We show that it is relatively consistent with ZFC to have PCF structures of heightδ, for all ordinalsδ<ω3.

We mainly investigate models of set theory with restricted choice, e.g., ZF + DC + the family of countable subsets of $$\lambda $$ is well ordered for every $$\lambda $$ (really local version for a given $$\lambda $$ ). We think that in this frame much of pcf theory, (and combinatorial set theory in general) can be generalized. We prove here, in particular, that there is a proper class of regular cardinals, every large enough successor of singular is not measurable (...) and we can prove cardinal inequalities. Solving some open problems, we prove that if $$\mu> \kappa = \textrm{cf}(\mu ) > \aleph _{0},$$ then from a well ordering of $${\mathscr {P}}({\mathscr {P}}(\kappa )) \cup {}^{\kappa >} \mu $$ we can define a well ordering of $${}^{\kappa } \mu.$$. (shrink)

We prove that a combinatorial consequence of the negation of the PCF conjecture for intervals, involving free subsets relative to set mappings, is not implied by even the strongest known large cardinal axiom.

. The PCF Trichotomy Theorem deals with sequences of ordinal functions on an infinite $\kappa$ modulo some ideal I. If a $<_I$ -increasing sequence of ordinal functions has regular length which is larger than $\kappa^+$ , then by the Trichotomy Theorem the sequence satisfies one of three structural conditions. It was of some interest to find out if the Trichotomy Theorem could hold also for sequences of length $\kappa^+$ . It is shown that this is not the case.

This is a survey paper giving a self-contained account of Shelah's theory of the pcf function pcf={cf:D is an ultrafilter on a}, where a is a set of regular cardinals such that a

Logic and Philosophy of Logic

Logic and Philosophy of Logic, Miscellaneous in Logic and Philosophy of Logic

Model Theory in Logic and Philosophy of Logic

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Let$\kappa $be a regular uncountable cardinal, anda cardinal greater than or equal to$\kappa $. Revisiting a celebrated result of Shelah, we show that ifis close to$\kappa $and(= the least size of a cofinal subset of) is greater than, thencan be represented (in the sense of pcf theory) as a pseudopower. This can be used to obtain optimal results concerning the splitting problem. For example we show that ifand, then no$\kappa $-complete ideal onis weakly-saturated.

There exists a family $\{B_\alpha\}_{\alpha of sets of countable ordinals such that: (1) max B α = α, (2) if α ∈ B β then $B_\alpha \subseteq B_\beta$ , (3) if λ ≤ α and λ is a limit ordinal then B α ∩ λ is not in the ideal generated by the $B_\beta, \beta , and by the bounded subsets of λ, (4) there is a partition {A n } ∞ n = 0 of ω 1 such that for (...) every α and every n, B α ∩ A n is finite. (shrink)

§1. Introduction. Among the most remarkable discoveries in set theory in the last quarter century is the rich structure of the arithmetic of singular cardinals, and its deep relationship to large cardinals. The problem of finding a complete set of rules describing the behavior of the continuum function 2ℵα for singular ℵα's, known as the Singular Cardinals Problem, has been attacked by many different techniques, involving forcing, large cardinals, inner models, and various combinatorial methods. The work on the singular cardinals (...) problem has led to many often surprising results, culminating in a beautiful theory of Saharon Shelah called the pcf theory. The most striking result to date states that if 2ℵn < ℵω for every n = 0, 1, 2, …, then 2ℵω < ℵω4.In this paper we present a brief history of the singular cardinals problem, the present knowledge, and an introduction into Shelah's pcf theory. In Sections 2, 3 and 4 we introduce the reader to cardinal arithmetic and to the singular cardinals problems. Sections 5, 6, 7 and 8 describe the main results and methods of the last 25 years and explain the role of large cardinals in the singular cardinals problem. In Section 9 we present an outline of the pcf theory.§2. The arithmetic of cardinal numbers. Cardinal numbers were introduced by Cantor in the late 19th century and problems arising from investigations of rules of arithmetic of cardinal numbers led to the birth of set theory. The operations of addition, multiplication and exponentiation of infinite cardinal numbers are a natural generalization of such operations on integers. Addition and multiplication of infinite cardinals turns out to be simple: when at least one of the numbers κ, λ is infinite then both κ + λ and κ·λ are equal to max {κ, λ}. In contrast with + and ·, exponentiation presents fundamental problems. In the simplest nontrivial case, 2κ represents the cardinal number of the power set P, the set of all subsets of κ. (Here we adopt the usual convention of set theory that the number κ is identified with a set of cardinality κ, namely the set of all ordinal numbers smaller than κ. (shrink)

