We introduce a class of forcing notions, called forcing notions of type S, which contains among other Sacks forcing, Prikry-Silver forcing and their iterations and products with countable supports. We construct and investigate some formalism suitable for this forcing notions, which allows all standard tricks for iterations or products with countable supports of Sacks forcing. On the other hand it does not involve internal combinatorial structure of conditions of iterations or products. We prove that (...) the class of forcing notions of type S is closed under products and certain iterations with countable supports. (shrink)

Recently, Gitik, Kanovei and the first author proved that for a classical Prikry forcing extension the family of the intermediate models can be parametrized by $\mathscr{P}(\omega)/\mathrm{finite}$. By modifying the standard Prikry tree forcing we define a Prikry-type forcing which also singularizes a measurable cardinal but which is minimal, i.e., there are \emph{no} intermediate models properly between the ground model and the generic extension. The proof relies on combining the rigidity of the tree structure with indiscernibility arguments resulting (...) from the normality of the associated measures. (shrink)

We determine exact consistency strengths for various failures of the Singular Cardinals Hypothesis (SCH) in the setting of the Zermelo-Fraenkel axiom system ZF without the Axiom of Choice (AC). By the new notion of parallel Prikry forcing that we introduce, we obtain surjective failures of SCH using only one measurable cardinal, including a surjective failure of Shelah's pcf theorem about the size of the power set of $\aleph _{\omega}$ . Using symmetric collapses to $\aleph _{\omega}$ , $\aleph _{\omega _{1}}$ (...) , or $\aleph _{\omega _{2}}$ , we show that injective failures at $\aleph _{\omega}$ , $\aleph _{\omega _{1}}$ , or $\aleph _{\omega _{2}}$ can have relatively mild consistency strengths in terms of Mitchell orders of measurable cardinals. Injective failures of both the aforementioned theorem of Shelah and Silver's theorem that GCH cannot first fail at a singular strong limit cardinal of uncountable cofinality are also obtained. Lower bounds are shown by core model techniques and methods due to Gitik and Mitchell. (shrink)

It is almost universally accepted that the Frege–Geach Point is necessary for explaining the inferential relations and compositional structure of truth-functionally complex propositions. I argue that this claim rests on a disputable view of propositional structure, which models truth-functionally complex propositions on atomic propositions. I propose an alternative view of propositional structure, based on a certain notion of simulation, which accounts for the relevant phenomena without accepting the Frege–Geach Point. The main contention is that truth-functionally complex propositions do not include (...) as their parts truth-evaluable propositions, but their simulations, which are neither forceful nor truth-evaluable. The view makes room for the idea that there is no such thing as the forceless expression of propositional contents and is attractive because it provides the resources for avoiding a fundamental problem generated by the Frege–Geach Point concerning the relation between forceless and forceful expressions of propositional contents. I further argue that the acceptance of the Frege–Geach Point mars Peter Hanks’ and François Recanati’s recent attempts to resist the widespread idea that assertoric force is extrinsic to the expression of propositional contents. Rejecting this idea, I maintain, requires a deeper break with the tradition than Hanks and Recanati have allowed for. (shrink)

We consider of constructing normal ultrafilters in extensions are here Easton support iterations of Prikry-type forcing notions. New ways presented. It turns out that, in contrast with other supports, seemingly unrelated measures or extenders can be involved here.

Professor l a mirarchi argues, In his "force and absolute motion in berkeley's philosophy of physics" (_journal of the history of ideas<d>, Volume 38, Pages 705-713), That I have misunderstood berkeley's treatment of inertial motion. I contend, Despite professor mirarchi's criticism, That while berkeley accepts the newtonian principle of inertia, He cannot accommodate it into his own radically contingent picture of the universe.

We study the Mathias–Prikry and Laver–Prikry forcings associated with filters on ω. We give a combinatorial characterization of Martinʼs number for these forcing notions and present a general scheme for analyzing preservation properties for them. In particular, we give a combinatorial characterization of those filters for which the Mathias–Prikry forcing does not add a dominating real.

We define and investigate versions of Silver and Mathias forcing with respect to lower and upper density. We focus on properness, Axiom A, chain conditions, preservation of cardinals and adding Cohen reals. We find rough forcings that collapse $$2^\omega $$ 2 ω to $$\omega $$ ω, while others are surprisingly gentle. We also study connections between regularity properties induced by these parametrized forcing notions and the Baire property.

