Trevor M. Wilson
W.H. Woodin showed that if $κ_1 < \cdots < κ_n$ are strong cardinals then two-step ${\bfΣ}^1_{n+3}$ generic absoluteness holds after collapsing $2^{2^{κ_n}}$ to be countable. We show that this number can be reduced to $2^{κ_n}$, and to $κ_n^+$ in the case $n = 1$, but cannot be further reduced to $κ_n$.
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We isolate a new large cardinal concept, "remarkability." Consistencywise, remarkable cardinals are between ineffable and omega-Erdos cardinals. They are characterized by the existence of "0^sharp-like" embeddings; however, they relativize down to L. It turns out that the existence of a remarkable cardinal is equiconsistent with L(R) absoluteness under proper forcings. In particular, said absoluteness does not imply Pi^1_1 determinacy.
Terminal Absoluteness of Collapse Forcings
本文探讨了坍缩forcing在项目性通用绝对性中的必要性，证明在大基数假设下，坍缩forcing的绝对性等价于任意forcing的绝对性，并指出在低项目性层次下结果在ZFC中成立，揭示了坍缩forcing的终端性。
We investigate properties of trees of height $ω_1$ and their preservation under subcomplete forcing. We show that subcomplete forcing cannot add a new branch to an $ω_1$-tree. We introduce fragments of subcompleteness which are preserved by subcomplete forcing, and use these in order to show that certain strong forms of rigidity of Suslin trees are preserved by subcomplete forcings. Finally, we explore under what circumstances subcomplete forcing preserves Aronszajn trees of height and width $ω_1$. We show that this is the case if CH fails, and if CH holds, then this is the case iff the bounde…
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…
We prove that the theory of the models constructible using finitely many cofinality quantifiers - $C_{λ_{1},...,λ_{n}}^{*}$ and $C_{<λ_{1},...,<λ_{n}}^{*}$ for $λ_{1},...,λ_{n}$ regular cardinals - is set-forcing absolute under the assumption of class many Woodin cardinals, and is independent of the regular cardinals used. Towards this goal we prove some properties of the generic embedding induced from the stationary tower restricted to $<μ$-closed sets.
The resurrection axioms are forcing axioms introduced recently by Hamkins and Johnstone, developing on ideas of Chalons and Velicković. We introduce a stronger form of resurrection axioms (the \emph{iterated} resurrection axioms $\textrm{RA}_α(Γ)$ for a class of forcings $Γ$ and a given ordinal $α$), and show that $\textrm{RA}_ω(Γ)$ implies generic absoluteness for the first-order theory of $H_{γ^+}$ with respect to forcings in $Γ$ preserving the axiom, where $γ=γ_Γ$ is a cardinal which depends on $Γ$ ($γ_Γ=ω_1$ if $Γ$ is any among the classes of countably closed, proper, semiproper, stationar…
We introduce a category whose objects are stationary set preserving complete boolean algebras and whose arrows are complete homomorphisms with a stationary set preserving quotient. We show that the cut of this category at a rank initial segment of the universe of height a super compact which is a limit of super compact cardinals is a stationary set preserving partial order which forces $MM^{++}$ and collapses its size to become the second uncountable cardinal. Next we argue that any of the known methods to produce a model of $MM^{++}$ collapsing a superhuge cardinal to become the second uncoun…
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。
We present a self-contained account of Woodin's extender algebra and its use in proving absoluteness results, including a proof of the $Σ^2_1$-absoluteness theorem. We also include a proof that the existence of an inner model with Woodin limit of Woodin cardinals implies the existence of divergent models of $\AD^+$.
We study the relationship between Amoeba forcing (the partial order which generically adds a measure one set of random reals) and projective measurability. Given a universe V of set theory and a forcing notion P in V we say that V is Sigma^1_n - P - absolute iff for every Sigma^1_n-sentence phi with parameters in V we have V models phi iff V^P models phi. We show that Sigma^1_4-Amoeba-absoluteness implies that forall a in omega^omega (omega_1^{L[a]} < omega_1^V), and hence Sigma^1_3-measurability. This answers a question of Haim Judah (private communication).