The following pcf results are proved:1. Assume thatκ>ℵ0κ>ℵ0is a weakly compact cardinal. Letμ>2κμ>2κbe a singular cardinal of cofinality κ. Then for every regularView the MathML sourceλ sup{suppcfσ⁎-complete|a⊆Reg∩and|a|<μ}.Turn MathJax onAs an application we show that:if κ is a measurable cardinal andj:V→Mj:V→Mis the elementary embedding by a κ-complete ultrafilter over κ, then for every τ the following holds:1. ifjjis a cardinal thenj=τj=τ;2. |j|=|j)||j|=|j)|;3. for any κ-complete ultrafilter W on κ, |j|=|jW||j|=|jW|.The first two items provide affirmative answers to questions from Gitik and Shelah (...) [2] and the third to a question of D. Fremlin. (shrink)

We deal with several pcf problems: we characterize another version of exponentiation: maximal number of κ-branches in a tree with λ nodes, deal with existence of independent sets in stable theories, possible cardinalities of ultraproducts and the depth of ultraproducts of Boolean Algebras. Also we give cardinal invariants for each λ with a pcf restriction and investigate further T D (f). The sections can be read independently, although there are some minor dependencies.

We consider simplified representation theorems in pcf-theory and, in particular, we prove that if ${\aleph_{\omega}^{\aleph_{0}} > \aleph_{\omega_{1}}\cdot2^{\aleph_{0}}}$ then there are cofinally many sequences of regular cardinals such that ${\aleph_{\omega_{1}+1}}$ is represented by these sequences modulo the ideal of finite subsets, using a topological approach to the pcf-structure.

In this essay I consider the fundamental features of Althusser’s reading of Machiavelli in its historical development, starting from the 1962 lecture course, passing through the 1972–76 course published with the title of Machiavel et nous as well as the writings of 1977–78, concluding with the group of writings written during the 1980s. I show that any teleological reading that sees in the final writings the truth finally revealed of the path of Althusser’s reading of Machiavelli should be rejected, and (...) instead present the thesis that Althusser, through the figure of Machiavelli, theoretically reworks his relationship with the Party. (shrink)

Cette note devait introduire à un public anglophone la traduction de la « Lettre de Louis Althusser datée du 18 mars 1966 et adressée au Comité central du PCF », elle est ici enrichie dans une version livrée au public français. Elle apporte le contexte historique et théorique nécessaire à la compréhension des interventions « anti-humanistes » de Louis Althusser qui questionne les choix politiques opérés par le PCF au cours des années 1960. Nulle part ailleurs, dans les écrits publiés (...) d’Althusser, nous ne voyons aussi clairement les enjeux politiques imbriqués dans son projet philosophique, ni comment ce projet évolua progressivement d’une entreprise « théoriciste » à la recherche d’une philosophie en lien plus étroit avec la pratique. Cette lettre nous paraît de la plus haute importance pour les spécialistes de l’œuvre d’Althusser, les historiens de la pensée marxiste, mais aussi plus largement à tous ceux qui s’intéressent à l’importance que peut revêtir le travail d’Althusser dans les débats actuels dans le champ de la philosophie marxiste. (shrink)

As an accompaniment to the translation into English of Louis Althusser's 'Letter to the Central Committee of the PCF, March 18th, 1966', this note provides the historical and theoretical context necessary to understand Althusser's 'anti-humanist' interventions into French Communist Party policy decisions during the mid-1960s. Because nowhere else in Althusser's published writings do we see as clearly the political stakes involved in his philosophical project, nor the way in which this project evolved from a 'theoreticist' pursuit into a more practical (...) one, the note also argues that the letter is of importance to Althusser scholars, to historians of Marxist thought, and to those interested in the relevance of Althusser's work to contemporary Marxist philosophy. (shrink)

We present a survey of combinatorial set theory relevant to the study of singular cardinals and their successors. The topics covered include diamonds, squares, club guessing, forcing axioms, and PCF theory.

