Forcing with ideals of closed sets

Forcing with quotients

We study an extensive connection between factor forcings of Borel subsets of Polish spaces modulo a sigma-ideal, and factor forcings of subsets of countable sets modulo an ideal.

Properness under closed forcing

For every uncountable regular $κ$, we give two examples of proper posets which turn improper in some $κ$-closed forcing extension.

A parallel to the null ideal for inaccessible lambda. Part I

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…

通用支配与理想化力迫

Universal domination and idealized forcing

该研究引入波兰空间可定义σ-理想的通用性质，将其作为理想化力迫固有性的基准并统一相关结果，借此回答了两个关于理想化力迫的问题。

A Characterization of the Boolean Prime Ideal Theorem in Terms of 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.

Laver forcing的性质与其所关联的co-ideal表达形式的Katetov序相关

Properties of Laver forcing associated with a co-ideal expressed via the Katetov order

本文研究了与co-ideal相关的Laver forcing的性质，分析其在Katetov序下的组合特性，并探讨了添加半Cohen实数而不添加Cohen实数的问题。

Forcing axioms and the complexity of non-stationary ideals

We study the influence of strong forcing axioms on the complexity of the non-stationary ideal on $ω_2$ and its restrictions to certain cofinalities. Our main result shows that the strengthening $MM^{++}$ of Martin's Maximum does not decide whether the restriction of the non-stationary ideal on $ω_2$ to sets of ordinals of countable cofinality is $Δ_1$-definable by formulas with parameters in $H(ω_3)$. The techniques developed in the proof of this result also allow us to prove analogous results for the full non-stationary ideal on $ω_2$ and strong forcing axioms that are compatible with CH. Fin…

Generics for Mathias forcing over general Turing ideals

In Mathias forcing, conditions are pairs $(D,S)$ of sets of natural numbers, in which $D$ is finite, $S$ is infinite, and $\max D < \min S$. The Turing degrees and computational characteristics of generics for this forcing in the special (but important) case where the infinite sets $S$ are computable were thoroughly explored by Cholak, Dzhafarov, Hirst, and Slaman~\cite{CDHS-2014}. In this paper, we undertake a similar investigation for the case where the sets $S$ are members of general countable Turing ideals, and give conditions under which generics for Mathias forcing over one ideal compute…

比较稠密理想的力迫方法

Comparing forcing approaches to dense ideals

本文分析了稠密理想力迫构造中的偏序集，证明了两种坍缩方法在大基数坍缩到 $ω_2$ 时等价，并给出了 $ω_2$ 上正规理想 $I$ 的简化构造。

Cofinalities of Marczewski-like ideals

We show that the cofinalities of both the Miller ideal m^0 (the sigma-ideal naturally related to Miller forcing) and the Laver ideal ell^0 (related to Laver forcing) are larger than the size of the continuum in ZFC.

More on real-valued measurable cardinals and forcing with ideals

Answering two questions of D. Fremlin [Real-valued measurable cardinals, in Set Theory of the Reals, H. Judah ed. 1993, 151-305 ] we show the following: (1) If c is real-valued measurable then the Maharam type of (c,P(c),sigma) is 2^c. (2) It is consistent to have k real-valued measurable but for every submodel V_1 with k measurable in it there are no k reals which are random over V_1.

Hechler's theorem for the meager ideal

We prove the following theorem: For a partially ordered set Q such that every countable subset has a strict upper bound, there is a forcing notion satisfying ccc such that, in the forcing model, there is a basis of the meager ideal of the real line which is order-isomorphic to Q with respect to set-inclusion. This is a variation of Hechler's classical result in the theory of forcing.

关于迭代强$(<λ)$-闭 stationary $λ^+$-cc 集合论 forcing 的注记

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。

A null ideal for inaccessibles

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.