In Part I of this series [A. Poveda, A. Rinot and D. Sinapova, Sigma-Prikry forcing I: The axioms, Canad. J. Math. 73(5) (2021) 1205–1238], we introduced a class of notions of forcing which we call [Formula: see text]-Prikry, and showed that many of the known Prikry-type notions of forcing that center around singular cardinals of countable cofinality are [Formula: see text]-Prikry. We showed that given a [Formula: see text]-Prikry poset [Formula: see text] and a [Formula: see text]-name (...) for a non-reflecting stationary set [Formula: see text], there exists a corresponding [Formula: see text]-Prikry poset that projects to [Formula: see text] and kills the stationarity of [Formula: see text]. In this paper, we develop a general scheme for iterating [Formula: see text]-Prikry posets and, as an application, we blow up the power of a countable limit of Laver-indestructible supercompact cardinals, and then iteratively kill all non-reflecting stationary subsets of its successor. This yields a model in which the singular cardinal hypothesis fails and simultaneous reflection of finite families of stationary sets holds. (shrink)

Are emotions like sneezes, unwilled, mechanical, or are they like judgments; are they entirely social constructions? Harré and Gillett believe that emotions are exclusively judgments. We argue that their view misses something important. Imagine a person quaking in anger. Both we and Harré and Gillett believe that he is angry only if he has made an implicit judgment, such as I have been transgressed against. But it is the quaking, not the judgment, that gives authenticity and force to the expression (...) of anger. The quaking does not clarify what the actor means but rather it clarifies the relation of the actor to the meaning of his display. What makes it a genuine expression of anger and not a joke or performance is that the quaking is beyond the will. Bodily displays are not necessary to make expressions authentic; anything that shows that the expression is beyond the will will do, for instance, obsessive thoughts, intrusions, or an inability to concentrate. For Harré and Gillett emotions both as displays and feelings do not merely embody judgments but are also speech acts. We argue that an expression, a feeling or flitting through the mind, cannot be a speech act since only the overt can fit into the convention, the strictures of a community. Nor is the display merely a speech act. Since for an emotional display to be genuine it must slip from the lips unbidden. Further, a speech act account makes the emotions arbitrary; they imply that the set of possible emotions is open. We think, on the other hand, that only some sorts of judgments can become part of an emotion; judgments that relate to things that are important enough in a particular culture that judgment display and feeling are linked together involuntarily. (shrink)

The core of this article is a comparative analysis of the double movement of the late nineteenth and early twentieth century with the double movement of the late twentieth and early twenty-first century. In both periods the movement toward allegedly self-regulating markets called forth a countermovement of protection. Nevertheless, important differences exist due, first, to differences in the nature of the hegemonic state and, second, to the greater role of subordinate forces in constraining the movement toward self-regulating markets in the (...) late twentieth century. (shrink)

It is known that there is a close relation between Prikry forcing and the iteration of ultrapowers: If U is a normal ultrafilter on a measurable cardinal κ and 〈Mn, jm,n | m ≤ n ≤ ω〉 is the iteration of ultrapowers of V by U, then the sequence of critical points 〈j0,n | n ∈ ω〉 is a Prikry generic sequence over Mω. In this paper we generalize this for normal precipitous filters.

Much of the reigning research on non-religion and non-belief focuses on demographics and personality characteristics. While this is a necessary foundation on which future research may be built upon, such data does not necessarily produce theory. In many ways the dominant cultural milieu of religions along with the benign intent of some researchers force a person who holds no belief in a God to assume an oppositional identity in relation to religion. This oppositional identity tautologically sets researchers up to continually (...) define its object by the absence of something. This something cannot always function as a normative point of reference in which to tell researchers what to look for. This article provides one such normative trajectory, termed “horizontal transcendence.”. (shrink)

The extender based Prikry forcing notion is being generalized to super compact extenders.

In this paper, we answer a question asked in Koepke et al. regarding a Mathias criteria for Tree-Prikry forcing. Also we will investigate Prikry forcing using various filters. For completeness and self inclusion reasons, we will give proofs of many known theorems.