This paper attempts to present and organize several problems in the theory of Singular Cardinals. The most famous problems in the area (bounds for the ℶ-function at singular cardinals) are well known to all mathematicians with even a rudimentary interest in set theory. However, it is less well known that the combinatorics of singular cardinals is a thriving area with results and problems that do not depend on a solution of the Singular Cardinals Hypothesis. We present here an annotated collection (...) of representative problems with some references. Where the problems are novel, attribution is attempted and it is noted where money is attached to particular problems. Three closely related themes are represented in these problems: stationary sets and stationary set reflection, variations of square and approachability, and the singular cardinals hypothesis. Underlying many of them are ideas from Shelah's PCF theory. Important subthemes were mutual stationarity, Aronszajn trees, and superatomic Boolean Algebras. The author notes considerable overlap between this paper and the unpublished report submitted to the Banff Center for the Workshop on Singular Cardinals Combinatorics, May 1–5, 2004. (shrink)

Avant même d'être condamné pour "contestation de crime contre l'humanité" au tournant du siècle, Roger Garaudy était déjà marginalisé dans les champs intellectuel et politique français. Après avoir incarné la résistance au néostalinisme dans le Parti communiste, le philosophe qui fut longtemps l'interlocuteur privilégié de Jean-Paul Sartre au sein du PCF en fut spectaculairement exclu en 1970 pour s'être opposé aux Soviétiques qui venaient d'écraser le Printemps de Prague. Celui que l'historiographie du communisme retient plus volontiers comme le "stalinien modèle", (...) qu'il fut en effet dans l'après-guerre, était pourtant devenu, malgré sa proximité avec Thorez, un critique de plus en plus affirmé du stalinisme après les révélations du XXe Congrès du Parti communiste de l'Union soviétique (PCUS) en 1956 et un acteur majeur de l'aggiornamento du PCF. Son combat pour rénover le communisme français et son goût précoce pour l'écologie et le dialogue des civilisations en firent quelque temps une figure respectée du progressisme français, mais celui qui répétait que le Socialisme était condamné s'il refusait de s'ouvrir à la "transcendance" vit son étoile pâlir après 1970 au point qu'il devint un intellectuel illégitime, celui que l'on condamne sans l'avoir lu. Ce livre explique ce parcours singulier et les raisons de cette "descente aux enfers" de l'ancien philosophe du Bureau politique, avant sa condamnation ultime."--Page 4 of cover. (shrink)

Since the work of Gödel and Cohen, which showed that Hilbert's First Problem was independent of the usual assumptions of mathematics, there have been a myriad of independence results in many areas of mathematics. These results have led to the systematic study of several combinatorial principles that have proven effective at settling many of the important independent statements. Among the most prominent of these are the principles diamond and square discovered by Jensen. Simultaneously, attempts have been made to find suitable (...) natural strengthenings of ZFC, primarily by Large Cardinal or Reflection Axioms. These two directions have tension between them in that Jensen's principles, which tend to suggest a rather rigid mathematical universe, are at odds with reflection properties. A third development was the discovery by Shelah of "PCF Theory", a generalization of cardinal arithmetic that is largely determined inside ZFC. In this paper we consider interactions between these three theories in the context of singular cardinals, focusing on the various implications between square and scales, and on consistency results between relatively strong forms of square and stationary set reflection. (shrink)

We use pcf theory to prove results on reflection at singular cardinals.

We deal here with colorings of the pair , when μ is a strong limit and singular cardinal. We show that there exists a coloring c with no refinement. It follows that the properties of colorings of when μ is singular differ in an essential way from the case of regular μ.