$σ$-continuity and related forcings

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 Approachability Ideal Without a Maximal Set

We develop a forcing poset with finite conditions which adds a partial square sequence on a given stationary set, with adequate sets of models as side conditions. We then develop a kind of side condition product forcing for simultaneously adding partial square sequences on multiple stationary sets. We show that certain quotients of such forcings have the $ω_1$-approximation property. We apply these ideas to prove, assuming the consistency of a greatly Mahlo cardinal, that it is consistent that the approachability ideal $I[ω_2]$ does not have a maximal set modulo clubs.

Hechler's theorem for the null ideal

We prove the following theorem: For a partially ordered set Q such that every countable subset has a strict upper bound, there is a forcing notion satisfying ccc such that, in the forcing model, there is a basis of the null ideal of the real line which is order-isomorphic to Q with respect to set-inclusion. This is a variation of Hechler's classical result in the theory of forcing, and the statement of the theorem for the meager ideal has been already proved by Bartoszynski and the author.

Strongly proper forcing and some problems of Foreman

We provide solutions to several problems of Foreman about ideals, several of which are closely related to Mitchell's notion of \emph{strongly proper} forcing. We prove: 1) Presaturation of a normal ideal implies projective antichain catching, enabling us to provide a solution to a problem from Foreman~\cite{MR2768692} about ideal projections which is more comprehensive and simpler than the solution obtained in \cite{MR3343538}. 2) We solve an older question from Foreman~\cite{MR819932} about the relationship between generic hugeness and generic almost hugeness. 3) Finally, we provide solutions…

Maximal almost disjoint families, determinacy, and forcing

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…

Filters and Ideal Independence

A family $\mathscr{I} \subseteq [ω]^ω$ such that for all finite $\{X_i\}_{i\in n}\subseteq \mathcal I$ and $A \in \mathscr{I} \setminus \{X_i\}_{i\in n}$, the set $A \setminus \bigcup_{i < n} X_i$ is infinite, is said to be ideal independent. An ideal independent family which is maximal under inclusion is said to be a maximal ideal independent family and the least cardinality of such family is denoted $\mathfrak{s}_{mm}$. We show that $\mathfrak{u}\leq\mathfrak{s}_{mm}$, which in particular establishes the independence of $\mathfrak{s}_{mm}$ and $\mathfrak{i}$. Given an arbitrary set $C$ of un…

On tree ideals

Let l^0 and m^0 be the ideals associated with Laver and Miller forcing, respectively. We show that add (l^0) < cov(l^0) and add (m^0) < cov(m^0) are consistent. We also show that both Laver and Miller forcing collapse the continuum to a cardinal <= h .

Prikry-type forcings after collapsing a huge cardinal

Some models of combinatorial principles have been obtained by collapsing a huge cardinal in the case of the successors of regular cardinals. For example, saturated ideals, Chang's conjecture, polarized partition relations, and transfer principles for chromatic numbers of graphs. In this paper, we study these in the case of the successors of singular cardinals. In particular, we show that Prikry forcing preserves the centeredness of ideals but kills the layeredness. We also study $\binom{μ^{++}}{μ^{+}}\to\binomκ{μ^{+}}_μ$ and $\mathrm{Tr}_{\mathrm{Chr}}(μ^{+++},μ^{+})$ in the extension by Prikr…

Ideal Independence, Free Sequences, and the Ultrafilter Number

We make use of a forcing technique for extending Boolean algebras. The same type of forcing was employed in [BK81], [Kos99], and elsewhere. Using and modifying a lemma of Koszmider, and using CH, we obtain an atomless BA, A such that f(A) = smm(A) < u(A), answering questions raised by [Mon08] and [Mon11].

Strolling through Paradise

With each of the usual tree forcings I (e.g., I = Sacks forcing S, Laver forcing L, Miller forcing M, Mathias forcing R, etc.) we associate a sigma--ideal i^0 on the reals as follows: A \in i^0 iff for all T \in I there is S \leq T (i.e. S is stronger than T or, equivalently, S is a subtree of T) such that A \cap [S] = \emptyset, where [S] denotes the set of branches through S. So, s^0 is the ideal of Marczewski null sets, r^0 is the ideal of Ramsey null sets (nowhere Ramsey sets) etc. We show (in ZFC) that whenever i^0, j^0 are two such ideals, then i^0 \sem j^0 \neq \emptyset. E.g., for I=S …

Note on omega-nw-nep forcing notions

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.

