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.

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

. 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.

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)

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.

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.

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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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)

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)

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.

The development of the forcing method has shown that several key questions regarding infinite sets cannot be settled under ZFC alone. The most widely supported view is that this undecidability simply reflects the limitations of ZFC in addressing all mathematical problems. This perspective has motivated an extensive search for new axioms—the so-called large cardinal axioms—which, when added to ZFC, yield a deeper and more robust understanding of the set-theoretic universe. Along this line, the dissertation is divided into three thematic blocks, (...) each of them framed within these extensions of ZFC: • Very large cardinals at the threshold of Kunen inconsistency, with a focus on elementarity and cardinal correctness (Chapter 2). • Generalized Descriptive Set Theory at singular strong limit cardinals of countable cofinality, with a focus on two regularity properties (Chapter 3). • Covering lemmas and Woodin’s HOD Dichotomy through the lens of Shelah’s pcf theory (Chapter 4). Specifically, Chapter 2 establishes an inconsistency result using tools from singular cardinal combinatorics and Shelah’s pcf theory, proving the nonexistence of cardinal preserving elementary embeddings into V and establishing thereby a limitation in the hierarchy of large cardinal axioms. The proof is based on the notion of good scales and its connection with Jónsson cardinals. Chapter 3 proves a consistency result obtained via a Prikry-type forcing construction, providing a singular-cardinal analogue of Solovay’s theorem. This thematic block is inspired by Woodin’s Axiom $I_0$ which provides the appropriate axiomatic framework to develop Generalized Descriptive Set Theory in generalized Baire/Cantor spaces at singular cardinals. Concretely, we construct a model of ZFC where $\kappa $ is a strong limit singular cardinal with countable cofinality, and every subset of ${}^\omega \kappa $ in $L(V_{\kappa +1})$ has both the $\kappa $ -Perfect Set Property and the $\vec {\mathcal {U}}$ -Baire Property. Chapter 4 further explores the study of “covering lemmas” and Woodin’s HOD Dichotomy, employing both the perspective and the tools of pcf theory. Specifically, the connection between the cover property and a new pcf-theoretic concept—called the scale property—is analyzed. This analysis builds a bridge between combinatorial principles in pcf theory and the structural behavior of HOD. Abstract prepared by Sebastiano Thei. E-mail: [email protected]. URL: https://air.uniud.it/handle/11390/1316344?mode=full. (shrink)

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)

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 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.

Cummings, Foreman, and Magidor proved that Jensen’s square principle is non-compact at $\aleph _\omega $, meaning that it is consistent that $\square _{\aleph _n}$ holds for all $n<\omega $ while $\square _{\aleph _\omega }$ fails. We investigate the natural question of whether this phenomenon generalizes to singulars of uncountable cofinality. Surprisingly, we show that under some mild ${{\mathsf {PCF}}}$ -theoretic hypotheses, the weak square principle $\square _\kappa ^*$ is in fact compact at singulars of uncountable cofinality.

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)

We consider mainly the following version of set theory: “ZF+DC and for every λ,λℵ0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\lambda, \lambda^{\aleph_0}}$$\end{document} is well ordered”, our thesis is that this is a reasonable set theory, e.g. on the one hand it is much weaker than full choice, and on the other hand much can be said or at least this is what the present work tries to indicate. In particular, we prove that for a sequence δ¯=⟨δs:s∈Y⟩,cf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} (...) \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{\delta} = \langle\delta_{s}: s \in Y\rangle, {\rm cf}}$$\end{document} large enough compared to Y, we can prove the pcf theorem with minor changes.We then deduce the existence of covering numbers and define and prove existence of a class of true successor cardinals. Using this we give some diagonalization arguments on Abelian groups, chosen as a characteristic case.We end by showing that some such consequences hold even in ZF above. (shrink)

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

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)

An elementary embedding [Formula: see text] between two inner models of [Formula: see text] is cardinal preserving if M and N correctly compute the class of cardinals. We look at the case [Formula: see text] and show that there is no nontrivial cardinal preserving elementary embedding from M into V, answering a question of Caicedo.

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)

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)

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)

We study the role of meeting numbers in pcf theory. In particular, Shelah's Strong Hypothesis is shown to be equivalent to the assertion that for any singular cardinal σ of cofinality ω, there is a size collection Q of countable subsets of σ with the property that for any infinite subset a of σ, there is a member of Q meeting a in an infinite set.

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.

