omega_1-preserving forcing notions of size at most omega_1 including all omega-proper forcing notions and some proper but not omega-proper forcing notions of size at most omega_1 do not create Kurepa tre…
We give some sufficient and necessary conditions on a forcing notion Q for preserving the forcing notion ([omega]^{aleph_0},supseteq^*) is proper. They cover many reasonable forcing notions.
For certain weak versions of the Axiom of Choice (most notably, the Boolean Prime Ideal theorem), we obtain equivalent formulations in terms of partial orders, and filter-like objects within them intersecting certain dense sets or antichains. This allows us to prove some consequences of the Boolean Prime Ideal theorem using arguments in the style of those that use Zorn's Lemma, or Martin's Axiom.
We present three natural combinatorial properties for class forcing notions, which imply the forcing theorem to hold. We then show that all known sufficent conditions for the forcing theorem (except for the forcing theorem itself), including the three properties presented in this paper, imply yet another regularity property for class forcing notions, namely that proper classes of the ground model cannot become sets in a generic extension, that is they do not have set-sized names in the ground model. We then show that over certain models of Gödel-Bernays set theory without the power set axiom, …
More notions of forcing add a square
研究在连续统假设下添加$\square_{\aleph_1}$的力迫概念$\mathbb P$，证明其可实现为$\aleph_2$-苏斯林树，更一般地，$\square_\lambda$可由$\lambda^+$-苏斯林树添加，构造统一且可扩展到不可达基数。
We continue investigations of forcing notions with strong ccc properties introducing new methods of building sweet forcing notions. We also show that quotients of topologically sweet forcing notions over Cohen reals are topologically sweet while the quotients over random reals do not have to be such.
We show that it is consistent with MA + the negation of CH, that the Forcing Axiom fails for all forcing notions in the class of omega^omega-bounding forcing notions with norms of "Norms on possibilities I: forcing with trees and creatures"(by Roslanowski and Shelah - math.LO/9807172).
The Steprans forcing notion arises as a quotient of Borel sets modulo the ideal of $σ$-continuity of a certain Borel not $σ$-continuous function. We give a characterization of this forcing in the language of trees and using this characterization we establish such properties of the forcing as fusion and continuous reading of names. Although the latter property is usually implied by the fact that the associated ideal is generated by closed sets, we show it is not the case with Steprans forcing. We also establish a connection between Steprans forcing and Miller forcing thus giving a new descripti…
The Maximality Principle MP is a scheme which states that if a sentence of the language of ZFC is true in some forcing extension V^P, and remains true in any further forcing extension of V^P, then it is true in all forcing extensions of V. A modified maximality principle MP_Gamma arises when considering forcing with a particular class Gamma of forcing notions. A parametrized form of such a principle, MP_Gamma(X), considers formulas taking parameters; to avoid inconsistency such parameters must be restricted to a specific set X which depends on the forcing class Gamma being considered. A strong…
In the first part of the paper, we show that if $ω\le κ< λ$ are cardinals, $κ^{<κ} = κ$, and $λ$ is weakly compact, then in $V[\M(κ,λ)]$ the tree property at $λ= κ^{++V[\M(κ,λ)]}$ is indestructible under all $κ^+$-cc forcing notions which live in $V[\Add(κ,λ)]$, where $\Add(κ,λ)$ is the Cohen forcing for adding $λ$-many subsets of $κ$ and $\M(κ,λ)$ is the standard Mitchell forcing for obtaining the tree property at $λ= (κ^{++})^{V[\M(κ,λ)]}$. This result has direct applications to Prikry-type forcing notions and generalized cardinal invariants. In the second part, we assume that $λ$ is superco…
On Cohen and Prikry Forcing Notions
本文证明用 κ 上的 Prikry 力迫可添加 κ⁺ 个 Cohen 子集，并引入非 Galvin 性质的强化，仅需单个可测基数即证明其相容性，同时讨论基于 extender 的 Prikry 力迫。
We give an example of iteration of length omega of (<kappa)-complete kappa^+-cc forcing notions with the limit collapsing kappa^+. The construction is decoded from the proof of Shelah [Proper and Improper Forcing, Appendix, Theorem 3.6(1)].
