Small forcings and Cohen reals

论 Cohen 与 Prikry 力迫概念

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。

Non Cohen Oracle c.c.c

The oracle c.c.c. is closely related to Cohen forcing. During an iteration we can ``omit a type''; i.e. preserve ``the intersection of a given family of Borel sets of reals is empty'' provided that Cohen forcing satisfies it. We generalize this to other cases. We replace Cohen by ``nicely'' definable c.c.c., do the parallel of the oracle c.c.c. and end with a criterion for extracting a subforcing (not a complete subforcing) of a given nicely one and satisfying the oracle.

How special are Cohen and random forcings i.e. Boolean algebras of the family of subsets of reals modulo meagre or null

The feeling that those two forcing notions-Cohen and Random-(equivalently the corresponding Boolean algebras Borel(R)/(meager sets), Borel(R)/(null sets)) are special, was probably old and widespread. A reasonable interpretation is to show them unique, or ``minimal'' or at least characteristic in a family of ``nice forcing'' like Borel. We shall interpret ``nice'' as Souslin as suggested by Judah Shelah [JdSh 292]. We divide the family of Souslin forcing to two, and expect that: among the first part, i.e. those adding some non-dominated real, Cohen is minimal (=is below every one), while among…

不可达基数下随机实数力迫的推广

Generalizing random real forcing for inaccessible cardinals

本文针对不可达基数$\u03bb$对应的$\u03bb^\u03bb$空间，在附加假设下构造了具备随机实数力迫性质的推广力迫，证明其保持基数与共尾性且不添加无支配实数，填补了随机实数力迫在高阶无穷空间的推广空白。

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。

A co-analytic Cohen indestructible maximal cofinitary group

Assuming that every set is constructible, we find a $Π^1_1$ maximal cofinitary group of permutations of $\mathbb N$ which is indestructible by Cohen forcing. Thus we show that the existence of such groups is consistent with arbitrarily large continuum. Our method also gives a new proof, inspired by the forcing method, of Kastermans' result that there exists a $Π^1_1$ maximal cofinitary group in $L$.

用不变测度力迫

Forcing with Invariant Measures

本文引入随机实数力迫的模型论推广，通过对称群不变测度力迫生成随机泛型结构，证明可数不变测度可唯一延拓到不可数集，并区分随机与Cohen泛型结构。

On Mathias generic sets

We present some results about generics for computable Mathias forcing. The $n$-generics and weak $n$-generics in this setting form a strict hierarchy as in the case of Cohen forcing. We analyze the complexity of the Mathias forcing relation, and show that if $G$ is any $n$-generic with $n \geq 3$ then it satisfies the jump property $G^{(n-1)} = G' \oplus \emptyset^{(n)}$. We prove that every such $G$ has generalized high degree, and so cannot have even Cohen 1-generic degree. On the other hand, we show that $G$, together with any bi-immune set $A \leq_T \emptyset^{(n-1)}$, computes a Cohen $n$…

在无选择模型上的力迫与通用绝对性

Forcing over choiceless models and generic absoluteness

该研究开发了无选择集合论模型上的力迫工具，规避了可数链条件力迫坍缩ω₁的问题，还证明了特定力迫类下的通用绝对性原则蕴含无限基数均有可数共尾性。

Sweet & Sour and other flavours of ccc forcing notions

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…

Forcing with ideals of closed sets

Let I be a sigma-ideal sigma-generated by a projective collection of closed sets. The forcing with I-positive Borel sets is proper and adds a single real r of an almost minimal degree: if s is a real in V[r] then s is Cohen generic over V or V[s]=V[r].

Laver Trees in the Generalized Baire Space

We prove that any suitable generalization of Laver forcing to the space $ κ^κ$, for uncountable regular $κ$, necessarily adds a Cohen $κ$-real. We also study a dichotomy and an ideal naturally related to generalized Laver forcing. Using this dichotomy, we prove the following stronger result: if $ κ^{<κ}=κ$, then every $<κ$-distributive tree forcing on $κ^κ$ adding a dominating $κ$-real which is the image of the generic under a continuous function in the ground model, adds a Cohen $κ$-real. This is a contribution to the study of generalized Baire spaces and answers a question from arXiv:1611.08…

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…

关联力迫关系

Relating forcing relations

本文通过模型论、布尔值模型和拓扑斯理论三种形式化方法，建立了力迫概念之间的等价关系，并给出了文献中未见的连接结果。

Mathias 力迫与 Silver 力迫的恰当与非恰当变体

Proper and Improper Variants of Mathias and Silver Forcing

该研究解答了 Mathias 与 Silver 力迫密度变体的恰当性及添加支配实数、Cohen 实数等问题，并将证明推广到满足条件的下半连续子测度参数化 Mathias 力迫。

