Jörg Brendle, Benedikt Loewe
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.
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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.
Independence relations in the Solovay model I
本文提出Solovay模型的新几何公理化，用于简洁无力迫地证明经典结果，并纯几何发展平衡力迫理论。
Independence relations in the Solovay model II
本文给出了Solovay模型的新公理化，并提出了平衡力迫理论的纯几何处理。
We prove that Solovay's set $Σ$ is generic over the ground model via a forcing notion whose order relation $\subseteq$-extends the given order relation.
What are the most general principles in set theory relating forceability and truth? As with Solovay's celebrated analysis of provability, both this question and its answer are naturally formulated with modal logic. We aim to do for forceability what Solovay did for provability. A set theoretical assertion psi is forceable or possible, if psi holds in some forcing extension, and necessary, if psi holds in all forcing extensions. In this forcing interpretation of modal logic, we establish that if ZFC is consistent, then the ZFC-provable principles of forcing are exactly those in the modal theory…
A $Σ$-construction of Solovay is extended to the case of intermediate sets which are not necessarily subsets of the ground model, with a more transparent description of the resulting forcing notion than in the classical paper of Grigorieff. As an application, we prove that, for a given name $t$ (not necessarily a name of a subset of the ground model), the set of all sets of the form $t[G]$ (the $G$-interpretation of $t$), $G$ being generic over the ground model, is Borel. This result was first established by Zapletal by a descriptive set theoretic argument.
Solovay's random-real forcing (1971) is the standard way of producing real-valued measurable cardinals. Following questions of Fremlin, by giving a new construction, we show that there are combinatorial, measure-theoretic properties of Solovay's model that do not follow from the existence of real-valued measurability.
We regard Forcing Notions P adding real numbers and the algebras of P-measurable sets. As for Cohen- and Random-Forcing we can show that each analytic set is P-measurable using Solovay's Unfolding Trick for infinite games. To show this we develop the notion of strategic fusion and give a proof of a uniform unfolding theorem which allows us to reduce the proof of analytic measurability to the definition of a certain function.
The Modal Logic of Provability and Forcing
本文研究可证性与力迫在模态逻辑中的交互，提出双模态逻辑 PF 及 PF^ω，并分别刻画一般情形与集合论 ω-模型中的对应逻辑。
Many of the most common reverse Easton iterations found in the large cardinal context, such as the Laver preparation, admit a gap at some small delta in the sense that they factor as P*Q, where P has size less than delta and Q is forced to be delta-strategically closed. In this paper, generalizing the Levy-Solovay theorem, I show that after such forcing, every embedding j:V[G]-->M[j(G)] in the extension which satisfies a mild closure condition is the lift of an embedding j:V-->M in the ground model. In particular, every ultrapower embedding in the extension lifts an embedding from the ground m…
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。
We give a necessary and sufficient condition for an atomless Boolean algebra to be countably generated, and use it to give new proofs of some some know facts due to Gaifman-Hales and Solovay and also due to Jech, Kunen and Magidor. We also show that Jensen's coding theorem can be used to provide cardinal preserving countably generated complete Boolean algebras of arbitrary large cardinality. This answers a question of Jech [5] from 1976.
We apply a general approach for distributions of binary isolating and semi-isolating formulas to the class of strongly minimal theories.
For suitable groups G we will show that one can add a Boolean algebra B by forcing in such a way that Aut(B) is almost isomorphic to G. In particular, we will give a positive answer to the following question due to J.Roitman: Is aleph_{omega} a possible number of automorphisms of a rich Boolean algebra?