Jean-Paul Sartre pointed to the ‘gap’ in Marxism where a theory of the individual should be. Three attempts to fill it vied in the context of an intense ideological debate within the Parti Communiste Français (PCF) which still resonates today. On the one hand, Louis Althusser’s denial of individual agency as traditionally understood, structuralist theory which proved difficult to apprehend, let alone apply, in a capitalist world. On the other, Roger Garaudy’s Marxist humanist explanation of personality, most likely unsatisfactory for (...) many Marxists on account of its spiritualistic leanings. Finally, Lucien Sève’s approach: very preliminary, often awkwardly formulated, and diluted over time, but a real alternative to structural Marxism and perhaps, in its final diluted form, not incompatible with a humanist approach after all. If the debate could be settled in his favour, the detailed practical work of applying the theory to psychology and other social sciences should take over with some urgency from here. (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)

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.

Port cities are internationally connected. Decisions and changes occurring in one city have a direct impact on port cities in other parts of the world. Studying these areas provides insight into social and spatial processes in which local communities and urban development are interconnected with global processes. The PortCityFutures project researches various themes within the port and urban areas. Asma Mehan is one of the researchers involved in this project since June 2020 and works at CADS. What exactly is PCF (...) and why is research in port areas important? An introduction to Asma Mehan and PCF. (shrink)

Serge Daney is widely recognised in his homeland as the most important French film critic after André Bazin. In a career devoted to criticism for Cahiers du cinéma and later Libération (where his remit widened to include other forms of journalism), including a key period as editor during the transition from the journal’s PCF and then Maoist phase beginning in 1973, Daney also held a lecturing position for a spell at the University of Paris, Paris III, La Censure. He was (...) a significant public intellectual and featured in several documentaries, including Claire Denis’ film Jacques Rivette – Le veilleur ([Jacques Rivette – The Night Watchman] 1990). From 1985 to 1990 Daney presented a programme on cinema on the radio station France Culture. Following the publication of a book on Haitian politics in 1973 under the assumed name Raymond Sapène, Daney’s journalism was collected in several volumes. He left Libération in 1981 to establish Trafic, a journal which, since his death from Aids in 1992, has continued his legacy. The only book-length English translation of Daney’s writings to date – Postcards from the Cinema (2007) – is of the posthumously published Persévérances. (shrink)

This paper provides a discussion of Jacques Rancière’s former teacher at École Normale Supérieure (ÉNS), then famous for fashioning Marxism with the philosophical gauge of structuralism, Louis Althusser (1918-1990). Perhaps a brief discussion on the relation between the two would render context to the origins of Rancière's philosophico-political praxis, specifically the humble beginnings of conceptualizing an egalitarian method out of his philosophical rupture with Althusserianism. Meanwhile, to reduce the philosophical enterprise of Althusser into its practical shortcomings and silence during the (...) revolutionary events of May 1968 in France would do an injustice to the magnitude of his contribution to contemporary French political theory and his major revisions in the theoretical direction of the Parti communiste français (PCF). Thus, the following discussions focus on sketching Althusser's theoretical foundations, which possibly clarifies the political decision he has made during May '68: a demand for organization over spontaneous revolutionary activity based on the authority of theoretical practice over the ideological activities—a decision that became the point of departure for Rancière's subversion of both mastery and the structural inequality Althusserianism entail. The whole piece is guided by the following two-fold question—a question, perhaps, akin to Badiou's inquiry: What were the philosophico-political interventions of Louis Althusser, and why did Rancière move away from his direction? (shrink)

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.

Club guessing principles were introduced by Shelah as a weakening of Jensen’s diamond. Most spectacularly, they were used to prove Shelah’s $\textsf{ZFC}$ bound on $2^{\aleph _\omega }$. These principles have found many other applications: in cardinal arithmetic and PCF theory; in the construction of combinatorial objects on uncountable cardinals such as Jónsson algebras, strong colourings, Souslin trees, and pathological graphs; to the non-existence of universals in model theory; to the non-existence of forcing axioms at higher uncountable cardinals; and many more.In (...) this paper, the first part of a series, we survey various forms of club guessing that have appeared in the literature, and then systematically study the various ways in which a club guessing sequence can be improved, especially in the way the frequency of guessing is calibrated.We include an expository section intended for those unfamiliar with club guessing and which can be read independently of the rest of the article. (shrink)

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.

