We survey recent developments in the theory of class forcing for- malized in the second-order set-theoretic setting.

In this article we show that Morse-Kelley class theory provides us with an adequate framework for class forcing. We give a rigorous definition of class forcing in a model \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$$$ \end{document} of MK, the main result being that the Definability Lemma can be proven without restricting the notion of forcing. Furthermore we show under which conditions the axioms are preserved. We conclude by proving that (...) Laver’s Theorem does not hold for class forcings. (shrink)

The class forcing theorem, which asserts that every class forcing notion ${\mathbb {P}}$ admits a forcing relation $\Vdash _{\mathbb {P}}$, that is, a relation satisfying the forcing relation recursion—it follows that statements true in the corresponding forcing extensions are forced and forced statements are true—is equivalent over Gödel–Bernays set theory $\text {GBC}$ to the principle of elementary transfinite recursion $\text {ETR}_{\text {Ord}}$ for class recursions of length $\text {Ord}$. It is also equivalent (...) to the existence of truth predicates for the infinitary languages $\mathcal {L}_{\text {Ord},\omega }$, allowing any class parameter A; to the existence of truth predicates for the language $\mathcal {L}_{\text {Ord},\text {Ord}}$ ; to the existence of $\text {Ord}$ -iterated truth predicates for first-order set theory $\mathcal {L}_{\omega,\omega }$ ; to the assertion that every separative class partial order ${\mathbb {P}}$ has a set-complete class Boolean completion; to a class-join separation principle; and to the principle of determinacy for clopen class games of rank at most $\text {Ord}+1$. Unlike set forcing, if every class forcing notion ${\mathbb {P}}$ has a forcing relation merely for atomic formulas, then every such ${\mathbb {P}}$ has a uniform forcing relation applicable simultaneously to all formulas. Our results situate the class forcing theorem in the rich hierarchy of theories between $\text {GBC}$ and Kelley–Morse set theory $\text {KM}$. (shrink)

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 [17], previously obtained in [8, 27], respectively. We prove that $C^{(n)}$ -extendible cardinals are preserved by forcing with standard Easton-support iterations for any possible $\Delta _2$ -definable behaviour of (...) the power-set function on regular cardinals. We show that one can force proper class-many disagreements between the universe and HOD with respect to the calculation of successors of regular cardinals, while preserving $C^{(n)}$ -extendible cardinals. We also show, assuming the GCH, that the class forcing iteration of Cummings–Foreman–Magidor for forcing $\diamondsuit _{\kappa ^+}^+$ at every $\kappa $ [10] preserves $C^{(n)}$ -extendible cardinals. We give an optimal result on the consistency of weak square principles and $C^{(n)}$ -extendible cardinals. In the last section prove another preservation result for $C^{(n)}$ -extendible cardinals under very general (not necessarily definable or weakly homogeneous) class forcing iterations. As applications we prove the consistency of $C^{(n)}$ -extendible cardinals with $\mathrm {{V}}=\mathrm {{HOD}}$, and also with $\mathrm {GA}$ (the Ground Axiom) plus $\mathrm {V}\neq \mathrm {HOD}$, the latter being a strengthening of a result from [14]. (shrink)

This serves as an introduction to a debate between Hans Kelsen and Otto Bauer concerning the nature and relative autonomy of the state, and the theories that informed the political practices of the Austro-Marxists and of the SDAP immediately after the fall of the monarchy and during the early years of the First Republic. Both pieces were published in Der Kampf, the SDAP’s theoretical journal, in which many key texts of Austro-Marxist thought appeared. The debate is of theoretical interest, particularly (...) with respect to Marxist state theory, but is potentially also of renewed relevance to an age of post-GFC austerity. Bauer’s distinction between functional oligarchy and functional democracy is identified as one such potential contribution to current debate and analysis. (shrink)

This serves as an introduction to a debate between Hans Kelsen and Otto Bauer concerning the nature and relative autonomy of the state, and the theories that informed the political practices of the Austro-Marxists and of the SDAP immediately after the fall of the monarchy and during the early years of the First Republic. Both pieces were published in Der Kampf, the SDAP’s theoretical journal, in which many key texts of Austro-Marxist thought appeared. The debate is of theoretical interest, particularly (...) with respect to Marxist state theory, but is potentially also of renewed relevance to an age of post-GFC austerity. Bauer’s distinction between functional oligarchy and functional democracy is identified as one such potential contribution to current debate and analysis. (shrink)