Since the work of Gödel and Cohen, which showed that Hilbert's First Problem was independent of the usual assumptions of mathematics, there have been a myriad of independence results in many areas of mathematics. These results have led to the systematic study of several combinatorial principles that have proven effective at settling many of the important independent statements. Among the most prominent of these are the principles diamond and square discovered by Jensen. Simultaneously, attempts have been made to find suitable (...) natural strengthenings of ZFC, primarily by Large Cardinal or Reflection Axioms. These two directions have tension between them in that Jensen's principles, which tend to suggest a rather rigid mathematical universe, are at odds with reflection properties. A third development was the discovery by Shelah of "PCF Theory", a generalization of cardinal arithmetic that is largely determined inside ZFC. In this paper we consider interactions between these three theories in the context of singular cardinals, focusing on the various implications between square and scales, and on consistency results between relatively strong forms of square and stationary set reflection. (shrink)

In a little remembered event in December 1961, Sartre entered into a debate with Roger Garaudy, as well as other representatives of the Parti Communiste Française (PCF), on the topic of the existence of a universal dialectical law applicable to nature as well as to human thought. In the debate, Sartre seeks to rebut the notion that humankind is merely an “alien addition“ to nature, as Engels maintained, and instead argues that individual subjectivity cannot be reduced to an object of (...) knowledge. This paper highlights the importance of the debate for both sides, but particularly for Sartre and his Critique of Dialectical Reason. (shrink)

A militância política de Jorge Amado no Partido Comunista é um capítulo à parte na história do comunismo no Brasil, que completou 100 anos no dia 25 de março. Ela também é algo marcante em sua obra, a ponto da crítica literária brasileira, de tradição uspiana, considerá-la uma forma de panfleto partidário da nossa esquerda. Nesta exposição, pretende-se realizar uma breve discussão acerca de suas lutas políticas, das quais se destacam as questões religiosa e racial, que levaram Jorge Amado a (...) criar um projeto de lei, no período que exerceu o mandato de deputado constituinte pelo PCB, em 1946, assegurando a liberdade religiosa no Brasil. Aqui, também se destaca a sua amizade com o povo de candomblé que lhe honrou com títulos, como o de Obá de Xangô e Ogã de Iansã; além da forma como a crítica o rotulava de regionalista e escritor panfletário. Contudo, esta é uma forma de homenagear tanto o mais famoso dos doze Obás de Xangô da Bahia, quanto o Partidão, nesse seu centenário. Em termos metodológicos, trata-se de uma leitura ensaística foucaultiana sobre esses fatos e acontecimentos que marcaram duplamente a vida de Jorge Amado e do Comunismo no Brasil, já que lançamos mão de uma parte da maquinaria teórica do militante comunista do PCF (Partido Comunista Francês), Michel Foucault, como vontade de verdade, comentário, discurso, autoria, obra, descontinuidade (na continuidade) e dispersão. (shrink)

We address three questions raised by Cummings and Foreman regarding a model of Gitik and Sharon. We first analyze the PCF-theoretic structure of the Gitik–Sharon model, determining the extent of good and bad scales. We then classify the bad points of the bad scales existing in both the Gitik–Sharon model and other models containing bad scales. Finally, we investigate the ideal of subsets of singular cardinals of countable cofinality carrying good scales.

We start by studying the relationship between two invariants isolated by Shelah, the sets of good and approachable points. As part of our study of these invariants, we prove a form of “singular cardinal compactness” for Jensen's square principle. We then study the relationship between internally approachable and tight structures, which parallels to a certain extent the relationship between good and approachable points. In particular we characterise the tight structures in terms of PCF theory and use our characterisation to prove (...) some covering results for tight structures, along with some results on tightness and stationary reflection. Finally, we prove some absoluteness theorems in PCF theory, deduce a covering theorem, and apply that theorem to the study of precipitous ideals. (shrink)

Questo saggio esplora la critica a Marx e al “marxismo” sviluppata da Simone Weil nel corso della sua opera. Vengono identificati tre nuclei principali della critica weiliana al marxismo: la validità del metodo materialistico per lo studio delle scienze sociali; il presunto “messianesimo” di Marx, inteso come la concezione della storia come un progresso intelligente verso il meglio; e l’identificazione del marxismo tout court con una filosofia determinista e meccanicista. Si cerca di definire a quale “Marx” effettivamente sia rivolta la (...) critica di Weil: si sostiene infatti che alcune delle sue argomentazioni siano riferibili alla “vulgata” marxiana, ossia a una lettura semplificata e schematica del marxismo, o alla “tendenza deteriore” del marxismo caratterizzata dall’economicismo e dal meccanicismo, dibattuta anche all’interno della Seconda Internazionale, o a posizioni di Engels e Lenin, o ancora all’operato del PCF o delle organizzazioni operaie, piuttosto che all’effettiva opera di Marx. (shrink)