The ethical terrain surrounding feeding at the end-of-life is emotionally charged and complex. In the literature, the recently introduced notion of “minimal comfort feeding” (MCF), defined as deliberate limitation of feeding or hydration, given only in response to visible signs of hunger or thirst, for a patient with a longer prognosis, has been proposed as a comfort measure at the end of life and as an alternative to voluntarily stopping eating and drinking (VSED) in advance care planning. However, the same (...) term may be applied to two morally distinct acts: 1) providing comfort in the context of active dying, and 2) a hastening death. In this paper, we argue for conceptual clarity around these two practices and their ethical valence, drawing on an Anscombean-Thomistic virtue ethics framework that deals with moral evaluation of human action and the doctrine of double effect. We present two clinical cases highlighting each of these ethical scenarios. Through the cases, we show how intentionality and some circumstances such as timing critically shape the morality of the act. We propose a new nomenclature with the aim of fostering ethical clarity of the morally distinct acts: “proportionate comfort feeding” (PCF) for MCF where nutrition and hydration is provided for comfort with the understanding that it may hasten death but is its foreseen, unintended outcome, and “minimal feeding to hasten death” (MFHD) for cases where MCF is chosen with the specific intention to hasten death. (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)

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 consider a question of Pereira as to whether the characteristic function of an internally approachable model can lead to free subsets for functions of the model. Pereira isolated the pertinent Approachable Free Subsets Property (AFSP) in his work on the $${\text {pcf}}$$ pcf -conjecture. A recent related property is the Approachable Bounded Subset Property (ABSP) of Ben-Neria and Adolf, and we here directly show it requires modest large cardinals to establish: Theorem If ABSP holds for an ascending sequence $$ (...) \langle \aleph _{n_{m}} \rangle _{m}$$ ⟨ ℵ n m ⟩ m $$( n_{m} \in \omega )$$ ( n m ∈ ω ) then there is an inner model with measurables $$\kappa < \aleph _{\omega }$$ κ < ℵ ω of arbitrarily large Mitchell order below $$\aleph _{\omega }$$ ℵ ω, that is: $$\sup \left\{ \alpha \mid {\exists }\kappa < \aleph _{\omega } o ( \kappa ) \ge \alpha \right\} = \aleph _{\omega }$$ sup α ∣ ∃ κ < ℵ ω o ( κ ) ≥ α = ℵ ω. A result of Adolf and Ben Neria then shows that this conclusion is in fact the exact consistency strength of ABSP for such an ascending sequence. Their result went via the consistency of the non-existence of continuous tree-like scales; the result of this paper is direct and avoids the use of PCF scales. (shrink)

Let S be the group of all permutations of the set of natural numbers. The cofinality spectrum CF(S) of S is the set of all regular cardinals λ such that S can be expressed as the union of a chain of λ proper subgroups. This paper investigates which sets C of regular uncountable cardinals can be the cofinality spectrum of S. The following theorem is the main result of this paper. Theorem. Suppose that $V \models GCH$ . Let C be (...) a set of regular uncountable cardinals which satisfies the following conditions. (a) C contains a maximum element. (b) If μ is an inaccessible cardinal such that $\mu = \sup(C \cap \mu)$ , then μ ∈ C. (c) If μ is a singular cardinal such that $\mu = \sup(C \cap \mu)$ , then μ + ∈ C. Then there exists a c.c.c. notion of forcing P such that $V^\mathbb{P} \models CF(S) = C$ . We shall also investigate the connections between the cofinality spectrum and pcf theory; and show that CF(S) cannot be an arbitrarily prescribed set of regular uncountable cardinals. (shrink)

El presente artículo recoge un texto inédito hasta la fecha del filósofo español Manuel Sacristán Sacristán (1925-1985) en torno al libro El futuro del Partido Comunista Francés, un ensayo de Waldeck Rochet (1969), entonces secretario general del PCF. En el mismo pueden observarse motivaciones políticas de Sacristán características de su tiempo que parecen prefigurar parte de la caracterización que más adelante realizará en torno al eurocomunismo, donde la firme defensa de los valores democráticos en el marxismo no es incompatible con (...) críticas a determinadas involuciones político-institucionales en el campo de los partidos comunistas europeos de la época. (shrink)

Plasmids are a major type of mobile genetic elements (MGEs) that mediate horizontal gene transfer. The stable maintenance of plasmids plays a critical role in the functions and survival for microbial populations. However, predicting and controlling plasmid persistence and abundance in complex microbial communities remain challenging. Computationally, this challenge arises from the combinatorial explosion associated with the conventional modeling framework. Recently, a plasmid‐centric framework (PCF) has been developed to overcome this computational bottleneck. This framework enables the derivation of a simple (...) metric, the persistence potential, to predict plasmid persistence and abundance. Here, we discuss how PCF can be extended to account for plasmid interactions. We also discuss how such model‐guided predictions of plasmid fates can benefit from the development of new experimental tools and data‐driven computational methods. (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)