In [5], Hjorth proved that for every countable ordinal $α$, there exists a complete $\mathcal{L}_{ω_1,ω}$-sentence $ϕ_α$ that has models of all cardinalities less than or equal to $\aleph_α$, but no models of cardinality $\aleph_{α+1}$. Unfortunately, his solution does not yield a single $\mathcal{L}_{ω_1,ω}$-sentence $ϕ_α$, but a set of $\mathcal{L}_{ω_1,ω}$-sentences, one of which is guaranteed to work. It was conjectured in [9] that it is independent of the axioms of ZFC which of these sentences has the desired property. In the present paper, we prove that this conjecture is true. More spec…
We show that for $Π_2$-properties of second or third order arithmetic as formalized in appropriate natural signatures the apparently weaker notion of forcibility overlaps with the standard notion of consistency (assuming large cardinal axioms). Among such $Π_2$-properties we mention: the negation of the Continuum hypothesis, Souslin Hypothesis, the negation of Whitehead's conjecture on free groups, the non-existence of outer automorphisms for the Calkin algebra, etc... In particular this gives an a posteriori explanation of the success forcing (and forcing axioms) met in producing models of su…
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。
On $(Σ^2_1)^{uB}$ Absoluteness Between V and HOD
本文结合 Woodin 的 Σ²₁ 基定理与 Vopěnka 定理，证明在存在真类多 Woodin 基数时，V 中为真的 (Σ²₁)^uB 命题在 HOD 中也为真，并探讨了反向绝对性及其与 Ultimate L 公理的关系。
$Σ^1_3$-absoluteness for ccc forcing means that for any ccc forcing $P$, ${H_{ω_1}}^V \prec_{Σ_2}{H_{ω_1}}^{V^P}$. "$ω_1$ inaccessible to reals" means that for any real $r$, ${ω_1}^{L[r]}<ω_1$. To measure the exact consistency strength of "$Σ^1_3$-absoluteness for ccc forcing and $ω_1$ is inaccessible to reals", we introduce a weak version of a weakly compact cardinal, namely, a (lightface) $Σ^1_2$-indescribable cardinal; $κ$ has this property exactly if it is inaccessible and $H_κ\prec_{Σ_2} H_{κ^+}$.
We introduce bounded category forcing axioms for well-behaved classes $Γ$. These are strong forms of bounded forcing axioms which completely decide the theory of some initial segment of the universe $H_{λ_Γ^+}$ modulo forcing in $Γ$, for some cardinal $λ_Γ$ naturally associated to $Γ$. These axioms naturally extend projective absoluteness for arbitrary set-forcing--in this situation $λ_Γ=ω$--to classes $Γ$ with $λ_Γ>ω$. Unlike projective absoluteness, these higher bounded category forcing axioms do not follow from large cardinal axioms, but can be forced under mild large cardinal assumptions o…
The exact strength of generic absoluteness for the universally Baire sets
本文研究了泛Baire集的泛绝对性条件$\sf{Sealing}$的精确一致性强度，利用Woodin的$\sf{Core\\ Model\\ Induction}$技术证明其与$\sf{LSA-over-uB}$等一致，并分离出对应的精确大基数理论。
Assuming that there is no inner model with a strong cardinal, the following is shown: any subset of ω_1 can be made Δ^1_3 (in the codes) by a reasonable set-forcing; there is a reasonable set-generic extension with a Δ^1_3 well-ordering of its reals; 2-step Σ^1_4 absoluteness fails w.r.t. set-sized reasonable forcings.
Aronszajn tree preservation and bounded forcing axioms
本文研究了力迫类中三个反射原理层级（有界力迫公理、Σ¹₁-绝对性和Aronszajn树保持）之间的关系，证明了对于不添加实数的力迫类，在2^ω层级上三者等价，并特别关注子完全力迫的子类。
We prove that various classical tree forcings -- for instance Sacks forcing, Mathias forcing, Laver forcing, Miller forcing and Silver forcing -- preserve the statement that every real has a sharp and hence analytic determinacy. We then lift this result via methods of inner model theory to obtain level-by-level preservation of projective determinacy (PD). Assuming PD, we further prove that projective generic absoluteness holds and no new equivalence classes classes are added to thin projective transitive relations by these forcings.
Generic Absoluteness Revisited
本文研究递归公理与拉弗泛型大基数公理的关系，证明从特定递归公理片段和紧拉弗泛型巨大基数存在性可推出更一般的泛型绝对性结论，且相关假设在多数常见偏序集类下蕴含地基公理的否定，区分了两类公理的对应片段。
Arithmetical Complexity and Absoluteness of Rigidity Phenomena for Ulam Sequences
本文研究Ulam序列的算术复杂性与绝对刚性现象，证明相关陈述在算术层级中处于低层，并在不同模型间具有绝对性。
We present several results relating the general theory of the stationary tower forcing developed by Woodin with forcing axioms. The main results is that the forcing axiom MM^{++} (also known as MM^{+ω_1}) decides the Π_2-theory of H_{ω_2} with respect to stationary set preserving forcings. We argue that this is a close to optimal generalization to H_{ω_2} of Woodin's absoluteness results for L(R).
