¿Qué ha pasado con el problema del cardinal del continuo después de Gödel (1938) y Cohen (1964)? Intentos de responder esta pregunta pueden encontrarse en los artículos de José Alfredo Amor (1946-2011), "El Problema del continuo después de Cohen (1964-2004)", de Carlos Di Prisco , "Are we closer to a solution of the continuum problem", y de Joan Bagaria, "Natural axioms of set and the continuum problem" , que se pueden encontrar en la biblioteca digital de mi blog de (...) Lógica Matemática y Fundamentos de la Matemática (ver). También en la entrada "The Continuum Hypothesis" de la web de Enciclopedia de Filosofía de la Universidad de Stanford existe información importante y actualizada al respecto. En esta breve nota se comenta sobre el tema de una manera divulgativa. (shrink)

Mostramos que a proposição "Só sei que nada sei", atribuída ao filósofo grego Sócrates, contém, em seu âmago, o axioma da seleção de Zermelo e a aritmética do infinito aleph-0.

El estudio de los "cardinales grandes" es uno de los principales temas de investigación de la teoría de conjuntos y de la teoría de modelos que ha contribuido con el desarrollo de dichas disciplinas. Existe una gran variedad de tales cardinales, por ejemplo cardinales inaccesibles, débilmente compactos, Ramsey, medibles, supercompactos, etc. Tres valiosos teoremas clásicos sobre cardinales medibles son los siguientes: (i) compacidad débil, (ii) Si κ es un cardinal medible, entonces κ es un cardinal inaccesible y existen (...) κ cardinales inaccesibles menores que κ , y (iii) Si existe un cardinal medible, entonces el axioma de constructibilidad (V=L) es falso. El objetivo de este artículo es presentar una demostración de cada uno de estos tres teoremas en el contexto de la teoría de modelos usando ideas del texto de Chang y Keisler (Model Theory). Tales demostraciones tienen en común el uso del método de construcción de modelos llamado ultraproductos, de lógicas infinitarias o fragmentos de la lógica de segundo orden, y del axioma de elección. Cardinales grandes y/o ultraproductos son importantes en teoría de conjuntos, teoría de modelos, análisis matemático, teoría de la medida, probabilidades, topología, análisis funcional, física, teoría de números, finanzas, etc. (shrink)

Es conocido que el Teorema de Completitud de Gödel, el Teorema del Colapso Transitivo de Mostowski y el Principio de Reflexión son resultados muy útiles en las investigaciones de Lógica matemática y/o los Fundamentos de la matemática. El objetivo de este trabajo es presentar algunas demostraciones clásicas de tales resultados: Dos del Teorema de Completitud de Gödel, una del Teorema del Colapso Transitivo de Mostowski y una del Principio de Reflexión. Se aspira que estas notas sean de utilidad para estudiar (...) dichas pruebas. Vale la pena resaltar que entre los métodos matemáticos que se utilizan en tales demostraciones se encuentran: (1) La técnica de construcción de modelos a partir de constantes (o "método del conjunto maximal consistente con un conjunto de testigos" de Henkin) y (2) el Principio de inducción matemática en varias versiones [ (2.1) Inducción matemática sobre los números naturales y (2.2) Inducción transfinita ((2.2.1) Inducción transfinita sobre conjuntos bien ordenados cualesquiera, por ejemplo sobre un cardinal infinito Alef_alfa, (2.2.2) Inducción transfinita sobre una clase de conjuntos bien ordenada, por ejemplo sobre la "clase de todos los ordinales, finitos o infinitos", y (2.2.3) Inducción transfinita sobre relaciones bien fundamentadas, por ejemplo sobre la "relación de pertenencia" entre conjuntos)]. (shrink)