Bounded Namba forcing axiom may fail

In a sigma-closed forcing extension, the bounded forcing axiom for Namba forcing fails. This answers a question of Justin Tatch Moore.

The covering number of the strong measure zero ideal can be above almost everything else

We show that certain type of tree forcings, including Sacks forcing, increases the covering of the strong measure zero ideal $\mathcal{SN}$. As a consequence, in Sacks model, such covering number is equal to the size of the continuum, which indicates that this covering number is consistently larger than any other classical cardinal invariant of the continuum. Even more, Sacks forcing can be used to force that $\mathrm{non}(\mathcal{SN})<\mathrm{cov}(\mathcal{SN})<\mathrm{cof}(\mathcal{SN})$, which is the first consistency result where more than two cardinal invariants associated with $\mathcal…

Proper forcing and rectangular Ramsey theorems

I prove forcing preservation theorems for products of definable partial orders preserving the cofinality of the meager or null ideal. Rectangular Ramsey theorems for related ideals follow from the proofs.

梯级饱和与力迫公理

Echeloned saturation and forcing axioms

针对Paul Larson的问题，研究结合不同力迫公理与集合论假设，证明特定条件下ℵ₁上无弱Laver理想，以及Baumgartner公理下ℵ₂上无Laver理想。

The weakly compact reflection principle need not imply a high order of weak compactness

The weakly compact reflection principle $\text{Refl}_{\text{wc}}(κ)$ states that $κ$ is a weakly compact cardinal and every weakly compact subset of $κ$ has a weakly compact proper initial segment. The weakly compact reflection principle at $κ$ implies that $κ$ is an $ω$-weakly compact cardinal. In this article we show that the weakly compact reflection principle does not imply that $κ$ is $(ω+1)$-weakly compact. Moreover, we show that if the weakly compact reflection principle holds at $κ$ then there is a forcing extension preserving this in which $κ$ is the least $ω$-weakly compact cardinal.…

Many Different Uniformity Numbers of Yorioka Ideals

Using a countable support product of creature forcing posets, we show that consistently, for uncountably many different functions the associated Yorioka ideals' uniformity numbers can be pairwise different. In addition we show that, in the same forcing extension, for two other types of simple cardinal characteristics parametrised by reals (localisation and anti-localisation cardinals), for uncountably many parameters the corresponding cardinals are pairwise different.

Bounded forcing axioms and Baumgartner's conjecture

We study the spectrum of forcing notions between the iterations of $σ$-closed followed by ccc forcings and the proper forcings. This includes the hierarchy of $α$-proper forcings for indecomposable countable ordinals as well as the Axiom A forcings. We focus on the bounded forcing axioms for the hierarchy of $α$-proper forcings and connect them to a hierarchy of weak club guessing principles. We show that they are, in a sense, dual to each other. In particular, these weak club guessing principles separate the bounded forcing axioms for distinct countable indecomposable ordinals. In the study o…

力迫公理与$\omega_1$上非驻定理想的可定义性

Forcing Axioms and the Definabilty of the Nonstationary Ideal on $ω_1$

该数理逻辑研究探讨力迫公理下$\omega_1$非驻定理想的可定义性，证明$\text{BMM}$加武丁基数、$(*)$-公理下其不可被$\Sigma_1(\omega_1)$定义，同时构造出$\text{BPFA}$、$\text{MA}$等公理下其可定义甚至饱和的集合论宇宙。

PFA and precipitousness of the nonstationary ideal

We apply Neeman's method of forcing with side conditions to show that PFA does not imply the precipitousness of the nonstationary ideal on $ω_1$.