Distinguishing Martin's axiom from its restrictions
该研究提出一种对强着色破坏极小的可数链条件力迫迭代方法，证明了Martin公理严格强于两类受限形式的Martin公理，给出了更精细的公理强度区分。
In light of the celebrated theorem of Vopěnka (1972), proving in ZFC that every set is generic over HOD, it is natural to inquire whether the set-theoretic universe $V$ must be a class-forcing extension of HOD by some possibly proper-class forcing notion in HOD. We show, negatively, that if ZFC is consistent, then there is a model of ZFC that is not a class-forcing extension of its HOD for any class forcing notion definable in HOD and with definable forcing relations there (allowing parameters). Meanwhile, S. Friedman (2012) showed, positively, that if one augments HOD with a certain ZFC-amena…
We present a notion of forcing that can be used, in conjunction with other results, to show that there is a Martin-Löf random set X such that X does not compute 0' and X computes every K-trivial set.
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^+.
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。
Given a Fraïssé class $\mathcal{K}$ and an infinite cardinal $κ,$ we define a forcing notion which adds a structure of size $κ$ using elements of $\mathcal{K}$, which extends the Fraïssé construction in the case $κ=ω.$
We note that some form of the condition "$p_1, p_2$ have a $\leq_{\mathbb{Q}}$-lub in $\mathbb{Q}$" is necessary in some forcing axiom for $λ$-complete $μ^+$-c.c. forcing notions. We also show some versions are really stronger than others.
We introduce a class of notions of forcing which we call $Σ$-Prikry, and show that many of the known Prikry-type notions of forcing that centers around singular cardinals of countable cofinality are $Σ$-Prikry. We show that given a $Σ$-Prikry poset $\mathbb P$ and a name for a non-reflecting stationary set $T$, there exists a corresponding $Σ$-Prikry poset that projects to $\mathbb P$ and kills the stationarity of $T$. Then, in a sequel to this paper, we develop an iteration scheme for $Σ$-Prikry posets. Putting the two works together, we obtain a proof of the following. Theorem. If $κ$ is the…
I survey an array of topics in set theory in the context of a novel class of forcing notions: subcomplete forcing. Subcompleteness was originally defined by Ronald Jensen. I have attempted to make the subject somewhat more approachable to set theorists, while showing various properties of subcomplete forcing which one might desire of a forcing class, drawing comparisons between subcomplete forcing and countably closed forcing. In particular, I look at the interaction between subcomplete forcing and $ω_1$-trees, preservation properties of subcomplete forcing, the subcomplete maximality principl…
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…
Sequential and distributive forcings without choice
本文研究无选择公理下集合论中$κ$-分配力迫与$κ$-序列力迫的等价性问题，证明该等价在弱选择公理下失效，给出拟正常性条件以刻画依赖选择公理的保持性，并分析两类力迫对不同选择公理的保持与违反情况。
We continue investigations of reasonable ultrafilters on uncountable cardinals defined in math.LO/0407498. We introduce stronger properties of ultrafilters and we show that those properties may be handled in lambda-support iterations of reasonably bounding forcing notions. We use this to show that consistently there are reasonable ultrafilters on an inaccessible cardinal lambda with generating system of size less than 2^lambda . We also show how reasonable ultrafilters can be killed by forcing notions which have enough reasonable completeness to be iterated with lambda-supports (and we show th…
David Aspero asks on the possibility of having Forcing axiom FA_{aleph_2}(K), where K is the class of forcing notions preserving stationarity of subsets of aleph_1 and of aleph_2. We answer negatively, in fact we show the negative result for any regular lambda>aleph_1 even demanding adding no new sequence of ordinals of length<lambda.
In this paper we introduce a tree-like forcing notion extending some properties of the random forcing in the context of the generalised Cantor space and study its associated ideal of null sets and notion of measurability. This issue was also addressed by Shelah ([11, Problem 0.5]) and concerns the definition of a forcing which is $κ^kappa$-bounding, $< κ$-closed and $κ^+$-cc, for $κ$ inaccessible.