Let λ ≤ κ be infinite cardinals and let Ω be a set of cardinality κ. The bounded permutation group B λ (Ω), or simply B λ , is the group consisting of all permutations of Ω which move fewer than λ points in Ω. We say that a permutation group G acting on Ω is a supplement of B λ if B λ G is the full symmetric group on Ω. In [7], Macpherson and Neumann claimed to have classified (...) all supplements of bounded permutation groups. Specifically, they claimed to have proved that a group G acting on the set Ω is a supplement of B λ if and only if there exists $\Delta \subset \Omega$ with $|\Delta| such that the setwise stabiliser G {Δ} acts as the full symmetric group on $\Omega\setminus\Delta$ . However I have found a mistake in their proof. The aim of this paper is to examine conditions under which Macpherson and Neumann's claim holds, as well as conditions under which a counterexample can be constructed. In the process we will discover surprising links with cardinal arithmetic and Shelah's recently developed pcf theory. (shrink)

In 1972 the Socialist Party and the Communist French Party signed the Programme Commune du Gouvenement d'Union de la Gauche. Six years marked the electoral agenda of the two “historical enemies” in search of political convergence that remained an open debate, also animated by the current of antitotalitarian thought that increased public disrepute towards the pro-Soviet left and the PCF. Political and ideological differences between alternative government guidelines were amplified by the publication of Alexander Solgenitsin’s Gulag Archipelago which will significantly (...) influence the philosophical discourse and French political dynamics of that historical ridge. (shrink)

Cet article vise à présenter les résultats d'une recherche portant sur la réception des premières études françaises sur le matérialisme dialectique, publiées en 1935 dans le recueil intitulé A la lumière du marxisme. L'analyse des nombreux compte-rendus et la reconstruction du débat autour de ce volume, à travers lequel est introduit en France le marxisme en tant que philosophie, jette une lumière nouvelle sur la genèse du "rationalisme moderne", dont se réclame, lors de sa fondation en 1939, la revue liée (...) au Pcf, La Pensée. On découvre ainsi que l'image figée du marxisme à la française, fondé sur la reprise de la tradition nationale matérialiste et rationaliste, telle qu'elle sera transmise à partir de l'après-guerre, est le résultat d'une lutte idéologique dans laquelle des positions différentes s'étaient affrontées. En effet, au début de la décennie les intellectuels réunis autour de la Commission scientifiques du Cercle de la Russie neuve avaient essayé d'envisager le marxisme principalement comme une méthodologie vivante pour l'avancement des sciences, y compris de la nature, et que dans cette appropriation épistémologique de la dialectique matérialiste la référence à Hegel y joue un rôle crucial pour l'appropriation et l'introduction de la philosophie de Marx dans le milieu académique. Cependant, à travers les recensions qui ont été faites de leurs premiers essaies, différentes pressions sont exercées, provenant principalement des instances de contrôle soviétiques, pour que l'étude de la "source française" du marxisme remplace la "source allemande". Malgré leur résistance initiale, dont témoignent les débats qui s'ensuivent, les "néo-marxistes" finissent par suivre ces invitations à abandonner la voie hégélienne à la dialectique matérialiste pour tracer une ligne nationale qui de Descartes mènerait jusqu'à Staline. Mots-clés: Marxisme. Matérialisme dialectique. Rationalisme moderne. Louis Althusser. Epistémologie marxiste. (shrink)