We introduce the notion of implicative algebra, a simple algebraic structure intended to factorize the model constructions underlying forcing and realizability (both in intuitionistic and classical logic). The salient feature of this structure is that its elements can be seen both as truth values and as (generalized) realizers, thus blurring the frontier between proofs and types. We show that each implicative algebra induces a (Set-based) tripos, using a construction that is reminiscent from the construction of a realizability tripos from a partial combinatory algebra. Relating this constructi…
We study sheaves in the context of a duality theory for lattice structure endowed with extra operations, and in the context of forcing in a topos. Using Sheaf duality theory of Comer for cylindric algebras, we give a representation theorem of of distributive bounded lattices expanded by modalities (functions distributing over joins) as the continuous sections of sheaves. Our representation is defined via a contravariant functor from an algebraic category to a category of sheaves. We show that if our category is a small site (cartesian closed with a stability condition on pullbacks), then we ca…
These notes present a compact and self-contained approach to iterated forcing with a particular emphasis on semiproper forcing. We tried to make our presentation accessible to any scholar who has some familiarity with forcing and boolean valued models and full details of all proofs are given. We focus our presentation using the boolean algebra language and defining an iteration system as a directed and commutative system of complete and injective homomorphisms between complete and atomless boolean algebras. It is well known that the boolean algebra approach to forcing and iterations is fully e…
We study the interplay between properties of measures on a Boolean algebra A and forcing names for ultrafilters on A. We show that several well known measure theoretic properties of Boolean algebras (such as supporting a strictly positive measure or carrying only separable measures) have quite natural characterizations in the forcing language. We show some applications of this approach. In particular, we reprove a theorem of Kunen saying that in the classical random model there are no towers of height $ω_2$.
Let $M$ be a transitive model of $ZFC$ and let ${\bf B}$ be a $M$-complete Boolean algebra in $M.$ (In general a proper class.) We define a generalized notion of forcing with such Boolean algebras, $^*$forcing. (A $^*$ forcing extension of $M$ is a transitive set of the form $M[{\bf G}]$ where ${\bf G}$ is an $M$-complete ultrafilter on ${\bf B}.$) We prove that 1. If ${\bf G}$ is a $^*$forcing complete ultrafilter on ${\bf B},$ then $M[{\bf G}]\models ZFC.$ 2. Let $H\sub M.$ If there is a least transitive model $N$ such that $H\in M,$ $Ord^M=Ord^N,$ and $N\models ZFC,$ then we denote $N$ by $…
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 prove rigidity results for large classes of corona algebras, assuming the Proper Forcing Axiom. In particular, we prove that a conjecture of Coskey and Farah holds for all separable $C^*$-algebras with the metric approximation property and an increasing approximate identity of projections.
The measure algebra adding $θ$-many random reals is $θ$-FAM-linked
本文利用概率论工具改进了 Shelah 的证明，证明了具有满足 $θ$-density 性质的严格正概率测度的完全布尔代数是 $θ$-FAM-linked 的，从而得出添加 $θ$ 个随机实数的测度代数是该力迫概念的新实例。
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…
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。
If $A_n$ is a sequence of C*-algebras, then the C*-algebra $\prod A_n / \bigoplus A_n$ is called a reduced product. We prove, assuming Todorcevic's Axiom and Martin's Axiom, that every isomorphism between two reduced products of separable, unital UHF algebras must be definable in a strong sense. As a corollary we deduce that two such reduced products $\prod A_n / \bigoplus A_n$ and $\prod B_n / \bigoplus B_n$ are isomorphic if and only if, up to an almost-permutation of $\mathbb{N}$, $A_n$ is isomorphic to $B_n$.
The Internal Modal Logic of Forcing
该研究通过为模态公式构建内部克里普克语义，将模态集合论与布尔值模型相连，计算模态有效性并分析特殊代数行为，最终证明在特定语义下正规逻辑\(\KTB\)是相关模型中有效模态公式集。
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].
Forcing with random variables in bounded arithmetics and set theory
本文研究了在有界算术和集合论中使用随机变量进行强制的方法，探讨了强制$B_{M,Ω}$在模型扩展中的作用及与基础模型整数的相互密度。
Iwasa investigated the preservation of various covering properties of opological spaces under Cohen forcing. By improving the argument in Iwasa's paper, we prove that the Rothberger property, the Menger property and selective screenability are also preserved under Cohen forcing and forcing with the measure algebra.
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…
The present paper addresses the problem of attainment of the supremums in various equivalent definitions of hereditary density hd and hereditary Lindelof degree hL of Boolean algebras. We partially answer two problems of J. Donald Monk (Problems 50 and 54 in his book "Cardinal Invariants on Boolean Algebras), showing consistency of different attainment behaviour and proving that (for the considered variants) this is the best result we can expect.