Despite being an established notion in the large cardinal hierarchy, results about Woodin cardinals are sparse in the literature. Here we gather known results about the preservation of Woodin cardinals under certain forcing extensions, as well as giving a template for preserving Woodin cardinals through forcing. Using this template, we form an indestructibility result under certain Easton iterations.
Towards a generic absoluteness theorem for Chang models
本文引入新方法建立含所有普遍贝尔实数集的模型的泛型绝对性结果，基于超紧基数等假设证明相关导出模型性质，还得出可定义幂集等结构的理论无法被力迫改变的结论。
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.
We give a brief survey on the interplay between forcing axioms and various other non-constructive principles widely used in many fields of abstract mathematics, such as the axiom of choice and Baire's category theorem. First of all we outline how, using basic partial order theory, it is possible to reformulate the axiom of choice, Baire's category theorem, and many large cardinal axioms as specific instances of forcing axioms. We then address forcing axioms with a model-theoretic perspective and outline a deep analogy existing between the standard Łoś Theorem for ultraproducts of first order s…
The landmark Levy-Solovay Theorem limits the kind of large cardinal embeddings that can exist in a small forcing extension. Here I announce a generalization of this theorem to a broad new class of forcing notions. One consequence is that many of the forcing iterations most commonly found in the large cardinal literature create no new weakly compact cardinals, measurable cardinals, strong cardinals, Woodin cardinals, strongly compact cardinals, supercompact cardinals, almost huge cardinals, huge cardinals, and so on.
We show that the Proper Forcing Axiom implies the Singular Cardinal Hypothesis. The proof is by interpolation and uses the Mapping Reflection Principle.
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.
We show that some of the most prominent large cardinal notions can be characterized through the validity of certain combinatorial principles at $ω_2$ in forcing extensions by the pure side condition forcing introduced by Neeman. The combinatorial properties that we make use of are natural principles, and in particular for inaccessible cardinals, these principles are equivalent to their corresponding large cardinal properties. Our characterizations make use of the concepts of internal large cardinals introduced in this paper, and of the classical concept of generic elementary embeddings.
We build a supercompact version of the forcing defined in \cite{gitik2019}. For each singular cardinal in the ground model with any fixed cofinality, which is a limit of supercompact cardinals, it is possible to force so that the size of the powerset of the singular cardinal is arbitrarily large, while preserving the singular cardinal. An important feature of this forcing is that it is possible to define the forcing so that the successor of the singular cardinal is collapsed, but all the cardinals above it are preserved.
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…
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…
Recently the second author introduced combinatorial principles that characterize supercompactness for inaccessible cardinals but can also hold true for small cardinals. We prove that the proper forcing axiom PFA implies these principles hold for $ω_2$. Using this, we argue to show that any of the known methods for forcing models of PFA from a large cardinal assumption requires a strongly compact cardinal. If one forces PFA using a proper forcing, then we get the optimal result that a supercompact cardinal is necessary.
A ccc-generically supercompact cardinal $κ$ can be smaller than or equal to the continuum. On the other hand, such a cardinal $κ$ still satisfies diverse largeness properties, like that it is a stationary limit of ccc-generically measurable cardinals (Theorem 4.1). This is in a strong contrast to $\cal P$-generically supercompact cardinals for the class $\cal P$ of all $σ$-closed posets, which can be $\aleph_n$ for any n>1.
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 …
This paper presents the main results in my Ph.D. thesis. In what follows several proofs of SCH are presented introducing a family of covering properties which implies both SCH and the failure of various forms of square. These covering properties are also applied to investigate models of strongly compact cardinals or of strong forcing axioms like MM or PFA.
The purpose of this paper is to present some results which suggest that the Singular Cardinals Hypothesis follows from the Proper Forcing Axiom. What will be proved is that a form of simultaneous reflection follows from the Set Mapping Reflection Principle, a consequence of PFA. While the results fall short of showing that MRP implies SCH, it will be shown that MRP implies that if SCH fails first at kappa then every stationary subset of S_{kappa^+}^omega = {a < kappa^+ : cf(a) = omega} reflects. It will also be demonstrated that MRP always fails in a generic extension by Prikry forcing.
The lottery preparation, a new general kind of Laver preparation, works uniformly with supercompact cardinals, strongly compact cardinals, strong cardinals, measurable cardinals, or what have you. And like the Laver preparation, the lottery preparation makes these cardinals indestructible by various kinds of further forcing. A supercompact cardinal kappa, for example, becomes fully indestructible by kappa-directed closed forcing; a strong cardinal kappa becomes indestructible by less-than-or-equal-kappa-strategically closed forcing; and a strongly compact cardinal kappa becomes indestructible …
We define extender sequences, generelizing measure sequences from Radin Forcing. Using the extender sequences we combine Gitik-Magidor forcing for adding many Prikry sequences with Radin forcing. This forcing satisfies Prikry like condition, destroys no cardinals, and has a kind of properness. Depending on the large cardinal we start with, this forcing can blow the power of a cardinal together with changing its' cofinality to a prescribe value. If this prescribed value is the cardinal itself then it remains regular and even can remain measurable.