In this article we introduce and study hyperclass-forcing in the context of an extension of Morse-Kelley class theory, called MK∗∗. We define this forcing by using a symmetry between MK∗∗ models and models of ZFC− plus there exists a strongly inaccessible cardinal. We develop a coding between β-models ℳ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal {M}$$ \end{document} of MK∗∗ and transitive models M+ of SetMK∗∗ which will allow us to go from ℳ\documentclass[12pt]{minimal} \usepackage{amsmath} (...) \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal {M}$$ \end{document} to M+ and vice versa. So instead of forcing with a hyperclass in MK∗∗ we can force over the corresponding SetMK∗∗ model with a class of conditions. For class-forcing to work in the context of ZFC− we show that the SetMK∗∗ model M+ can be forced to look like Lκ∗[X]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$L_{\kappa ^*}[X]$$ \end{document}, where κ∗ is the height of M+, κ strongly inaccessible in M+ and X ⊆ κ. Over such a model we can apply definable class forcing and we arrive at an extension of M+ from which we can go back to the corresponding β-model of MK∗∗, which will in turn be an extension of the original ℳ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal {M}$$ \end{document}. Our main result combines hyperclass forcing with coding methods of Beller et al. and Friedman to show that every β-model of MK∗∗ can be extended to a minimal such model of MK∗∗ with the same ordinals. A simpler version of the proof also provides a new and analogous minimality result for models of second-order arithmetic. (shrink)

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)

We consider the modality “ $\varphi $ is true in every $\sigma $ -centered forcing extension,” denoted $\square \varphi $, and its dual “ $\varphi $ is true in some $\sigma $ -centered forcing extension,” denoted $\lozenge \varphi $, which give rise to the notion of a principle of $\sigma $ -centered forcing. We prove that if ZFC is consistent, then the modal logic of $\sigma $ -centered forcing, i.e., the ZFC-provable principles of $\sigma $ -centered (...) forcing, is exactly $\mathsf {S4.2}$. We also generalize this result to other related classes of forcing. (shrink)

The technique of forcing is almost ubiquitous in set theory, and it seems to be based on technicalities like the concepts of genericity, forcing names and their evaluations, and on the recursively defined forcing predicates, the definition of which is particularly intricate for the basic case of atomic first order formulas. In his [3], the first author has provided an axiomatic framework for set forcing over models of $\mathrm {ZFC}$ that is a collection of guiding principles (...) for extensions over which one still has control from the ground model, and has shown that these axiomatics necessarily lead to the usual concepts of genericity and of forcing extensions, and also that one can infer from them the usual recursive definition of forcing predicates. In this paper, we present a more general such approach, covering both class forcing and set forcing, over various base theories, and we provide additional details regarding the formal setting that was outlined in [3]. (shrink)

We define a class of first-order formulas $\mathsf{P}^{\ast }$ which exactly contains formulas $\varphi$ such that satisfaction of $\varphi$ in any classical structure attached to a node of a Kripke model of intuitionistic predicate logic deciding atomic formulas implies its forcing in that node. We also define a class of $\mathsf{E}$-formulas with the property that their forcing coincides with their classical satisfiability in Kripke models which decide atomic formulas. We also prove that any formula with this (...) property is an $\mathsf{E}$-formula. Kripke models of intuitionistic arithmetical theories usually have this property. As a consequence, we prove a new conservativity result for Peano arithmetic over Heyting arithmetic. (shrink)