理想化Miller空集的基数不变量

Cardinal invariants of idealized Miller null sets

本文研究了理想化Miller空集的基数不变量，通过计算典型Borel理想参数化的$σ$-理想，扩展了Cichoń的最大值。

Uniformization and Internal Absoluteness

Measurability with respect to ideals is tightly connected with absoluteness principles for certain forcing notions. We study a uniformization principle that postulates the existence of a uniformizing function on a large set, relative to a given ideal. We prove that for all $σ$-ideals $I$ such that the ideal forcing $\mathbb{P}_I$ of Borel sets modulo $I$ is proper, this uniformization principle is equivalent to an absoluteness principle for projective formulas with respect to $\mathbb{P}_I$ that we call internal absoluteness. In addition, we show that it is equivalent to measurability with res…

Borel chromatic numbers of closed graphs and forcing with uniform trees

In this work, we continue the tradition initiated by Geschke, 2011 of viewing the uncountable Borel chromatic number of analytic graphs as cardinal invariants of the continuum. We show that various uncountable Borel chromatic numbers of closed graphs can be consistently different, as well as consistently equal to the continuum. This is done using arguments that are typical to Axiom A forcing notions.

模态逻辑的集合论翻译：通过力迫法

A Set-Theoretic Translation of Modal Logic via Forcing

本文通过力迫法为模态逻辑的扩张理论建立集合论翻译，证明其可靠性与完备性，并构建相应的Kripke模型。

Combinatorics on Ideals and Axiom A

Throughout this abstract let U be a fixed p-point ultrafilter and let I be the dual ideal. Grigorieff forcing is P(U)={p:omega to 2|dom(p) is an element of I} ordered by reverse inclusion. It is well known that Grigorieff forcing is proper. The main result of this paper is the following: THEOREM: Gregorieff forcing does not satisfy Axiom A. To prove this we use the following game, denoted G(U), for two players playing alternatively: Player I plays a partition of omega, {J_n| n<omega}, such that for all n<omega, J_n is an element of I; At the nth turn Player II plays a finite subset F_n of J_n.…

滤子、理想独立性与理想Mrówka空间

Filters, ideal independence and ideal Mrówka spaces

本文研究了理想独立族，证明了极大理想独立族的等价条件，确立了基数不变量间的独立性与一致性，并引入了相关的自然滤子与Mrówka空间类比。

Kurepa-trees and Namba-forcing

We show that compact cardinals and {\rm MM} are sensitive to $λ$-closed forcings for arbitrarily large $λ$. This is done by adding 'regressive' $λ$-Kurepa-trees in either case. We argue that the destruction of regressive Kurepa-trees with {\rm MM} requires the use of Namba forcing.

Destroying Saturation while Preserving Presaturation at an Inaccessible; an Iterated Forcing Argument

We prove that a large class of presaturated ideals at inaccessible cardinals can be de-saturated while preserving their presaturation, answering both a question of Foreman and of Cox and Eskew. We do so by iterating a generalized version of Baumgartner and Taylor's forcing to add a club with finite conditions along an inaccessible cardinal, and invoking Foreman's Duality Theorem.

Indestructibility of the tree property

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…

A friendly iteration forcing that the four cardinal characteristics of $\mathcal{E}$ can be pairwise different

Let $\mathcal{E}$ be the $σ$-ideal generated by the closed measure zero sets of reals. We use an ultrafilter-extendable matrix iteration of ccc posets to force that, for $\mathcal{E}$, their associated cardinal characteristics (i.e.\ additivity, covering, uniformity and cofinality) are pairwise different.

$F_σ$ equivalence relations and Laver forcing

Following the topic of the book Canonical Ramsey Theory on Polish Spaces by V. Kanovei, M. Sabok and J. Zapletal we study Borel equivalences on Laver trees. Here we prove that equivalence relations Borel reducible to an equivalence relation on $2^ω$ given by some $F_σ$ $P$-ideal on $ω$ can be canonized to the full equivalence relation or to the identity relation. This has several corollaries, e.g. Silver type dichotomy for the Laver ideal and equivalences Borel reducible to equivalence relations given by $F_σ$ $P$-ideals.