We introduce weakenings of two of the more prominent open problems in the classification of $\mathrm{C}^*$-algebras, namely the quasidiagonality problem and the UCT problem. We show that the a positive solution of the conjunction of the two weaker problems implies a positive solution of the original quasidiagonality problem as well as allows us to give a local, finitary criteria for the MF problem, which asks whether every stably finite $\mathrm{C}^*$-algebra is MF.
Kripke-Joyal forcing for type theory and uniform fibrations
本文提出将 Kripke-Joyal 力迫推广到类型论的新方法，用以连接预层范畴中的代数结构与内部类型论判断，并应用于刻画同伦类型论中的代数弱分解系统。
It was realized early on that topologies can model constructive systems, as the open sets form a Heyting algebra. After the development of forcing, in the form of Boolean-valued models, it became clear that, just as over ZF any Boolean-valued model also satisfies ZF, so do Heyting-valued models satisfy IZF, which stands for Intuitionistic ZF, the most direct constructive re-working of the ZF axioms. In this paper, we return to topologies, and introduce a variant model, along with a correspondingly revised forcing or satisfaction relation. The purpose is to prove independence results related to…
We prove that if $\mathcal{A}$ is a $σ$-complete Boolean algebra in a model $V$ of set theory and $\mathbb{P}\in V$ is a proper forcing with the Laver property preserving the ground model reals non-meager, then every pointwise convergent sequence of measures on $\mathcal{A}$ in a $\mathbb{P}$-generic extension $V[G]$ is weakly convergent, i.e. $\mathcal{A}$ has the Vitali--Hahn--Saks property in $V[G]$. This yields a consistent example of a whole class of infinite Boolean algebras with this property and of cardinality strictly smaller than the dominating number $\mathfrak{d}$. We also obtain a…
Our original aim was, in Abelian group theory to prove the consistency of: lambda is strong limit singular and for some properties of abelian groups which are relatives of being free, the compactness in singular fails. In fact this should work for R-modules, etc. As in earlier cases part of the work is analyzing how to move between the set theory and the algebra. Set theoretically we try to force a universe which satisfies G.C.H. and diamond holds for many stationary sets but, for every regular uncountable lambda, in some sense anything which "may" hold for some stationary set, does hold for s…
Let mu be singular of uncountable cofinality. If mu>2^{cf(mu)}, we prove that in P=([mu]^mu,supseteq) as a forcing notion we have a natural complete embedding of Levy(aleph_0, mu^+) (so P collapses mu^+ to aleph_0) and even Levy(aleph_0, U_{J^{bd}_kappa}(mu)) . The ``natural'' means that the forcing ({p in [mu]^mu :p closed}, supseteq) is naturally embedded and is equivalent to the Levy algebra. If mu <2^{cf(mu)} we have weaker results.
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…
This is an introduction to the set-theoretic method of forcing, including its application in proving the independence of the Continuum Hypothesis from the Zermelo-Fraenkel axioms of set theory. I presuppose no particular mathematical background beyond some familiarity with set theory and mathematical logic - in particular, no algebra is presupposed, though it can be useful. The goal is to have a document that makes this material accessible to mathematics graduate students in all fields, and to philosophers with an interest in set theory and mathematical logic but no other mathematical backgrou…
The hidden-variables premise is shown to be equivalent to the existence of generic filters for algebras of commuting propositions and for certain more general propositional systems. The significance of this equivalence is interpreted in light of the theory of generic filters and boolean-valued models in set theory (the method of forcing). The apparent stochastic nature of quantum observation is derived for these hidden-variables models.
Every $\mathrm{C}^*$-algebra, regardless of its density character, can be embedded into the Calkin algebra in a forcing extension of the universe obtained without collapsing any cardinal.
