Some Natural Internal Forcing Schemata Extending ZFC

A product forcing model in which the Russell-nontypical sets satisfy ZFC strictly between HOD and the universe

A set is nontypical in the Russell sense, if it belongs to a countable ordinal definable set. The class HNT of all hereditarily nontypical sets satisfies all axioms of ZF and the double inclusion HOD $\subseteq$ HNT $\subseteq$ V holds. Solving a problem recently proposed by Tzouvaras, a generic extension L$[a,x]$ of L, by two reals $a,x$, is presented in which L=HOD $\subsetneqq$ L$[a]$=HNT $\subsetneqq$ V=L$[a,x]$, so that HNT is a model of ZFC strictly between HOD and the universe.

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…

The set-theoretic universe $V$ is not necessarily a class-forcing extension of HOD

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…

Small forcings and Cohen reals

We prove it consistent relative to ZFC that all nontrivial forcings of size $\aleph _1$ add a Cohen real.

The modal logic of $σ$-centered forcing and related forcing classes

We consider the modality "$φ$ is true in every $σ$-centered forcing extension", denoted $\squareφ$, and its dual "$φ$ is true in some $σ$-centered forcing extension", denoted $\lozengeφ$ (where $φ$ is a statement in set theory), which give rise to the notion of a "principle of $σ$-centered forcing". We prove that if ZFC is consistent, then the modal logic of $σ$-centered forcing, i.e. the ZFC-provable principles of $σ$-centered forcing, is exactly $\mathsf{S4.2}$. We also generalize this result to other related classes of forcing.

Solovay-type characterizations for forcing-algebras

We give characterizations for the (in ZFC unprovable) sentences ``Every $\Sb{1}{2}$--set is measurable" and ``Every $\Db{1}{2}$--set is measurable" for various notions of measurability derived from well--known forcing partial orderings.

Stranger Things about Forcing without AC

Typically, set theorists reason about forcing constructions in the context of ZFC. We show that without AC, several simple properties of forcing posets fail to hold, one of which answers Miller's question from arXiv:0704.3998.

Set-Theoretic Geology

A ground of the universe V is a transitive proper class W subset V, such that W is a model of ZFC and V is obtained by set forcing over W, so that V = W[G] for some W-generic filter G subset P in W . The model V satisfies the ground axiom GA if there are no such W properly contained in V . The model W is a bedrock of V if W is a ground of V and satisfies the ground axiom. The mantle of V is the intersection of all grounds of V . The generic mantle of V is the intersection of all grounds of all set-forcing extensions of V . The generic HOD, written gHOD, is the intersection of all HODs of all s…

强制对exacting与ultraexacting基数的影响

The effects of forcing on exacting and ultraexacting cardinals

本文研究集合论强制对exacting与ultraexacting基数的影响，证明I2嵌入产生具有真类exacting基数的ZFC模型，并探讨HOD猜想下相关性质及强制扩张。

借助 ZFC$^-$ 或二阶 Peano 算术得到 Jensen $\varDelta^1_3$ 实数

Jensen $\varDelta^1_3$ reals by means of ZFC$^-$ or second order Peano arithmetic

本文解决 Jensen 构造在二阶 Peano 算术或 ZFC$^-$ 内重现的难题，通过在 forcing 构造中仅消灭可定义反链来规避 CCC 证明障碍，并定义了极小 $\varPi^1_2$ 单例模型。

含本原元素集合论的公理化与力迫

Axiomatization and Forcing in Set Theory with Urelements

本文系统研究含本原元素集合论中的公理层级，并提出新的力迫名称定义以证明力迫基本定理，进而揭示力迫对公理的破坏与恢复以及地面模型可定义性的失效现象。

The method of forcing

The purpose of this article is to give a presentation of the method of forcing aimed at someone with a minimal knowledge of set theory and logic. The emphasis will be on how the method can be used to prove theorems in ZFC.

Killing GCH everywhere by a cofinality-preserving forcing notion over a model of GCH

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 $V_2$.

Formalization of Forcing in Isabelle/ZF

We formalize the theory of forcing in the set theory framework of Isabelle/ZF. Under the assumption of the existence of a countable transitive model of ZFC, we construct a proper generic extension and show that the latter also satisfies ZFC. In doing so, we remodularized Paulson's ZF-Constructibility library.