Creative Class, Creative Economy, and the Wisdom Society as a Solution to their Controversy The paper briefly introduces the notion of creativity, linking the concepts of creative class and the related creative economy that are considered by Florida and his followers as the driving force of the current social and economic development. The concept of creative economy and its quantification in form of the Creative Class Index 3T or the Euro-Creativity Index were submitted to strong critique.The critics (...) overturn some key issues of the concept of creative class with an indication that the creative people give priority in choosing the work from the place; and there is no or almost nil correlation of a perceptual representation of the persons of homosexual orientation and the high-tech growth which confirms the opinion that the assertion about higher creativity of homosexual persons is a myth only. The purpose of the article is the examination of propriety of the concept of creative class and its quantification at one hand, and the rightfulness of their critique at the other. The analysis of the axiology of the concept of creative class is the method of examination. From point of view of axiology probably the most significant characteristic of the creative class is its accompanying destruction of thee social capital of creative people and it substitution of the creative capital. The rise of creative class takes place in parallel with the implosion of values at which basis our Jewish-Christian civilization is established. To this Florida expresses rather vague attitude by a statement that all socio-cultural changes that accompany the rise of the creative class will have no favorable impact on society.Florida labeled creativity as a basic human law, furthermore, he announced the need for a new live style in which the quality of life will increase and people become better men. The concept of creativity as an instrument of urban planning evolved into a left-liberal ideology with elements of social engineering that is the most important finding of the axiological analysis. The second finding is the need to accept the meaning of creativity. By implication of the two findings there arises a proposal for wisdom as a new driving force of social and economic development. The conclusion is the contribution is a requirement for formulation of the concept of wisdom society that is currently likely absent in the academic world, and the statement that if the West will not create the wisdom society, it will lose the raison d'être of its existence. (shrink)

Zeilinger's observation that phenomena of the Aharonov-Bohm type lead to non-dispersive, i.e., energy-independent, phase shifts in interferometers is generalized in a new proof which shows that the precise condition for nondispersivity is a force-free interaction. The converse theorem is disproved by a conceptual counter-example. Applications to several nondispersive interference phenomena are reviewed briefly. Those fall into two classes which are objectively distinct from each other in that in the first class phase shifts depend only on the topology of the (...) interfering beam paths, while in the second class force-free physical interactions take place at identifiable points along the path. Apparent disagreements in the literature about the topological nature of the phenomena in the second class stem from differing definitions. (shrink)

This article studies three most basic systems of truth as well as their subsystems over set theory ZF possibly with AC or the axiom of global choice GC, and then correlates them with subsystems of Morse–Kelley class theory MK. The article aims at making an initial step towards the axiomatic study of truth in set theory in connection with class theory. Some new results on the side of class theory, such as conservativity, forcing and some forms (...) of the reflection principle, are also presented. The equivalence results among systems of truth and classes obtained in this article are summarized in Theorem 104 , Theorem 107 and Theorem 108. (shrink)

This chapter is a discussion of Class and Resemblance Nominalism. According to the traditional versions of these theories, properties are classes of particulars. Thus, the property of being red is the class of red particulars, and the property of being square is the class of square particulars. Several objections have been advanced against these theories, and one of the most powerful of such objections is the so-called Coextension Difficulty, according to which Class and Resemblance Nominalism have (...) to wrongly identify distinct but coextensive properties. After discussing the nature of Class and Resemblance Nominalism, some problems of these theories and how they can be deal with, the superiority of Class and Resemblance Nominalism over other theories of properties and the superiority of Resemblance Nominalism over Class Nominalism, I concentrate on the Coextension Difficulty. With respect to this I argue that Class and Resemblance Nominalism can be developed in such a way that they admit properties without identifying them with classes, and this allows them to avoid the Coextension Difficulty, since such versions of these theories are not forced to wrongly identify distinct but coextensive properties. Thus the Coextension Difficulty is not a lethal problem for Class and Resemblance Nominalism. (shrink)

Illocutionary Force and Romanian Orthodox Sermons: An Application of Speech Act Theory to Some Romanian Orthodox Sermons The aim of the paper is to analyze religious discourse with the use of the instruments of semantics and pragmatics. Essentially, it sets out to identify the linguistic elements which enable the illocutionary force in the Romanian orthodox sermons, especially in the discourse of some important figures which have influenced and still influence the Romanian orthodox theology and the religious life in Romania: Father (...) Cleopa, Father Nicolae Steinhardt, and, nowadays, Father Teofil Pârâianu. It is usually assumed that a sermon implies, on the one hand, a kerugmatik action, that of annunciation, and, on the other hand, a translating action of the biblical induce the human will to practice the words in order to Christian precepts. Mutatis mutandis, the perlocutionary effect depends on the illocutionary act and of the illocutionary force. In theory, the illocutionary force of an utterance is strictly motivated by the pragmatic. But there are linguistic elements which function as efficient markers of the illocutionary force, for example performative verbs, verbal moods and some adverbs. The criteria proposed by Searle in order to differentiate the illocutionary acts resulted in the identification of several types inside this class. Searle's taxonomy serves the research purposes: the identification of illocutionary acts found in this kind of religious discourse and, finally, outlining a pattern of manifestation of the illocutionary force inside the sermon. (shrink)