Let $(R, \sim )$ be the Rado graph, $Emb (R)$ the monoid of its self-embeddings, $Π(R)=\{ f[R]: f\in Emb (R)\}$ the set of copies of $R$ contained in $R$, and ${\mathcal I}_R$ the ideal of subsets of $R$ which do not contain a copy of $R$. We consider the poset $( Π(R ), \subset )$, the algebra $P (R)/{\mathcal I _R}$, and the inverse of the right Green's pre-order on $Emb (R)$, and show that these pre-orders are forcing equivalent to a two step iteration of the form $P \ast π$, where the poset $P$ is similar to the Sacks perfect set forcing: adds a generic real, has the $\aleph _0$-covering p…
Boolean-valued models of set theory were independently introduced by Scott, Solovay and Vopěnka in 1965, offering a natural and rich alternative for describing forcing. The original method was adapted by Takeuti, Titani, Kozawa and Ozawa to lattice-valued models of set theory. After this, Löwe and Tarafder proposed a class of algebras based on a certain kind of implication which satisfy several axioms of ZF. From this class, they found a specific 3-valued model called PS3 which satisfies all the axioms of ZF, and can be expanded with a paraconsistent negation *, thus obtaining a paraconsistent…
We compare the forcing related properties of a complete Boolean algebra B with the properties of the convergences $λ_s$ (the algebraic convergence) and $λ_{ls}$ on B generalizing the convergence on the Cantor and Aleksandrov cube respectively. In particular we show that $λ_{ls}$ is a topological convergence iff forcing by B does not produce new reals and that $λ_{ls}$ is weakly topological if B satisfies condition $(\hbar)$ (implied by the ${\mathfrak t}$-cc). On the other hand, if $λ_{ls}$ is a weakly topological convergence, then B is a $2^{\mathfrak h}$-cc algebra or in some generic extensi…
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。
Given a nonunital $\mathrm{C}^*$-algebra $A$ one constructs its corona algebra $\mathcal M(A)/A$. This is the noncommutative analog of the Čech-Stone remainder of a topological space. We analyze the two faces of these algebras: the first one is given assuming CH, and the other one arises when Forcing Axioms are assumed. In their first face, corona $\mathrm{C}^*$-algebras have a large group of automorphisms that includes nondefinable ones. The second face is the Forcing Axiom one; here the automorphism group of a corona $\mathrm{C}^*$-algebra is as rigid as possible, including only definable el…
This is a survey on the amalgamated limit, a limit construction for complete Boolean algebras in iterated forcing theory, which generalizes both the direct limit and the two-step amalgamation. We focus in particular on examples of the amalgamated limit from the literature and on the topological amalgamated limit for compact Hausdorff spaces.
The Proper Forcing Axiom implies all automorphisms of every Calkin algebra associated with an infinite-dimensional complex Hilbert space and the ideal of compact operators are inner. As a means of the proof we introduce the notion of Polish $ω_1$-trees and cohrerent families of Polish spaces and prove some uniformization results.
We use the technique of "classical realizability" to build new models of ZF + DC in which R is not well ordered. This gives new relative consistency results, probably not obtainable by forcing. This gives also a new method to get programs from proofs of arithmetical formulas with dependent choice.
Boolean ultrapowers extend the classical ultrapower construction to work with ultrafilters on any complete Boolean algebra, rather than only on a power set algebra. When they are well-founded, the associated Boolean ultrapower embeddings exhibit a large cardinal nature, and the Boolean ultrapower construction thereby unifies two central themes of set theory---forcing and large cardinals---by revealing them to be two facets of a single underlying construction, the Boolean ultrapower.
We prove that the statement `For all Borel ideals I and J on $ω$, every isomorphism between Boolean algebras $P(ω)/I$ and $P(ω)/J$ has a continuous representation' is relatively consistent with ZFC. In this model every isomorphism between $P(ω)/I$ and any other quotient $P(ω)/J$ over a Borel ideal is trivial for a number of Borel ideals I on $ω$. We can also assure that the dominating number is equal to $\aleph_1$ and that $2^{\aleph_1}>2^{\aleph_0}$. Therefore the Calkin algebra has outer automorphisms while all automorphisms of $P(ω)/Fin$ are trivial. Proofs rely on delicate analysis of name…
Using the proof-program (Curry-Howard) correspondence, we give a new method to obtain models of ZF and relative consistency results in set theory. We show the relative consistency of ZF + DC + there exists a sequence of subsets of R the cardinals of which are strictly decreasing + other similar properties of R. These results seem not to have been previously obtained by forcing.