The downward directed grounds hypothesis and very large cardinals

A transitive model $M$ of ZFC is called a ground if the universe $V$ is a set forcing extension of $M$. We show that the grounds of $V$ are downward set-directed. Consequently, we establish some fundamental theorems on the forcing method and the set-theoretic geology. For instance, (1) the mantle, the intersection of all grounds, must be a model of ZFC. (2) $V$ has only set many grounds if and only if the mantle is a ground. We also show that if the universe has some very large cardinal, then the mantle must be a ground.

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。

通过ssp力迫添加穿过多个正则基数的共尾可数序列

Adding cofinal countable sequences through multiple regular cardinals by ssp forcing

本研究提出一种基于ZFC的初等平稳集保持力迫构造方法，可使指定的大于ω₁的正则基数集元素都具有ω共尾性，同时保证基模其他不可数正则基数不会在力迫扩张中获得可数共尾性。

Well-Ordered Model Universes

In this paper we show how to build a model of $ZFC$ such that all its inner models satisfying the Axiom of Choice are well-ordered with respect to inclusion, and that said ordering is of arbitrary height (including possibly $Ord$ high). We do this by iterating $κ$-Sacks forcing for ever-increasing $κ$, while showing that such forcings do not add any unexpected intermediate inner models.

The Ground Axiom

A new axiom is proposed, the Ground Axiom, asserting that the universe is not a nontrivial set-forcing extension of any inner model. The Ground Axiom is first-order expressible, and any model of ZFC has a class-forcing extension which satisfies it. The Ground Axiom is independent of many well-known set-theoretic assertions including the Generalized Continuum Hypothesis, the assertion V=HOD that every set is ordinal definable, and the existence of measurable and supercompact cardinals. The related Bedrock Axiom, asserting that the universe is a set-forcing extension of a model satisfying the Gr…

NNR Revisited

We show if we use countable support iteration of forcing notions not adding reals that satisfy additional conditions, then the limit forcing does not add reals. As a result we prove that we can amalgamate two earlier methods and prove the consistency with ZFC + GCH of two statements gotten separately earlier: Souslin hypothesis and non-club guessing. We also answer a question of Justin Moore by proving the consistency of one further case of "strong failure of club guessing" with GCH.

Consistency Strengths of Modified Maximality Principles

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…

A Framework for Forcing Constructions at Successors of Singular Cardinals

We describe a framework for proving consistency results about singular cardinals of arbitrary cofinality and their successors. This framework allows the construction of models in which the Singular Cardinals Hypothesis fails at a singular cardinal of uncountable cofinality, while its successor enjoys various combinatorial properties. As a sample application, we prove the consistency (relative to that of ZFC plus a supercompact cardinal) of there being a strong limit singular cardinal $κ$ of uncountable cofinality where SCH fails and for which there is a collection of graphs on $κ^+$ whose size…

力迫法CTM进路的形式化验证

The formal verification of the ctm approach to forcing

本研究借助Isabelle证明助手与Isabelle/ZF库，形式化验证了ZFC可数传递模型的力迫兼纳扩张构造，并明确了替换公理模式中21个关键实例的作用，为连续统假设相关独立性结果提供了机器验证基础。

The Ground Axiom (GA)

A new axiom is proposed, the Ground Axiom, asserting that the universe is not a nontrivial set forcing extension of any inner model. The Ground Axiom is first-order expressible, and any model of ZFC has a class forcing extension which satisfies it. The Ground Axiom is independent of many well-known set-theoretic assertions including the Generalized Continuum Hypothesis, the assertion V=HOD that every set is ordinal definable, and the existence of measurable and supercompact cardinals. The related Bedrock Axiom, asserting that the universe is a set forcing extension of a model satisfying the Gr…

Resurrection axioms and uplifting cardinals

We introduce the resurrection axioms, a new class of forcing axioms, and the uplifting cardinals, a new large cardinal notion, and prove that various instances of the resurrection axioms are equiconsistent over ZFC with the existence of an uplifting cardinal.