This is an introductory paper to a series of results linking generic absoluteness results for second and third order number theory to the model theoretic notion of model companionship. Specifically we develop here a general framework linking Woodin's generic absoluteness results for second order number theory and the theory of universally Baire sets to model companionship and show that (with the required care in details) a $Π_2$-property formalized in an appropriate language for second order number theory is forcible from some $T\supseteq\mathsf{ZFC}+$large cardinals if and only if it is consi…
On the strength of ultrafilters above choiceless large cardinals and their Prikry forcings
本文研究无选择大基数上超滤子强度与普利克里力迫，证明超滤子的大基数强度界，拓展普利克里力迫理论并限制可坍缩/奇异化基数，提出张量普利克里系统以构造无选择集合论的一致性结果（如所有不可数基数奇异的模型）。
Let $D$ be an infinite discrete set of measurable cardinals. It is shown that generalized Prikry forcing to add a countable sequence to each cardinal in $D$ is subcomplete. To do this it is shown that a simplified version of generalized Prikry forcing which adds a point below each cardinal in $D$, called generalized diagonal Prikry forcing, is subcomplete. Moreover, the generalized diagonal Prikry forcing associated to $D$ is subcomplete above $μ$, where $μ$ is any regular cardinal below the first limit point of $D$.
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.
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。
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。
The Necessary Maximality Principle for c.c.c. forcing asserts that any statement about a real in a c.c.c. extension that could become true in a further c.c.c. extension and remain true in all subsequent c.c.c. extensions, is already true in the minimal extension containing the real. We show that this principle is equiconsistent with the existence of a weakly compact cardinal.
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.
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。
Suppose $κ$ is a singular strong limit cardinal of countable cofinality and let $\langle κ_{n}: n<ω\rangle$ be an incrasing sequence of regular cardinals cofinal in $κ$. We show that if $cf(2^κ)= κ^+$, then forcing with the full product $\prod_{n<ω}Add(κ_n,1)$ collapses $2^κ$ into $κ^+$. This result gives a consistent positive answer to a question of Sy Friedman. We also give a new proof of a result due to Shelah by showing that if the sequence carries a scale of length $κ^+,$ then forcing with $\prod_{n<ω}Add(κ_n,1)$ adds a generic filter for $Add(κ^+, 1)$, and indeed \[ \prod_{n<ω}Add(κ_n,1)…
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.
Answering a question of Usuba, we show that an extendible cardinal can be preserved by a set forcing that is not a small forcing.
This dissertation includes many theorems which show how to change large cardinal properties with forcing. I consider in detail the degrees of inaccessible cardinals (an analogue of the classical degrees of Mahlo cardinals) and provide new large cardinal definitions for degrees of inaccessible cardinals extending the hyper-inaccessible hierarchy. I showed that for every cardinal $κ$, and ordinal $α$, if $κ$ is $α$-inaccerssible, then there is a $\mathbb{P}$ forcing that $κ$ which preserves that $α$-inaccessible but destorys that $κ$ is $(α+1)$-inaccessible. I also consider Mahlo cardinals and d…
We construct a variety of inner models exhibiting features usually obtained by forcing over universes with large cardinals. For example, if there is a supercompact cardinal, then there is an inner model with a Laver indestructible supercompact cardinal. If there is a supercompact cardinal, then there is an inner model with a supercompact cardinal κfor which 2^κ=κ^+, another for which 2^κ=κ^++ and another in which the least strongly compact cardinal is supercompact. If there is a strongly compact cardinal, then there is an inner model with a strongly compact cardinal, for which the measurable c…
On forcing axioms and weakenings of the Axiom of Choice
本文证明H_κ与DC_κ等选择公理弱化形式的力迫公理等价物，并借助针对特定稠密集族的新模板给出选择公理的新表述。
Canonical reconstruction and forcing absoluteness of standard structures
本文通过分离一个简单的保持原则，研究代数或拓扑代数结构具有标准形式 F(X) 的力迫绝对性，并应用于对称群、全变换幺半群等，回答 Noah Schweber 提出的问题。
It is known that the set of possible cofinalities $\mathrm{pcf}(A)$ has good properties if $A$ is a progressive interval of regular cardinals. In this paper, we give an interval of regular cardinals $A$ such that $\mathrm{pcf}(A)$ has no good properties in the presense of a measurable cardinal, or in generic extensions by Prikry-type forcing.
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…
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.
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