Henle, Mathias, and Woodin proved in [21] that, provided that${\omega }{\rightarrow }({\omega })^{{\omega }}$holds in a modelMof ZF, then forcing with$([{\omega }]^{{\omega }},{\subseteq }^*)$overMadds no new sets of ordinals, thus earning the name a “barren” extension. Moreover, under an additional assumption, they proved that this generic extension preserves all strong partition cardinals. This forcing thus produces a model$M[\mathcal {U}]$, where$\mathcal {U}$is a Ramsey ultrafilter, with many properties of the original modelM. This begged the question of how important the (...) Ramseyness of$\mathcal {U}$is for these results. In this paper, we show that several classes of$\sigma $-closed forcings which generate non-Ramsey ultrafilters have the same properties. Such ultrafilters include Milliken–Taylor ultrafilters, a class of rapid p-points of Laflamme,k-arrow p-points of Baumgartner and Taylor, and extensions to a class of ultrafilters constructed by Dobrinen, Mijares, and Trujillo. Furthermore, the class of Boolean algebras$\mathcal {P}({\omega }^{{\alpha }})/{\mathrm {Fin}}^{\otimes {\alpha }}$,$2\le {\alpha }<{\omega }_1$, forcing non-p-points also produce barren extensions. (shrink)

The Lévy-Solovay Theorem [8] 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.

It seems evident that class differences and class struggle continue to exist in socialist societies; that is to say, in societies like the Soviet Union and China, which have undergone socialist revolutions and in which private property in the means of production has been largely abolished. I shall not attempt to prove this proposition here; rather it will form my starting point. For my purpose in this paper is to show how the phenomenon of class in socialist (...) society can be understood and interpreted in Marxist terms; and, in particular, to explain and expound Mao Zedong's attempt to do so. For one of Mao's most striking and important contributions to Marxism was his recognition that `contradictions among the people' continue to exist in socialist society, and his attempt to explain them within the theoretical framework of historical materialism. Marx outlines his account of historical development in the following well-known words: It is not the consciousness of men that determines their being, but on the contrary it is their social being that determines their consciousness. At a certain stage of their development, the material productive forces of society come into conflict with the existing relations of production or - what is merely a legal expression for the same thing - with the property relations within the framework of which they have hitherto operated. From forms of development of the productive forces these relations turn into their fetters. At that point an era of social revolution begins. With the change in the economic foundation the whole immense superstructure is more slowly or more rapidly transformed. (Marx, Preface to A Contribution to the Critique of Political Economy) It has been common to interpret these words as expressing a simple form of economic or even technological determinism which would rule out the very possibility of class divisions continuing to be a fundamental feature of socialist society. For, according to this account, a socialist society, by abolishing the private ownership of the means of production, thereby abolishes the material and economic basis of class differences; and so classes are destined to die out in socialist society 2 as the forces of production are developed. According to this interpretation, which I shall call the `traditional' account, in Marx's account of historical development all the emphasis is placed upon the development of the productive forces.. (shrink)

For a large natural class of forcing notions, we prove general equivalence theorems between forcing absoluteness statements, regularity properties, and transcendence properties over and the core model . We use our results to answer open questions from set theory of the reals.

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 [Formula: see text]-algebras with the metric approximation property and an increasing approximate identity of projections.

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 [Formula: see text]-algebras with the metric approximation property and an increasing approximate identity of projections.