Upward closure and amalgamation in the generic multiverse of a countable model of set theory

I prove several theorems concerning upward closure and amalgamation in the generic multiverse of a countable transitive model of set theory. Every such model $W$ has forcing extensions $W[c]$ and $W[d]$ by adding a Cohen real, which cannot be amalgamated in any further extension, but some nontrivial forcing notions have all their extensions amalgamable. An increasing chain $W[G_0]\subseteq W[G_1]\subseteq\cdots$ has an upper bound $W[H]$ if and only if the forcing had uniformly bounded essential size in $W$. Every chain $W\subseteq W[c_0]\subseteq W[c_1]\subseteq\cdots$ of extensions adding Co…

How much sweetness is there in the universe?

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.

More notions of forcing add a Souslin tree

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.

Uncountable trees and Cohen $κ$-reals

We investigate some versions of amoeba for tree-forcings in the generalized Cantor and Baire spaces. This answers [10, Question 3.20] and generalizes a line of research that in the standard case has been studied in [11], [13], and [7]. Moreover, we also answer questions posed in [3] by Friedman, Khomskii, and Kulikov, about the relationships between regularity properties at uncountable cardinals. We show $Σ_1^1$-counterexamples to some regularity properties related to trees without club splitting. In particular we prove a strong relationship between the Ramsey and the Baire properties, in slig…

Descriptive set theory and forcing; How to prove theorems about Borel sets the hard way

These are lecture notes from a course I gave at the University of Wisconsin during the Spring semester of 1993. Part 1 is concerned with Borel hierarchies. Section 13 contains an unpublished theorem of Fremlin concerning Borel hierarchies and MA. Section 14 and 15 contain new results concerning the lengths of Borel hierarchies in the Cohen and random real model. Part 2 contains standard results on the theory of Analytic sets. Section 25 contains Harrington's Theorem that it is consistent to have $Π^1_2$ sets of arbitrary cardinality. Part 3 has the usual separation theorems. Part 4 gives some …

Combinatorial properties of classical forcing notions

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 …

Cohen 扩张中的完全二部图划分关系

Complete Bipartite Partition Relations in Cohen Extensions

本文研究了添加 $ω_2$ 个 Cohen 实数对 $ω_2$ 上图的影响，证明了在 CH 模型中经力迫后 $ω_2 \to (ω_2, ω: ω)^2$ 成立，并指出该结果在某种意义上是最优的。

The grounded Martin's axiom

We introduce a variant of Martin's axiom, called the grounded Martin's axiom, which asserts that the universe is a ccc forcing extension in which Martin's axiom holds for posets in the ground model. This principle already implies several of the combinatorial consequences of Martin's axiom. The new axiom is shown to be consistent with the failure of Martin's axiom and a singular continuum. We prove that the grounded Martin's axiom is preserved in a strong way when adding a Cohen real and that adding a random real to a model of Martin's axiom preserves the grounded version (even though it destro…

Small forcing creates neither strong nor Woodin cardinals

After small forcing, almost every strongness embedding is the lift of a strongness embedding in the ground model. Consequently, small forcing creates neither strong nor Woodin cardinals.

On ground model definability

Laver, and Woodin independently, showed that models of ${\rm ZFC}$ are uniformly definable in their set-forcing extensions, using a ground model parameter. We investigate ground model definability for models of fragments of ${\rm ZFC}$, particularly of ${\rm ZF}+{\rm DC}_δ$ and of ${\rm ZFC}^-$, and we obtain both positive and negative results. Generalizing the results of Laver and Woodin, we show that models of ${\rm ZF}+{\rm DC}_δ$ are uniformly definable in their set-forcing extensions by posets admitting a gap at $δ$, using a ground model parameter. In particular, this means that models of…

Small Forcing Makes Any Cardinal Superdestructible

Small forcing always ruins the indestructibility of an indestructible supercompact cardinal. In fact, after small forcing, any cardinal $κ$ becomes superdestructible---any further ${<}κ$-closed forcing which adds a subset to $κ$ will destroy the measurability, even the weak compactness, of $κ$. Nevertheless, after small forcing indestructible cardinals remain resurrectible, but never strongly resurrectible.