A generalization of Příkrý's forcing is analyzed which adjoins to a model of ZFC a set of order type at most ω below each member of a discrete set of measurable cardinals. A characterization of generalized Příkrý generic sequences reminiscent of Mathias' criterion for Příkrý genericity is provided, together with a maximality theorem which states that a generalized Příkrý sequence almost contains every other one lying in the same extension.This forcing can be used to falsify the covering lemma (...) for a higher core model if there is an inner model with infinitely many measurable cardinals – changing neither cardinalities nor cofinalities. Another application is an alternative proof of a theorem of Mitchell stating that if the core model contains a regular limit θ of measurable cardinals, then there is a model in which every set of measurable cardinals of K bounded in θ has an indiscernible sequence but there is no such sequence for the entire set of measurables of K below θ. (shrink)

Much of the reigning research on non-religion and non-belief focuses on demographics and personality characteristics. While this is a necessary foundation on which future research may be built upon, such data does not necessarily produce theory. In many ways the dominant cultural milieu of religions along with the benign intent of some researchers force a person who holds no belief in a God to assume an oppositional identity in relation to religion. This oppositional identity tautologically sets researchers up to continually (...) define its object by the absence of something. This something cannot always function as a normative point of reference in which to tell researchers what to look for. This article provides one such normative trajectory, termed “horizontal transcendence.”. (shrink)

If $U$ is a normal ultrafilter on a measurable cardinal $\kappa$, then the intersection of the $\omega$ first iterated ultrapowers of the universe by $U$ is a Prikry generic extension of the $\omega$th iterated ultrapower.

Let \ be an infinite discrete set of measurable cardinals. It is shown that generalized Prikry forcing to add a countable sequence to each cardinal in \ is subcomplete. To do this it is shown that a simplified version of generalized Prikry forcing which adds a point below each cardinal in \, called generalized diagonal Prikry forcing, is subcomplete. Moreover, the generalized diagonal Prikry forcing associated to \ is subcomplete above \, where \ is any regular (...) cardinal below the first limit point of \. (shrink)

Journal of Mathematical Logic, Volume 24, Issue 03, December 2024. We study which [math]-distributive forcing notions of size [math] can be embedded into tree Prikry forcing notions with [math]-complete ultrafilters under various large cardinal assumptions. An alternative formulation — can the filter of dense open subsets of a [math]-distributive forcing notion of size [math] be extended to a [math]-complete ultrafilter.

Abstract(1)We show that it is possible to add $\kappa ^+$ -Cohen subsets to $\kappa $ with a Prikry forcing over $\kappa $. This answers a question from [9].(2)A strengthening of non-Galvin property is introduced. It is shown to be consistent using a single measurable cardinal which improves a previous result by S. Garti, S. Shelah, and the first author [5].(3)A situation with Extender-based Prikry forcings is examined. This relates to a question of H. Woodin.

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 κ then every stationary subset of reflects. It will also be (...) demonstrated that MRP always fails in a generic extension by Prikry forcing. (shrink)

We show that Gitik’s short extender gap-2 forcing is of Prikry type.

We study which [Formula: see text]-distributive forcing notions of size [Formula: see text] can be embedded into tree Prikry forcing notions with [Formula: see text]-complete ultrafilters under various large cardinal assumptions. An alternative formulation — can the filter of dense open subsets of a [Formula: see text]-distributive forcing notion of size [Formula: see text] be extended to a [Formula: see text]-complete ultrafilter.

We prove a Mathias-type criterion for the Magidor iteration of Prikry forcings.

Applying the seed concept to Prikry tree forcing P μ , I investigate how well P μ preserves the maximality property of ordinary Prikry forcing and prove that P μ Prikry sequences are maximal exactly when μ admits no non-canonical seeds via a finite iteration. In particular, I conclude that if μ is a strongly normal supercompactness measure, then P μ Prikry sequences are maximal, thereby proving, for a large class of measures, a conjecture of W. Hugh Woodin's.

We continue [4] and study sets in generic extensions by the Magidor forcing and by the Prikry forcing with non-normal ultrafilters.

We study non-homogeneity of quotients of Prikry and tree Prikry forcings with non-normal ultrafilters over some natural distributive forcing notions.

In this paper we investigate the structure of uncountable maximal antichains of Silver forcing and show that they have to be at least of size d, where d is the dominating number. Part of this work can be used to show that the additivity of the Silver forcing ideal has size at least the unbounding number b. It follows that every reasonable amoeba Silver forcing adds a dominating real.

Extender-based Prikry–Magidor forcing for overlapping extenders is introduced. As an application, models with strong forms of negations of the Shelah Weak Hypothesis for various cofinalities are constructed.