The article focuses on the temporal and epistemic economy connected to the transatlantic travels of the categorical triad of ‘race-class-gender’. It looks at conditions and forces that have fuelled the dynamics of the discourse on differences and inequality among women and analyses feminist discourse and its aporias as a particular environment for the travels of theories. Furthermore, it follows the changes the triad of ‘race-class-gender’ undergoes on its transatlantic route from the United States to a German-speaking context and (...) it outlines the theoretical challenges connected to an intersectional perspective that aims to overcome a theoretical stagnation that itself finds symptomatic expression in the ritual citing of ‘race-class-gender’. (shrink)

Scholars see the gendered division of household labor as a stronghold of gender inequality. We explore changes in household labor and gender relations when conservative, working-class families experience employment disruptions. Using data from 49 qualitative interviews conducted with men and women following the forced unemployment of breadwinning husbands, we observe some change in gendered household labor but conclude that a significant degendering of housework is thwarted by institutional-, interactive-, and individual-level processes. At the institutional level, the lack of well-paying (...) jobs and the persistent gendering of household tasks discourage change. At the individual level, challenges to gendered identities encourage a reinforcement of traditional gender ideologies. At the interactional level, women’s responsibility for care work and the meaning of paid work for unemployed husbands forestall the adjustment of tasks. (shrink)

Classes of forcings which add a real by forcing with branching conditions are examined, and conditions are found which guarantee that the generic real is of minimal degree over the ground model. An application is made to almost-disjoint coding via a real of minimal degree.

We use model theoretic forcing to study and generalize the construction of ()-generic models introduced by Kueker and Laskowski. We characterize the ()-generic models in terms of forcing and introduce a more general class of models, called essential forcing generics, which have many of the same properties.

Hamkins and Löwe proved that the modal logic of forcing is S4.2 . In this paper, we consider its modal companion, the intermediate logic KC and relate it to the fatal Heyting algebra H ZFC of forcing persistent sentences. This Heyting algebra is equationally generic for the class of fatal Heyting algebras. Motivated by these results, we further analyse the class of fatal Heyting algebras.

If T has only countably many complete types, yet has a type of infinite multiplicity then there is a c.c.c. forcing notion Q such that, in any Q-generic extension of the universe, there are non-isomorphic models M 1 and M 2 of T that can be forced isomorphic by a c.c.c. forcing. We give examples showing that the hypothesis on the number of complete types is necessary and what happens if `c.c.c.' is replaced by other cardinal-preserving adjectives. We (...) also give an example showing that membership in a pseudo-elementary class can be altered by very simple cardinal-preserving forcings. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Force and Translation; Or, The Polymorphous Body of LanguageElissa MarderOr un corps verbal ne se laisse pas traduire ou transporter dans une autre langue. Il est cela même que la traduction laisse tomber. Laisser tomber le corps, telle est même l’énergie essentielle de la traduction. Quand elle réinstitue un corps, elle est poésie.—Jacques Derrida, “Freud et la scène de l’écriture”The materiality of a word cannot be translated or carried (...) over into another language. Materiality is precisely that which translation relinquishes. To relinquish materiality: such is the driving force of translation. And when that materiality is reinstated, translation becomes poetry.—Jacques Derrida, “Freud and the Scene of Writing”Babel Revisited (The Mother Tongue and The Translation Machine)On January 20, 2012, Lawrence Summers, former president of Harvard and former Secretary of the U.S. Treasury, boldly announced a six-point vision for reforming the U.S. education system in a New York Times Education supplement editorial modestly entitled, “What you (Really) Need to Know.” For the [End Page 1] purposes of this discussion, I will limit myself here to a brief examination of what he says about the study of foreign languages and the merits of machine translation in the item listed as “number 5”:5. The world is much more open, and events abroad affect the lives of Americans more than ever before. This makes it essential that the educational experience breed cosmopolitanism—that students have international experiences, and classes in the social sciences draw on examples from around the world. It seems logical, too, that more in the way of language study be expected of students. I am not so sure.English’s emergence as the global language, along with the rapid progress in machine translation and the fragmentation of languages spoken around the world, make it less clear that the substantial investment necessary to speak a foreign tongue is universally worthwhile. While there is no gainsaying the insights that come from mastering a language, it will over time become less essential in doing business in Asia, treating patients in Africa or helping resolve conflicts in the Middle East.For those of us who still earn our living by teaching literary texts written in foreign languages in foreign language departments in the university, Summers’s breezy assessment that the university of the future need not (indeed ought not) include extensive foreign language study is, sadly, not exactly news. On the contrary, there is nothing particularly innovative or original about the position he advocates here: his vision of a future “global” American university—in which virtually all speculative activities are conducted solely in English—has already begun to be implemented in various colleges and universities around the world. If his remarks are nonetheless noteworthy, it is because of his apparent willingness—even eagerness—to defend a policy that university officials often officially still pretend to abhor. Indeed, Summers’s position goes far beyond the more familiar pragmatic voices that have concluded that the reduction of foreign language instruction is a necessary and unavoidable consequence of the unfortunate economic reality that the American university as we once knew it has become financially untenable and hence unable to accommodate areas of study that can be construed as nonessential. Instead, Summers goes a step farther: he actively advocates against investing precious university resources in foreign language study on ideological, political, quasi-philosophical, or dare I say it, perhaps even theological, grounds. By appealing to the value of what he calls here “cosmopolitanism,” he clearly implies that English’s status as “global language” discharges English speakers in the U.S. university system from any potential obligation to any other language on the globe. [End Page 2]According to Summers, English is optimally positioned to be a vehicle for genuine “cosmopolitanism” because, as a so-called “global language,” it constitutes a receptive, inclusive, and welcoming “common ground” for multicultural exchange that would be otherwise impossible given the sheer volume and overwhelming diversity of all the other languages in the world. In other words, in his opinion, recourse to English actually facilitates and enhances—rather than inhibits—genuine multicultural encounters as it offsets the potentially... (shrink)