来自决定性模型的共尾强Chang猜想

The Cofinal Strong Chang Conjecture from Models of Determinacy

通过修改Woodin的证明，表明从决定性模型使用P_max力迫得到的模型满足更强的共尾版本的强Chang猜想，并由此证明Namba力迫是半正常的且Θ^{UB}=ω_3的一致性，以及Woodin (*)_{UB}公理蕴含该猜想。

Steel's Programme: Evidential Framework, the Core and Ultimate-L

We address Steel's Programme to identify a 'preferred' universe of set theory and the best axioms extending ZFC by using his multiverse axioms MV and the 'core hypothesis'. In the first part, we examine the evidential framework for MV, in particular the use of large cardinals and of 'worlds' obtained through forcing to 'represent' alternative extensions of ZFC. In the second part, we address the existence and the possible features of the core of MV_T (where T is ZFC+Large Cardinals). In the last part, we discuss the hypothesis that the core is Ultimate-L, and examine whether and how, based on …

依赖选择、恰当性与泛型绝对性

Dependent Choice, Properness, and Generic Absoluteness

本文证明依赖选择原理足以发展恰当力迫理论，并在大基数下保证Chang模型的泛型绝对性，从而确立ZF+DC作为经典数学和集合论重要部分的自然基础。

$Σ_n$-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。

Combinatorial dichotomies and cardinal invariants

Assuming the P-ideal dichotomy, we attempt to isolate those cardinal characteristics of the continuum that are correlated with two well-known consequences of the proper forcing axiom. We find a cardinal invariant $\mathfrak{x}$ such that the statement that $\mathfrak{x} > ω_{1}$ is equivalent to the statement that 1, $ω$, $ω_{1}$, $ω\times ω_{1}$, and ${\left[ω_{1}\right]}^{< ω}$ are the only cofinal types of directed sets of size at most ${\aleph}_{1}$. We investigate the corresponding problem for the partition relation $ω_{1} \rightarrow (ω_{1}, α)^2$ for all $α< ω_{1}$. To this effect, we i…

Alternative Cichoń Diagrams and Forcing Axioms Compatible with CH

This dissertation surveys several topics in the general areas of iterated forcing, infinite combinatorics and set theory of the reals. There are two parts. In the first half I consider alternative versions of the Cichoń diagram. First I show that for a wide variety of reduction concepts there is a Cichoń diagram for effective cardinal characteristics relativized to that reduction. As an application I investigate in detail the Cichoń diagram for degrees of constructibility relative to a fixed inner model of ZFC. Then I study generalizations of cardinal characteristics to the space of functions …

开着色公理

The Open Coloring Axiom

本文系统阐释了Todorcěvić提出的开着色公理，给出其若干应用，并通过一个技术性引理证明该公理与ZFC相对一致，同时证明它蕴含b=ℵ₂从而与连续统假设不相容。

孪生模型：被迫同构的非同构模型。第一部分

Twins: non-isomorphic models forced to be isomorphic. Part I

本文研究哪些完备理论存在非同构模型在力迫扩张后变得同构，给出具有独立性质的理论在力迫不添加新ω-序列时的充分条件，并重点讨论向树添加新分支的力迫偏序，属于分类纲领与Main Gap纲领的一部分。

Sacks模型中的Menger集与consonant集

Menger and consonant sets in the Sacks model

本文通过迭代Sacks力迫和拓扑博弈，证明Cantor立方体中基数为连续统的全不完美Menger集的存在性独立于ZFC，并分析了Sacks模型中Hurewicz与consonant子集的结构。

自然性与可定义性 III

Naturality and Definability III

本文探究范畴论自然性与模型论可定义性的关联，证明温和条件下自然性蕴含可定义性，构造出满足可一致化构造均为弱自然的ZFC传递模型，还证明公式表示的自然构造可无额外参数一致可定义，回答了Hodges与Shelah的相关问题。

论不可破坏的强猜测模型

On Indestructible Strongly Guessing Models

本文提出了GM+原则的强化版本SGM+，并在超紧基数假设下证明了其与ZFC的一致性。该原则连同温和的基数算术假设表明，任何增加ω₂新子集的力迫必然增加实数或崩塌基数，部分回答了Abraham问题并扩展了Todorčević的结果。

演绎数学的基础

A foundation for deductive mathematics

本文指出了ZFC公理体系在标准数学实践中的不一致性，通过引入“相容极限公理”构建了与极大模型对应的集合论基础，证明主流实践隐含于该理论中，并指出过去百年的公理集合论与标准实践无关。