El objetivo de este trabajo es abordar la problemática del marxismo desde el pensamiento de Jean Paul Sartre y el de Maurice Merleau-Ponty a partir de su dimensión teórica y de su dimensión práctica. Con esta finalidad el presente estudio se encuentra estructurado en cuatro apartados. En el primero de ellos, se aborda la relación intrínseca entre filosofía e historia para ambos autores. El segundo apartado tiene como eje la discusión en torno al comunismo real y la forma en que (...) Sartre y Merleau-Ponty tomarán posición con respecto a las políticas de la URSS y del PCF. En el tercer apartado, se expone la forma en que, en la Crítica de la razón dialéctica, Sartre concibe al marxismo como la filosofía insuperable de su tiempo. El cuarto y último apartándose centra en la comprensión del marxismo como momento de la a-filosofía que lleva a cabo Merleau-Ponty en una de sus últimas notas de curso, lo que significa, entre otras cosas, una oposición explícita a la afirmación que Sartre realiza en la Crítica. (shrink)

We replace Shelah's notion of true cofinality by the notion of the bounding number for an arbitrary partial order and begin to develop a theory similar to Shelah's pcf theory, which gives many analog results, including the existence of the so-called generators, for the more general case of products of partial orders. The development can be strictly divided into an ideal theoretical and a combinatorial part. We also show that pcf theory is a special case of this more general theory (...) and conclude with some remarks about Shelah's function pp, which also show that there are some differences between pcf theory and the presented theory of bounding numbers. (shrink)

As Dag Normann has recently shown, the fully abstract model for PCF of hereditarily-sequential functionals is not ω-complete . This is also applicable to a potentially wider class of models such as the recently constructed by the author fully abstract model for PCF+=PCF+pif . Here we will present an outline of a general approach to this kind of ‘natural’ domains which, although being non-dcpos, allow considering ‘naturally’ continuous functions . There is also an appropriate version of ‘naturally’ algebraic and ‘naturally’ (...) bounded complete ‘natural’ domains which serves as the non-dcpo analogue of the well-known concept of Scott domains, or equivalently, the complete f-spaces of Ershov. It is shown that this special version of ‘natural’ domains, if considered under ‘natural’ Scott topology, exactly corresponds to the class of f-spaces, not necessarily complete. (shrink)

Noetherian type and Noetherian π-type are two cardinal functions which were introduced by Peregudov in 1997, capturing some properties studied earlier by the Russian School. Their behavior has been shown to be akin to that of the cellularity, that is the supremum of the sizes of pairwise disjoint non-empty open sets in a topological space. Building on that analogy, we study the Noetherian π-type of κ-Suslin Lines, and we are able to determine it for every κ up to the first (...) singular cardinal. We then prove a consequence of Changʼs Conjecture for ℵω regarding the Noetherian type of countably supported box products which generalizes a result of Lajos Soukup. We finish with a connection between PCF theory and the Noetherian type of certain Pixley–Roy hyperspaces. (shrink)

En Occident, Sartre a été le maître à penser d'une génération où les rapports avec le communisme se trouvaient au coeur des débats intellectuels et politiques. Or, si le philosophe a eu des relations souvent mauvaises avec le PCF, il a revendiqué ses liens avec l'URSS entre 1952 et 1968, malgré une pause provoquée par l'intervention militaire soviétique à Budapest. Sartre s'est même rendu onze fois en URSS, le plus souvent avec Simone de Beauvoir, et ces séjours, qui allaient de (...) quelques heures à plus d'un mois, sont au coeur de ce livre grâce aux rapports rédigés par les accompagnateurs soviétiques du couple. Le philosophe y rencontrait des auteurs officiels qu'il considérait parfois comme des amis et dont les parcours permettent de dresser les contours d'une histoire culturelle de l'URSS. Mais ces voyages et ces contacts s'inscrivaient aussi dans des pratiques établies par Moscou dès le début des années 1920 pour influencer, séduire et manipuler des Occidentaux. Le cas de Sartre est d'autant plus intéressant que le philosophe a créé et incarné la théorie de l'engagement : pour lui, chacun a la responsabilité de changer le monde et doit agir en ce sens ; chacun est responsable de ses actes et sera jugé sur ceux-ci. La question se pose donc : avec l'URSS, pour quoi Sartre s'engageait-il et avec qui?"--Page 4 of cover. (shrink)

The purpose of the present paper is to present new methods of blowing up the power of a singular cardinal κ of cofinality ω. New PCF configurations are obtained. The techniques developed here will be used in a subsequent paper to construct a model with a countable set which pcf has cardinality ℵ1.