This brief work valuably shows how a distinguished historian ascertains the causes of his historical facts. Koht, a Norwegian European historian, eschews any philosophy of history, claiming only that the nature of man is permanent through historical change. Drawing from his own historical research he discusses the significance of the different forces of history. These are religion, economics, class consciousness, the power of the state, war, revolt, science, and internationalism. No one force or cause is primary.—A. B. D.

We develop the theory of partial satisfaction relations for structures that may be proper classes and define a satisfaction predicate ($\models^*$) appropriate to such structures. We indicate the utility of this theory as a framework for the development of the metatheory of first-order predicate logic and set theory, and we use it to prove that for any recursively enumerable extension $\Theta$ of ZF there is a finitely axiomatizable extension $\Theta'$ of GB that is a conservative extension of $\Theta$. We also (...) prove a conservative extension result that justifies the use of $\models^*$ to characterize ground models for forcing constructions. (shrink)

This paper criticises the view that, according to Marx, “productive forces” determine “relations of production” and that the growth of the former basically determines the course of history. The particular version of this account discussed is that to be found in G.A. Cohen’s Karl Marx's Theory of History: A Defence. The main part of this criticism in:-volves a presentation of what, it is suggested, was in fact Marx's conception of "productive forces", "relations of production" and their relations, and an identification (...) of class-struggle as the primary factor at least for the historically significant periods with which Marx was concerned. (shrink)

A discussion of the philosophical prospects for basing a neo-Fregean theory of classes on a principle that attempts to articulate the limitation-of-size conception.

From Italy’s capital to its northern borders runs an old racist ad-age: “Africa begins in Naples.” In a range of texts and films beginning in the late 1950s, Pier Paolo Pasolini shifted that proverbial frontier further north, to places like Pietralata, or Rebibbia, or Tiburtina: that is, to the edges of the Eternal City itself, to the borgate disinherited from authority both urban and institutional. The very notion of “Africa,” Pasolini writes in 1961,is the concept of an extremely complex subproletarian (...) condition that has yet to be utilized as a real revolutionary force. Perhaps I am best able to define this concept if we identify Africa with the entire world of Bandung [Indonesia, site of a... (shrink)

The class of stable posets is defined and investigated. We give a forcing construction of a universe of set theory which satisfies a weak form of Martin's Axiom and $2^{\aleph_0} > \aleph_1$ and yet some propositions which follow from CH hold in this universe.