La Hipótesis del Continuo (HC) es la suposición de que dado el primer transfinito, ℵ0 , el cardinal de los naturales; el siguiente cardinal, ℵ1 , es el cardinal de los números reales. La Hipótesis Generalizada del Continuo es que dado un transfinito ℵn , el siguiente transfinito es el conjunto potencia de dicho transfinito ℵn+1 = P(ℵn). Desde la teoría intuitiva de conjuntos cantoriana se ha tratado conocer si en dicha teoría se sigue HC. Sin embargo (...) Cantor no pudo probar HC. Tiempo después Gödel y Cohen probarán que HC es independiente a la teoría de conjuntos ZF ó ZFC, si se acepta eleccíon. Después del trabajo de Cohen, Gödel planteo un programa para extender, axiomáticamente, ZFC con el cual se pudiera conocer sí en ZFC HC se sigue, o se sigue ¬HC. Hoy en día aun se desconoce una extensíon HC ó ¬HC en ZFC. Gutiérrez plantea que de hecho posiblemente hoy en día no hay extensíon axiomática filosóficamente adecuada que pueda determinar si en ZFC se sigue HC ó ¬HC. Desde la perspectiva de Gutiérrez, dado que no existe una extensíon adecuada en ZFC para determinar HC, se puede decir que HC es absolutamente indecidible. Esta conferencia se tratará de entender los conceptos de absoluta indecidibilidad planteada por Gutiérrez, el porqué considera que sucede esto con respecto a HC en ZFC y algunas posibles soluciones que podrían explorarse. (shrink)

Permissivist metaontology proposes answering customary existence questions in the affirmative. Many of the existence questions addressed by ontologists concern the existence of theoretical entities which admit precise formal specification. This causes trouble for the permissivist, since individually consistent formal theories can make pairwise inconsistent demands on the cardinality of the universe. We deploy a result of Gabriel Uzquiano’s to show that this possibility is realised in the case of two prominent existence debates and propose rejecting permissivism in favour of substantive (...) ontology conducted on a cost–benefit basis. (shrink)

José Ferreirós y Abel Lasalle Casanave, El árbol de los números: cognición, lógica y práctica matemática, Editorial Universidad de Sevilla, Sevilla, 2015, 256 pp.

The cardinal role that notions of respect and self-respect play in Rawls’s A Theory of Justice has already been abundantly examined in the literature. In contrast, it has hardly been noticed that these notions are also central to Michael Walzer’s Spheres of Justice. Respect and self-respect are not only central topics of his chapter “Recognition”, but constitute a central aim of a “complex egalitarian society” and of Walzer’s theory of justice. This paper substantiates this thesis and elucidates Walzer’s criticism (...) of Rawls that we need to distinguish between “self-respect” and “self-esteem”. (shrink)

The thesis of Weak Unrestricted Composition says that every pair of objects has a fusion. This thesis has been argued by Contessa and Smith to be compatible with the world being junky and hence to evade an argument against the necessity of Strong Unrestricted Composition proposed by Bohn. However, neither Weak Unrestricted Composition alone nor the different variants of it that have been proposed in the literature can provide us with a satisfying answer to the special composition question, or so (...) we will argue. We will then go on to explore an alternative family of purely mereological rules in the vicinity of Weak Unrestricted Composition, Cardinal Composition: A plurality of pairwise non-overlapping objects composes an object iff the objects in the plurality are of cardinality smaller than $$\kappa $$ κ. As we will show, all the instances for infinite $$\kappa $$ κ s determine fusion and are compatible with junk, and every instance for a $$\kappa > \aleph _0$$ κ > ℵ 0 is furthermore compatible with gunk and dense chains of parthood. (shrink)

We reexamine some of the classic problems connected with the use of cardinal utility functions in decision theory, and discuss Patrick Suppes's contributions to this field in light of a reinterpretation we propose for these problems. We analytically decompose the doctrine of ordinalism, which only accepts ordinal utility functions, and distinguish between several doctrines of cardinalism, depending on what components of ordinalism they specifically reject. We identify Suppes's doctrine with the major deviation from ordinalism that conceives of utility functions (...) as representing preference differences, while being non- etheless empirically related to choices. We highlight the originality, promises and limits of this choice-based cardinalism. (shrink)

The cardinal role that notions of respect and self-respect play in Rawls’s A Theory of Justice has already been abundantly examined in the literature. However, it has hardly been noticed that these notions are also central for Michael Walzer’s Spheres of Justice. Respect and self-respect are not only central topics of his chapter on “recognition”, but constitute a central aim of his whole theory of justice. This paper substantiates this thesis and elucidates Walzer’s criticism of Rawls’s that we need (...) to distinguish between “self-respect” and “self-esteem”. (shrink)