Hajnal–Máté 图、Cohen 实数与不相交类型猜测

Hajnal--Máté graphs, Cohen reals, and disjoint type guessing

本文研究 Hajnal–Máté 图在添加单个 Cohen 实数模型中的存在性，证明此类模型必含无三角形 Hajnal–Máté 图，并提出独立于 ZFC 的不相交类型猜测，用于构造有限子图色数任意缓慢增长的 Hajnal–Máté 图。

How to have more things by forgetting how to count them

Cohen's first model is a model of Zermelo--Fraenkel set theory in which there is a Dedekind-finite set of real numbers, and it is perhaps the most famous model where the Axiom of Choice fails. We force over this model to add a function from this Dedekind-finite set to some infinite ordinal $κ$. In the case that we force the function to be injective, it turns out that the resulting model is the same as adding $κ$ Cohen reals to the ground model, and that we have just added an enumeration of the canonical Dedekind-finite set. In the case where the function is merely surjective it turns out that …

Mitchell-style forcing, with small working parts and collections of models as side conditions, and gap-one simplified morasses

We give a modification of Mitchell's technique for adding objects of size $ω_2$ with conditions with finite working parts in which the collections of models used as side conditions are very highly structured, arguably making them more wieldy. We use one such forcing (essentially a `pure side conditions' forcing) to answer affirmatively the question, asked independently by Shelah and Velleman in the late 1980s, as to whether a $(κ^+,1)$-simplified morass can be added by a forcing with working parts of size $<κ$.

Can a small forcing create Kurepa trees?

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…

Covering properties of $ω$-mad families

We prove that CH implies the existence of a Cohen-indestructible mad family such that the Mathias forcing associated to its filter adds dominating reals, while $\mathfrak b=\mathfrak c$ is consistent with the negation of this statement as witnessed by the Laver model for the consistency of Borel's conjecture.

Proper forcing and $L({\mathbb R})$

We present two ways in which the model $L({\mathbb R})$ is canonical assuming the existence of large cardinals. We show that the theory of this model, with {\em ordinal} parameters, cannot be changed by small forcing; we show further that a set of ordinals in $V$ cannot be added to $L({\mathbb R})$ by small forcing. The large cardinal needed corresponds to the consistency strength of $AD^{L({\mathbb R})}$; roughly $ω$ Woodin cardinals.

作为访问小扩张的局部方法的力迫法

Forcing as a Local Method of Accessing Small Extensions

本文在固定集合论宇宙V中，将小扩张视为广义可计算度，形式化研究定义度子类的方法复杂性，并给出力迫法复杂性的刻画。

Forcing consequences of PFA together with the continuum large

We develop a new method for building forcing iterations with symmetric systems of structures as side conditions. Using our method we prove that the forcing axiom for the class of all the small finitely proper posets is compatible with a large continuum.

Methods in Higher Forcing Axioms (Workshop Notes)

Methods of Higher Forcing Axioms was a small workshop in Norwich, taking place between 10--12 of September, 2019. The goal was to encourage future collaborations, and create more focused threads of research on the topic of higher forcing axioms. This is an improved version of the notes taken during the meeting by Asaf Karagila.

实数集相对于某些集合论子域的超越度

The transcendence degree of the reals over certain set-theoretical subfields

本文研究了在添加有限个Cohen实数后，实数集相对于某些集合论子域的超越度问题，并证明其最大性。

Specializing Trees with Small Approximations I

Assuming $\rm PFA$, we shall use internally club $ω_1$-guessing models as side conditions to show that for every tree $T$ of height $ω_2$ without cofinal branches, there is a proper and $\aleph_2$-preserving forcing notion with finite conditions which specialises $T$. Moreover, the forcing has the $ω_1$-approximation property.

不可线性与小条件

Unthreadability with Small Conditions

本文在CH下引入一种forcing，添加一个具有可数条件的square(aleph_2,aleph_0)序列，并在弱不可捉性基数假设下证明在forcing扩展中square(aleph_2,<aleph_0)和square_{aleph_1,aleph_0}均失效。

$Σ_n$-correct 力迫公理

$Σ_n$-correct Forcing Axioms

本文提出了扩展ZFC的$Σ_n$-correct力迫公理族，证明了不同层级公理与对应大基数的相容性，还研究了其力迫保持性、与其他公理的关系及数学推论。

Convergence of measures after adding a real

We prove that if $\mathcal{A}$ is an infinite Boolean algebra in the ground model $V$ and $\mathbb{P}$ is a notion of forcing adding any of the following reals: a Cohen real, an unsplit real, or a random real, then, in any $\mathbb{P}$-generic extension $V[G]$, $\mathcal{A}$ has neither the Nikodym property nor the Grothendieck property. A similar result is also proved for a dominating real and the Nikodym property.

Cohen 模型中 Calkin 代数的较小 masa

Small masas of the Calkin algebra in the Cohen model

本文在 Cohen 模型下研究 Calkin 代数的极大交换子代数，证明其密度可小于连续统，并刻画了由布尔代数构造的 masa 的同构与酉等价类型，同时给出 ZFC 中不存在密度小于 𝔭 的 masa 的结论。