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)

Journal of Mathematical Logic, Volume 22, Issue 03, December 2022. 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 [math]-Prikry, and showed that many of the known Prikry-type notions of forcing that center around singular cardinals of countable cofinality are [math]-Prikry. We showed that given a [math]-Prikry poset [math] and a (...) [math]-name for a non-reflecting stationary set [math], there exists a corresponding [math]-Prikry poset that projects to [math] and kills the stationarity of [math]. In this paper, we develop a general scheme for iterating [math]-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)

This note is a continuation of our former paper ''Complexity of the r-query tautologies in the presence of a generic oracle.'' We give a very short direct proof of the nonexistence of t-generic oracles, a result obtained first by Dowd. We also reconstitute a proof of Dowd's result that the class of all r-generic oracles in his sense has Lebesgue measure one.

Cette édition numérique a été réalisée à partir d'un support physique, parfois ancien, conservé au sein du dépôt légal de la Bibliothèque nationale de France, conformément à la loi n° 2012-287 du 1er mars 2012 relative à l'exploitation des Livres indisponibles du XXe siècle. Pages de début Introduction I - Questions générales 1 - Sur le concept de politique 2 - Politique et classes sociales 3 - Sur le concept de pouvoir II - L'État capitaliste 1 - Le problème 2 (...) - Typologie et type d'État capitaliste 3 - L'État absolutiste, État de transition 4 - Sur les modèles de la révolution bourgeoise III - Traits fondamentaux de l'État capitaliste Présentation 1 - L'État capitaliste et les intérêts des classes dominées 2 - L'État capitaliste et les idéologies 3 - L'État capitaliste et la force 4 - L'État capitaliste et les classes dominantes IV - L'unité du pouvoir et l'autonomie relative de l'État capitaliste 1 - Le problème et sa position théorique par les classiques du marxisme 2 - Quelques mésinterprétations et leurs conséquences 3 - L'État capitaliste et le champ de la lutte des classes 4 - L'État capitaliste et les classes dominantes 5 - Le problème dans les formes d'État et dans les formes de régime : l'exécutif et le législatif V - Sur la bureaucratie et les élites 1 - Le problème et les théories des élites 2 - La position marxiste et la question d'appartenance de classe de l'appareil d'État 3 - État capitaliste - Bureaucratisme - Bureaucratie4 - La bureaucratie et la lutte des classes Pages de fin. (shrink)