(CONTENIDO: LA FILOSOFÍA DE ALTHUSSER A 50 AÑOS DE LIRE LE CAPITAL Pedro Karczmarczyk, 3; DISCURSO Y DECRETO: SPINOZA ALTHUSSER Y PÊCHEUX Warren Montag 11; ALTHUSSER LECTOR DE GRAMSCI Vittorio Morfino 43 LAS ABSTRACCIONES, ENTRE LA IDEOLOGÍA Y LA CIENCIA João Quartim de Moraes 67 ELOGIO DEL TEORICISMO. PRÁCTICA TEÓRICA E INCONSCIENTE FILOSÓFICO EN LA PROBLEMÁTICA ALTHUSSERIANA, Natalia Romé 85 MARXISMO Y FEMINISMO: EL RECOMIENZO DE UNA PROBLEMÁTICA1 115 Luisina Bolla* / Pedro Karczmarczyk* 115 RRESEÑAS El materialismo de Althusser. Más (...) allá del telos y del Eschaton de Vittorio Morfino, por Valentín Huarte 153; El sujeto en cuestión, Pedro Karczmarczyk (ed.) por Constanza Storani; 159 La posición materialista. El pensamiento de L. Althusser entre la práctica teórica y la práctica política, de Natalia Romé, por Ingrid Sarchman y Carolina Collazo 164. (shrink)

There are multiple formal characterizations of the natural numbers available. Despite being inter-derivable, they plausibly codify different possible applications of the naturals – doing basic arithmetic, counting, and ordering – as well as different philosophical conceptions of those numbers: structuralist, cardinal, and ordinal. Nevertheless, some influential philosophers of mathematics have argued for a non-egalitarian attitude according to which one of those characterizations is more “legitmate” in virtue of being “more basic” or “more fundamental”. This paper addresses two related issues. (...) First, we review some of these non-egalitarian arguments, lay out a laundry list of different, legitimate, notions of relative priority, and suggest that these arguments plausibly employ different such notions. Secondly, we argue that given a metaphysical-cum-epistemological gloss suggested by Frege’s foundationalist epistomology, the ordinals are plausibly more basic than the cardinals. This is just one orientation to relative priority one could take, however. Ultimately, we subscribe to an egalitarian attitude towards these formal characterizations: they are, in some sense, equally “legitimate”. (shrink)

The independence phenomenon in set theory, while pervasive, can be partially addressed through the use of large cardinal axioms. A commonly assumed idea is that large cardinal axioms are species of maximality principles. In this paper, I argue that whether or not large cardinal axioms count as maximality principles depends on prior commitments concerning the richness of the subset forming operation. In particular I argue that there is a conception of maximality through absoluteness, on which large (...) class='Hi'>cardinal axioms are restrictive. I argue, however, that large cardinals are still important axioms of set theory and can play many of their usual foundational roles. (shrink)

There are long-standing doubts about whether data from subjective scales—for instance, self-reports of happiness—are cardinally comparable. It is unclear how to assess whether these doubts are justified without first addressing two unresolved theoretical questions: how do people interpret subjective scales? Which assumptions are required for cardinal comparability? This paper offers answers to both. It proposes an explanation for scale interpretation derived from philosophy of language and game theory. In short: conversation is a cooperative endeavour governed by various maxims (Grice (...) 1989); because subjective scales are vague and individuals want to make themselves understood, scale interpretation is a search for a focal point (Schelling 1960). A specific focal point it hypothesised; if this hypothesis is correct, subjective data will be cardinally comparable. Four individually necessary and jointly sufficient conditions for cardinal comparability are specified. The paper then argues this hypothesis can be empirically be tested, makes an initial attempt to do so using subjective well-being data, and concludes it is supported. Numerous areas for further research are identified including, at the end of the paper, how certain tests could be used to ‘correct’ subjective data if they are not cardinal. (shrink)

As a representative of the papacy Bellarmine was an extremely moderate one. In fact Sixtus V in 1590 had the first volume of his Disputations placed on the Index because it contained so cautious a theory of papal power, denying the Pope temporal hegemony. Bellarmine did not represent all that Hobbes required of him either. On the contrary, he proved the argument of those who championed the temporal powers of the Pope faulty. As a Jesuit he tended to maintain the (...) relative autonomy of the state, denying the temporal powers ascribed by radical papalists and Augustinians. Their argument was generally framed as a syllogism: Christ, who possessed direct temporal power as both God and man, exercised it on earth; the Pope is the vicar of Christ; therefore the Pope possesses and may exercise direct temporal jurisdiction. Bellarmine simply denied that Christ had exercised the temporal power, which as God, it is true, he possessed. Moreover, he drew up and circulated a list of patristic passages collected under the title De Regno Christi quale sit, to prove to the Pope the orthodoxy of his position. (shrink)