An ℵ1-Souslin tree is a complicated combinatorial object whose existence cannot be decided on the grounds of ZFC alone. But fifteen years after Tennenbaum and Jech independently devised notions of forcing for introducing such a tree, Shelah proved that already the simplest forcing notion—Cohen forcing—adds an ℵ1-Souslin tree. In this article, we identify a rather large class of notions of forcing that, assuming a GCH-type hypothesis, add a λ+-Souslin tree. This class includes Prikry, Magidor, and Radin (...) forcing. (shrink)

We study the Magidor iteration of Prikry forcings, and the resulting normal measures on κ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\kappa}$$\end{document}, the first measurable cardinal in a generic extension. We show that when applying the iteration to a core model below 0¶\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${0^{\P}}$$\end{document}, then there exists a natural correspondence between the normal measures on κ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\kappa}$$\end{document} in the ground model, and those (...) of the generic extension. (shrink)

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 are added to thin projective transitive relations by these forcings.

Thought: A Journal of Philosophy, EarlyView.

In this paper we investigate some properties of forcing which can be considered “nice” in the context of singularizing regular cardinals to have an uncountable cofinality. We show that such forcing which changes cofinality of a regular cardinal, cannot be too nice and must cause some “damage” to the structure of cardinals and stationary sets. As a consequence there is no analogue to the Prikry forcing, in terms of “nice” properties, when changing cofinalities to be uncountable.

The dissertation under comment is a contribution to the area of Set Theory concerned with the interactions between the method of Forcing and the so-called Large Cardinal axioms.The dissertation is divided into two thematic blocks. In Block I we analyze the large-cardinal hierarchy between the first supercompact cardinal and Vopěnka’s Principle. In turn, Block II is devoted to the investigation of some problems arising from Singular Cardinal Combinatorics.We commence Part I by investigating the Identity Crisis phenomenon in the region (...) comprised between the first supercompact cardinal and Vopěnka’s Principle. As a result, we generalize Magidor’s classical theorems [2] to this higher region of the large-cardinal hierarchy. Also, our analysis allows to settle all the questions that were left open in [1]. Finally, we conclude Part I by presenting a general theory of preservation of $C^{}$ -extendible cardinals under class forcing iterations. From this analysis we derive several applications. For instance, our arguments are used to show that an extendible cardinal is consistent with “ $^{\mathrm {HOD}}<\lambda ^+$, for every regular cardinal $\lambda $.” In particular, if Woodin’s HOD Conjecture holds, and therefore it is provable in ZFC + “There exists an extendible cardinal” that above the first extendible cardinal every singular cardinal $\lambda $ is singular in HOD and $^{\textrm {{HOD}}}=\lambda ^+$, there may still be no agreement at all between V and HOD about successors of regular cardinals.In Part II and Part III we analyse the relationship between the Singular Cardinal Hypothesis with other relevant combinatorial principles at the level of successors of singular cardinals. Two of these are the Tree Property and the Reflection of Stationary sets, which are central in Infinite Combinatorics.Specifically, Part II is devoted to prove the consistency of the Tree Property at both $\kappa ^+$ and $\kappa ^{++}$, whenever $\kappa $ is a strong limit singular cardinal witnessing an arbitrary failure of the SCH. This generalizes the main result of [3] in two senses: it allows arbitrary cofinalities for $\kappa $ and arbitrary failures for the SCH.In the last part of the dissertation we introduce the notion of $\Sigma $ -Prikry forcing. This new concept allows an abstract and uniform approach to the theory of Prikry-type forcings and encompasses several classical examples of Prikry-type forcing notions, such as the classical Prikry forcing, the Gitik-Sharon poset, or the Extender Based Prikry forcing, among many others.Our motivation in this part of the dissertation is to prove an iteration theorem at the level of the successor of a singular cardinal. Specifically, we aim for a theorem asserting that every $\kappa ^{++}$ -length iteration with support of size $\leq \kappa $ has the $\kappa ^{++}$ -cc, provided the iterates belong to a relevant class of $\kappa ^{++}$ -cc forcings. While there are a myriad of works on this vein for regular cardinals, this contrasts with the dearth of investigations in the parallel context of singular cardinals. Our main contribution is the proof that such a result is available whenever the class of forcings under consideration is the family of $\Sigma $ -Prikry forcings. Finally, and as an application, we prove that it is consistent—modulo large cardinals—the existence of a strong limit cardinal $\kappa $ with countable cofinality such that $\mathrm {SCH}_\kappa $ fails and every finite family of stationary subsets of $\kappa ^+$ reflects simultaneously. (shrink)

In this paper we analyse some notions of amoeba for tree forcings. In particular we introduce an amoeba-Silver and prove that it satisfies quasi pure decision but not pure decision. Further we define an amoeba-Sacks and prove that it satisfies the Laver property. We also show some application to regularity properties. We finally present a generalized version of amoeba and discuss some interesting associated questions.