In Part I of this series, we introduced a class of notions of forcing which we call Sigma-Prikry, and showed that many of the known Prikry-type notions of forcing that center around singular cardinals of countable cofinality are Sigma-Prikry. We showed that given a Sigma-Prikry poset P and a P-name for a non-reflecting stationary set T, there exists a corresponding Sigma-Prikry poset that projects to P and kills the stationarity of T. In this paper, we develop a general scheme for iterating Sigma-Prikry posets and, as an application, we blow up the power of a countable limit of Laver-indestruc…
It is well known to generalize the meagre ideal replacing aleph_0 by a (regular) cardinal lambda > aleph_0 and requiring the ideal to be lambda^+-complete. But can we generalize the null ideal? In terms of forcing, this means finding a forcing notion similar to the random real forcing, replacing aleph_0 by lambda, so requiring it to be (<lambda)-complete. Of course, we would welcome additional properties generalizing the ones of the random real forcing. Returning to the ideal (instead forcing) we may look at the Boolean Algebra of lambda-Borel sets modulo the ideal. Surprisingly we get an posi…
We study the notion of $\mathcal J$-MAD families where $\mathcal J$ is a Borel ideal on $ω$. We show that if $\mathcal J$ is an arbitrary $F_σ$ ideal, or is any finite or countably iterated Fubini product of $F_σ$ ideals, then there are no analytic infinite $\mathcal J$-MAD families, and assuming Projective Determinacy there are no infinite projective $\mathcal J$-MAD families; and under the full Axiom of Determinacy + $V=\mathbf{L}(\mathbb{R})$ there are no infinite $\mathcal J$-mad families. These results apply in particular when $\mathcal J$ is the ideal of finite sets $\mathrm{Fin}$, which…
In this paper, we investigate connections between structures present in every generic extension of the universe $V$ and computability theory. We introduce the notion of {\em generic Muchnik reducibility} that can be used to to compare the complexity of uncountable structures; we establish basic properties of this reducibility, and study it in the context of {\em generic presentability}, the existence of a copy of the structure in every extension by a given forcing. We show that every forcing notion making $ω_2$ countable generically presents some countable structure with no copy in the ground …
We analyze the forcing notion $\mathcal P$ of finite matrices whose rows consists of isomorphic countable elementary submodels of a given structure of the form $H_θ$. We show that forcing with this poset adds a Kurepa tree $T$. Moreover, if $\mathcal P_c$ is a suborder of $\mathcal P$ containing only continuous matrices, then the Kurepa tree $T$ is almost Souslin, i.e. the level set of any antichain in $T$ is not stationary in $ω_1$.
Using a theorem from pcf theory, we show that for any singular cardinal nu, the product of the Cohen forcing notions on kappa, kappa < nu adds a generic for the Cohen forcing notion on nu^+. This solves Problem 5.1 in Miller's list (attributed to Rene David and Sy Friedman).
We give a model where there is a ccc Souslin forcing which does not satisfy the Knaster condition. Next, we present a model where there is a sigma-linked not sigma-centered Souslin forcing such that all its small subsets are sigma-centered but Martin Axiom fails for this order. Furthermore, we construct a totally nonhomogeneous Souslin forcing and we build a Souslin forcing which is proper but not ccc that does not contain a perfect set of mutually incompatible conditions. Finally we show that ccc Sigma^1_2-notions of forcing may not be indestructible ccc.
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 that if $cf(2^{\aleph_0})=\aleph_1,$ then any non-trivial $\aleph_1$-closed forcing notion of size $\leq 2^{\aleph_0}$ is forcing equivalent to $Add(\aleph_1, 1),$ the Cohen forcing for adding a new Cohen subset of $ω_1.$ We also produce, relative to the existence of suitable large cardinals, a model of $ZFC$ in which $2^{\aleph_0}=\aleph_2$ and all $\aleph_1$-closed forcing notion of size $\leq 2^{\aleph_0}$ collapse $\aleph_2,$ and hence are forcing equivalent to $Add(\aleph_1, 1).$ These results answer a question of Scott Williams from 1978. We also extend a result of Todorcevic and…
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.
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