We prove a compactness theorem for pseudopower operations of the form \}\) where \\le {{\,\mathrm{cf}\,}}\). Our main tool is a result that has Shelah’s cov versus pp Theorem as a consequence. We also show that the failure of compactness in other situations has significant consequences for pcf theory, in particular, implying the existence of a progressive set A of regular cardinals for which \\) has an inaccessible accumulation point.

I will give a brief overview of Saharon Shelah’s work in mathematical logic. I will focus on three transformative contributions Shelah has made: stability theory, proper forcing and PCF theory. The first is in model theory and the other two are in set theory.

We survey the use of club guessing and other PCF constructs in the context of showing that a given partially ordered class of objects does not have a largest, or a universal, element.

Parmi les courants de pensée française de l'après-guerre, le marxisme joua un rôle très important, en imposant les thèmes par rapport auxquels les philosophes se situèrent. Les débats sur le marxisme ne sont pas séparables de l'existence du pays qui prétendait le réaliser dans toutes les sphères de sa vie. La politique comme une action collective en vue de la transformation de la société par les débats et par l'émancipation n'a pas été développée, ni dans les pays qu'on disait socialistes, (...) ni dans les pays dits 'bourgeois'. En même temps, nous pensons que les intellectuels français de gauche dans l'après-guerre ont considérablement contribué au renouveau de la pensée socialiste. Il y avait ceux qui adhéraient au PCF, et d'autres, comme Jean-Paul Sartre, qui devenaient 'compagnons de route'. D'autres intellectuels encore, comme Cornelius Castoriadis et Claude Lefort, voulaient former des groupuscules indépendants en critiquant la politique du PCF stalinien et du PCI trotskiste. La crise du socialisme réel, les luttes anti-bureaucratiques dans les pays d'Europe de l'Est et la découverte de la vérité sur les camps soviétiques, ont provoqué des ruptures de certains de ces intellectuels avec le marxisme. Les changements démocratiques en ont conduit certains à envisager la perspective d'une révolution démocratique au-delà du 'socialisme réel'."--Page 4 of cover. (shrink)

We show that to every recursive total continuous functional $\Phi$ there is a PCF-definable representative $\Psi$ of $\Phi$ in the hierarchy of partial continuous functionals, where PCF is Plotkin's programming language for computable functionals. PCF-definable is equivalent to Kleene's S1-S9-computable over the partial continuous functionals.

We prove that the Kleene schemes for primitive recursion relative to the μ-operator, relativized to some nondeterministic objects, have the same power to express total functionals when interpreted over the partial continuous functionals and over the Kleene-Kreisel continuous functionals. Relating the former interpretation to Niggl's ℳω we prove Nigg's conjecture that ℳω is strictly weaker than Plotkin's PCF + PA.

We relate two sequential models of PCF: the sequential algorithm model due to Berry and Curien and the strongly stable model due to Bucciarelli and the author. More precisely, we show that all the morphisms araising in the strongly stable model of PCF are sequential in the sense that they are the “extensional projections” of some sequential algorithms. We define a model of PCF where morphisms are “extensional” sequential algorithms and prove that any equation between PCF terms which holds in (...) this model also holds in the strongly stable model. (shrink)

We prove that for every singular cardinal μ of cofinality ω, the complete Boolean algebra contains a complete subalgebra which is isomorphic to the collapse algebra CompCol. Consequently, adding a generic filter to the quotient algebra collapses μ0 to 1. Another corollary is that the Baire number of the space U of all uniform ultrafilters over μ is equal to ω2. The corollaries affirm two conjectures of Balcar and Simon. The proof uses pcf theory.

We show that the consistency of the theories “ holds below ” + “there is an injection ” + “both and fail” and + “ holds below ” + “there is an injection ” + “ satisfies the tree property” follow from the appropriate supercompactness hypotheses. These provide answers in a choiceless context to certain long‐standing open questions concerning, weak square, approachability, and the tree property. There is nothing special about, and the injection into can be from any ordinal λ (...) (which means that pcf theory can fail badly in the models constructed if ). (shrink)