Ramsey degrees of ultrafilters, pseudointersection numbers, and the tools of topological Ramsey spaces

This paper investigates properties of $σ$-closed forcings which generate ultrafilters satisfying weak partition relations. The Ramsey degree of an ultrafilter $\mathcal{U}$ for $n$-tuples, denoted $t(\mathcal{U},n)$, is the smallest number $t$ such that given any $l\ge 2$ and coloring $c:[ω]^n\rightarrow l$, there is a member $X\in\mathcal{U}$ such that the restriction of $c$ to $[X]^n$ has no more than $t$ colors. Many well-known $σ$-closed forcings are known to generate ultrafilters with finite Ramsey degrees, but finding the precise degrees can sometimes prove elusive or quite involved, at …

On splitting trees

We investigate two variants of splitting tree forcing, their ideals and regularity properties. We prove connections with other well-known notions, such as Lebesgue measurablility, Baire- and Doughnut-property and the Marczewski field. Moreover, we prove that any \emph{absolute} amoeba forcing for splitting trees necessarily adds a dominating real, providing more support to Spinas' and Hein's conjecture that $\add(\ideal{I}_\spl) \leq \mathfrak{b}$.

Oracle-树强制与十三元素情形下的Wilfs不等式

Oracle-Tree Forcing and Wilfs Inequality for Thirteen Left Elements

本文研究filter-Laver强制及其商与基数性质，证明网格贫乏理想覆盖数与贫乏理想一致，并应用该框架给出十三元素数值半群的Wilfs不等式c≤13e的完整算术证明。

PFA与非平稳理想的可定义性

PFA and the definability of the nonstationary ideal

该研究在含超紧基数的ZFC模型基础上构造出满足PFA的模型，证明ω₁上的非平稳理想在H_{ℵ₂}中可由参数Π₁-定义，解决了PFA与非平稳理想可定义性的关联问题。

分离强零测度理想的基数特征

Separating cardinal characteristics of the strong measure zero ideal

该研究针对实数强零测度理想$\mathcal{SN}$，给出有限支撑迭代后控制其加性的力迫概念性质，构造力迫扩张使$\mathcal{SN}$的四个基数特征两两不同且满足Cichoń极大性。

Specializing trees and answer to a question of Williams

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…

Superdestructibility: a dual to Laver indestructibility

After small forcing, any < kappa-closed forcing will destroy the supercompactness, even the strong compactness, of kappa .

On cardinal characteristics associated with the strong measure zero ideal

Let $\mathcal{SN}$ be the strong measure zero $σ$-ideal. We prove a result providing bounds for $\mathrm{cof}(\mathcal{SN})$ which implies Yorioka's characterization of the cofinality of the strong measure zero. In addition, we use forcing matrix iterations to construct a model of ZFC that satisfies $\mathrm{add}(\mathcal{SN})=\mathrm{cov}(\mathcal{SN})<\mathrm{non}(\mathcal{SN})<\mathrm{cof}(\mathcal{SN})$.

Indestructibility properties of remarkable cardinals

Remarkable cardinals were introduced by Schindler, who showed that the existence of a remarkable cardinal is equiconsistent with the assertion that the theory of $L(\mathbb R)$ is absolute for proper forcing. Here, we study the indestructibility properties of remarkable cardinals. We show that if $κ$ is remarkable, then there is a forcing extension in which the remarkability of $κ$ becomes indestructible by all $\ltκ$-closed $\leqκ$-distributive forcing and all two-step iterations of the form ${\rm Add}(κ,θ)*\dot{\mathbb R}$, where $\dot{\mathbb R}$ is forced to be $\ltκ$-closed and $\leqκ$-di…