We propose an extension of Aczel's constructive set theory CZF by an axiom for inductive types and a choice principle, and show that this extension has the following properties: it is interpretable in Martin-Lof's type theory (hence acceptable from a constructive and generalised-predicative standpoint). In addition, it is strong enough to prove the Set Compactness Theorem and the results in formal topology which make use of this theorem. Moreover, it is stable under the standard constructions from algebraic set theory, namely exact completion, realizability models, forcing as well as more gene…
Extending Martín Escardó's effectful forcing technique, we give a new proof of a well-known result: Brouwer's monotone bar theorem holds for any bar that can be realized by a functional of type $(\mathbb{N} \to \mathbb{N}) \to \mathbb{N}$ in Gödel's System T. Effectful forcing is an elementary alternative to standard sheaf-theoretic forcing arguments, using ideas from programming languages, including computational effects, monads, the algebra interpretation of call-by-name $λ$-calculus, and logical relations. Our argument proceeds by interpreting System T programs as well-founded dialogue tree…
Solution to Dixmier's Problem about spectra of C*-algebras
本文解决了Dixmier关于单C*-代数谱的1967年问题，证明了在某些单C*-代数（如CAR代数与SL_3(Z)的约化交叉积）之间不存在Borel可定义的单射，并推广到可数amenable群与自由群的酉对偶，同时给出核非I型代数谱的Borel可定义双射存在性。
We construct a measure on omega-one^2 over the ground model in the forcing extension of a measure algebra, and investigate when measure theoretic properties of some measurable colouring of omega-one^2 imply the existence of an uncountable subset of omega-one whose square is homogeneous. This gives a new proof of the fact that, under a suitable axiomatic assumption, there are no Souslin (omega-one,omega-one) gaps in the Boolean algebra L^0(nu)/Fin when nu is a separable measure.
In this article, we consider the notion of almost irredundant sets: A subset $\mathcal{X}$ of a C*-algebra $\mathcal{A}$ is called almost irredundant if and only if for every $a\in \mathcal{X}$, the element $a$ does not belong to the norm-closure of $$\{\sum_{i=1}^n λ_i \prod_{j=1}^{n_i}a_{i,j}: \textrm{ where } a_{i,j} \in \mathcal{X}\setminus\{a\} \textrm{ and} \sum |λ_i|\leq 1\}.$$ Since every almost irrredundant set is in particular a discrete set, it follows that the density of $\mathcal{A}$ is an upper bound for the size of almost irredundant sets. We prove that under the Proper Forcing …
Every $\mathrm{AW}^*$-Algebra is Normal
本文用集合论方法证明每个AW*-代数都是正规的，解决Wright悬置46年的问题，并建立因子情形到一般情形的力迫传递原理。
We introduce and study Polish topologies on various spaces of countable enumerated groups, where an enumerated group is simply a group whose underlying set is the set of natural numbers. Using elementary tools and well known examples from combinatorial group theory, combined with the Baire category theorem, we obtain a plethora of results demonstrating that several phenomena in group theory are generic. In effect, we provide a new topological framework for the analysis of various well known problems in group theory. We also provide a connection between genericity in these spaces, the word prob…
By Glimm's dichotomy, a separable, simple $\textrm{C}^*$-algebra has continuum-many unitarily inequivalent irreducible representations if, and only if, it is non-type I while all of its irreducible representations are unitarily equivalent if, and only if, it is type I. Naimark asked whether the latter equivalence holds for all $\textrm{C}^*$-algebras. In 2004, Akemann and Weaver gave a negative answer to Naimark's problem, using Jensen's diamond axiom $\diamondsuit_{\aleph_1}$, a powerful diagonalization principle that implies the Continuum Hypothesis ($\mathsf{CH}$). By a result of Rosenberg,…
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