The Theme. Strong forcing axioms like Martin’s Maximum give a reasonably satisfactory structural analysis of $H(\omega _2)$. A broad program in modern Set Theory is searching for strong forcing axioms beyond $\omega _1$. In other words, one would like to figure out the structural properties of taller initial segments of the universe. However, the classical techniques of forcing iterations seem unable to bypass the obstacles, as the resulting forcings axioms beyond $\omega _1$ have not thus far been (...) strong enough! However, with his celebrated work on generalised side conditions, I. Neeman introduced us to a novel paradigm to iterate forcings. In particular, he could, among other things, reprove the consistency of the Proper Forcing Axiom using an iterated forcing with finite supports. In 2015, using his technology of virtual models, Veličković built up an iteration of semi-proper forcings with finite supports, hence reproving the consistency of Martin’s Maximum, an achievement leading to the notion of a virtual model.In this thesis, we are interested in constructing forcing notions with finitely many virtual models as side conditions to preserve three uncountable cardinals. The thesis constitutes six chapters and three appendices that amount to 118 pages, where Section 1 is devoted to preliminaries, and Section 2 is a warm-up about the scaffolding poset of a proper forcing. In Section 3, we present the general theory of virtual models in the context of forcing with sets of models of two types, where we, e.g., define the “meet” between two virtual models and prove its properties.The main results are joint with Boban Veličković, and partly appeared in Guessing models and the approachability ideal, J. Math. Log. 21 (2021).Pure Side Conditions. In Section 4, we use two types of virtual models (countable and large non-transitive ones induced by a supercompact cardinal, which we call Magidor models) to construct our forcing with pure side conditions. The forcing covertly uses a third type of models that are transitive. We also add decorations to the conditions to add many clubs in the generic $\omega _2$. In contrast to Neeman’s method, we do not have a single chain, but $\alpha $ -chains, for an ordinal $\alpha $ with $V_\alpha \prec V_\lambda $. Thus, starting from suitable large cardinals $\kappa <\lambda $, we construct a forcing notion whose conditions are finite sets of virtual models described earlier. The forcing is strongly proper, preserves $\kappa $, and has the $\lambda $ -Knaster property. The relevant quotients of the forcing are strongly proper, which helps us prove strong guessing model principles. The construction is generalisable to a ${<}\mu $ -closed forcing, for any given cardinal $\mu $ with $\mu ^{<\mu }=\mu <\kappa $.The Iteration Theorem. In Section 5, we use the forcing with pure side conditions to iterate a subclass of proper and $\aleph _2$ -c.c. forcings and obtain a forcing axiom at the level of $\aleph _2$. The iterable class is closely related to Asperó–Mota’s forcing axiom for finitely proper forcings.Guessing Model Principles. Section 6 encompasses the main parts of the thesis. We prove the consistency of the guessing principle $\mathrm {GMP}^+(\omega _3,\omega _1)$ that states for any cardinal ${\theta \geq \omega _3}$, the set of $\aleph _2$ -sized elementary submodels M of $H(\theta )$, which are the union of an $\omega _1$ -continuous $\in $ -chain of $\omega _1$ -guessing, I.C. models is stationary in $\mathcal P_{\omega _3}(H(\theta ))$. The consistency and consequences of this principle are demonstrated in the following diagram. We also prove that one can obtain the above guessing models in a way that the $\omega _1$ -sized $\omega _1$ -guessing models remain $\omega _1$ -guessing model in any outer transitive model with the same $\omega _1$, and we denote this principle by $\rm{SGMP}^+(\omega_3,\omega_1)$.In the following diagram, $\mathrm{TP}$ stands for the tree property; $w\mathrm{KH}$ stands for the weak Kurepa Hypothesis; $\mathrm{MP}$ stands for Mitchell property, i.e., the approachability ideal is trivial modulo the nonstationary ideal; $\mathrm{AP}$ stands for the approachability property; $\mathrm {AMTP}(\kappa ^+)$ states that if $2^\kappa <\aleph _{\kappa ^+}$, then every forcing which adds a new subset of $\kappa ^+$ whose initial segments are in the ground model, collapses some cardinal $\leq 2^{\kappa }$. The dotted arrow denotes the relative consistency, while others are logical implications.Appendices. Appendix A includes merely the above diagram. Appendix B presents a proof of the Mapping Reflection Principle with finite conditions under $\mathrm {PFA}$. Appendix C contains open problems. Finally, the thesis’s bibliography consists of 42 items.Abstract prepared by Rahman MohammadpourE-mail: [email protected]: https://theses.hal.science/tel-03209264. (shrink)

In this paper I will communicate some new consequences of the Proper Forcing Axiom. First, the Bounded Proper Forcing Axiom implies that there is a well ordering of R which is Σ 1 -definable in (H(ω 2 ), ∈). Second, the Proper Forcing Axiom implies that the class of uncountable linear orders has a five element basis. The elements are X, ω 1 , ω 1 * , C, C * where X is any suborder of (...) the reals of size ω 1 and C is any Countryman line. Third, the Proper Forcing Axiom implies the Singular Cardinals Hypothesis at κ unless stationary subsets of S κ + ω reflect. The techniques are expected to be applicable to other open problems concerning the theory of H(ω 2 ). (shrink)

The purpose of this article is to investigate how sex segregation, social class, and gender are analytically related to occupational stratification. Recent discussions of women and men in the labor force revolve around whether a sex-segregated model in which sex of the worker affects placement, a pure social class model using classical criteria, or a gendered social class model in which social organizational processes of a gendered social class structure affect positioning in the stratification system. This (...) article addresses the influence that social class and gender have on stratification in the labor market. Data from the 1972-82 General Social Survey on 371 occupations, representing 7,541 male and female workers, are used to formulate a social class scheme based on all workers' characteristics. In comparing the models, the analyses show that an integrated gendered social class perspective is consistent with the empirical differentiation found among women and men in the labor force. (shrink)

I analyze the hierarchies of the bounded and the weak bounded forcing axioms, with a focus on their versions for the class of subcomplete forcings, in terms of implications and consistency strengths. For the weak hierarchy, I provide level-by-level equiconsistencies with an appropriate hierarchy of partially remarkable cardinals. I also show that the subcomplete forcing axiom implies Larson’s ordinal reflection principle atω2, and that its effect on the failure of weak squares is very similar to that of (...) Martin’s Maximum. (shrink)