It is shown that cardinals below a real-valued measurable cardinal can be split into finitely many intervals so that the powers of cardinals from the same interval are the same. This generalizes a theorem of Prikry [9]. Suppose that the forcing with a κ-complete ideal over κ is isomorphic to the forcing of λ-Cohen or random reals. Then for some τ<κ, λτ2κ and λ2<κ implies that 2κ=2τ= cov. In particular, if 2κ<κ+ω, then λ=2κ. This answers a question from (...) [3]. If A0, A1,..., An,… are sets of reals, then there are disjoint sets B0, B1,..., Bn,… such that BnAn and μ*=μ* for every n<ω, where μ* is the Lebes gue outer measure. For finitely many sets the result is due to N. Lusin. Let be a σ-centered forcing notion and An ¦n<ω subsets of P witnessing this. If P, An's and the relation of compatibility are Borel, then P adds a Cohen real. The forcing with a κ-complete ideal over a set X, ¦X¦κ cannot be isomorphic to a Hechler real forcing. This result was claimed in [3], but the proof given there works only for X of cardinality κ. (shrink)

We present a version for κ-distributive ideals over a regular infinite cardinal κ of some of the combinatorial results of Mathias on happy families. We also study an associated notion of forcing, which is a generalization of Mathias forcing and of Prikry forcing.

The basic assumption of the paper is the recognition that the complexity of the challenges related to population ageing forces the development of cooperative links in the area of the silver economy between public policy entities representing various sectors. In other words, there is a need for more intensive and better-coordinated cooperation between organisations in the commercial sector, public sector, non-governmental sector, informal sector and social economy sector (e.g., cooperatives).

We show how to construct Gitik’s short extenders gap-3 forcing using a morass, and that the forcing notion is of Prikry type.

We construct a model where every increasing ω-sequence of regular cardinals carries a mutually stationary sequence which is not tightly stationary, and show that this property is preserved under a class of Prikry-type forcings. Along the way, we give examples in the Cohen and Prikry models of ω-sequences of regular cardinals for which there is a non-tightly stationary sequence of stationary subsets consisting of cofinality ω_1 ordinals, and show that such stationary sequences are mutually stationary in the presence of interleaved (...) supercompact cardinals. (shrink)

Es conocido que el método de forcing es una de las técnicas de construcción de modelos más importantes de la Teoría de conjuntos en la actualidad, siendo el mismo muy útil para investigar problemas de matemática y/o de fundamentos de la matemática. El destacado matemático Joan Bagaria afirma lo siguiente sobre el método de forcing en su artículo "Paul Cohen y la técnica del forcing" (Gaceta de la Real Sociedad Matemática Española, Vol. 2, Nº 3, 1999, págs (...) 543-553) : "Aunque Cohen recibió la medalla Fields por su demostración de la independencia de la hipótesis del continuo y del axioma de elección, su contribución va mucho más allá de estos problemas. Su nuevo método, el forcing, no sólo ha permitido resolver un sinfín de problemas importantes en prácticamente todas las áreas de las matemáticas, si no que ha cambiado para siempre nuestra concepción de la matemática como ciencia". El objetivo principal del siguiente trabajo es estudiar el método de forcing describiendo algunas de sus aplicaciones (forcing de Cohen, forcing aleatorio, forcing de Mathias, forcing de Sacks, forcing de Silver, etc.), y ofreciendo una aproximación a sus fundamentos metamatemáticos. Se aspira que estas notas sirvan de apoyo para aprender dicho método. (shrink)

The First World War forced the Russian intelligentsia to rethink its values—values that had been constructed in the nineteenth century. Distancing itself from pacifism and cultural relativism, it began to search for a moral meaning to the war that broke out in 1914—i.e. to defend the war as morally right and having a higher spiritual purpose. Russian philosophers were central to these debates, as they tried to interpret the war, and the relationship between war and peace, from a metaphysical point (...) of view. The paper will look at philosophical, ethical and religious aspects of these debates, and some of the philosophers who participated in them. (shrink)

Consider where κ is an uncountable regular cardinal. By a result of Shelah's we have for almost all witnessing this. Here we consider the question if there could be a similar result for. We will discuss some basic facts implying that this cannot hold in general. We will use these facts to construct an interesting example of a pseudo Prikry forcing, answering a question of Sinapova.