In this paper, I introduce an iterative method aimed at exploring numbers within the interval [0, 1]. Beginning with a foundational set, S0, a series of algorithms are employed to expand and refine this set. Each algorithm has its designated role, from incorporating irrational numbers to navigating non-deterministic properties. With each successive iteration, our set grows, and after infinite iterations, its cardinality is proposed to align with that of the real numbers. This work is an initial exploration into this approach, (...) and further investigation is encouraged to understand its full implications and potential. (shrink)

Wittgenstein’s atomist picture, as embodied in his Tractatus, is initially very appealing. However, it faces the famous colour-exclusion problem. In this paper, I shall explain when the atomist picture can be defended in the face of that problem; and, in the light of this, why the atomist picture should be rejected. I outline the atomist picture in Section 1. In Section 2, I present a very simple necessary and sufficient condition for the tenability of the atomist picture. The condition is: (...) logical space is a power of two. In Sections 3 and 4, I outline the colour-exclusion problem, and then show how the cardinality-condition supplies a response to exclusion problems. In Section 5, I explain how this amounts to a distillation of a proposal due to Moss, which goes back to Carruthers. And in Section 6, I show how all this vindicates Wittgenstein’s ultimate rejection of the atomist picture. The brief reason is that we have no guarantee that there are any solutions to a given exclusion problem but, if there are any, then there are far too many. (shrink)

Se puede considerar que Berkeley y Hume son antecedentes filosóficos del Formalismo Matemático. Ambos sostienen una visión instrumentalista y no-realista de las matemáticas. En la conferencia se explora las diferencias y similitudes de ambos autores, así como el porque se les puede considerar ser antecesores del Formalismo.

Creationism about fictional entities requires a principle connecting what fictions say exist with which fictional entities really exist. The most natural way of spelling out such a principle yields inconsistent verdicts about how many fictional entities are generated by certain inconsistent fictions. Avoiding inconsistency without compromising the attractions of creationism will not be easy.

An overview of what Frege accomplishes in Part II of Grundgesetze, which contains proofs of axioms for arithmetic and several additional results concerning the finite, the infinite, and the relationship between these notions. One might think of this paper as an extremely compressed form of Part II of my book Reading Frege's Grundgesetze.

It will be shown in this article that an ontological approach for some problems related to the interpretation of Quantum Mechanics (QM) could emerge from a re-evaluation of the main paradox of early Greek thought: the paradox of Being and non-Being, and the solutions presented to it by Plato and Aristotle. More well known are the derivative paradoxes of Zeno: the paradox of motion and the paradox of the One and the Many. They stem from what was perceived by classical (...) philosophy to be the fundamental enigma for thinking about the world: the seemingly contradictory results that followed from the co-incidence of being and non-being in the world of change and motion as we experience it, and the experience of absolute existence here and now. The most clear expression of both stances can be found, again following classical thought, in the thinking of Heraclitus of Ephesus and Parmenides of Elea. The problem put forward by these paradoxes reduces for both Plato and Aristotle to the possibility of the existence of stable objects as a necessary condition for knowledge. Hence the primarily ontological nature of the solutions they proposed: Plato’s Theory of Forms and Aristotle’s metaphysics and logic. Plato’s and Aristotle’s systems are argued here to do on the ontological level essentially the same: to introduce stability in the world by introducing the notion of a separable, stable object, for which a ‘principle of contradiction’ is valid: an object cannot be and not-be at the same place at the same time. So it becomes possible to forbid contradiction on an epistemological level, and thus to guarantee the certainty of knowledge that seemed to be threatened before. After leaving Aristotelian metaphysics, early modern science had to cope with these problems: it did so by introducing “space” as the seat of stability, and “time” as the theater of motion. But the ontological structure present in this solution remained the same. Therefore the fundamental notion ‘separable system’, related to the notions observation and measurement, themselves related to the modern concepts of space and time, appears to be intrinsically problematic, because it is inextricably connected to classical logic on the ontological level. We see therefore the problems dealt with by quantum logic not as merely formal, and the problem of ‘non-locality’ as related to it, indicating the need to re-think the notions system, entity, as well as the implications of the operation ‘measurement’, which is seen here as an application of classical logic (including its ontological consequences) on the material world. (shrink)

In this article, a possible generalization of the Löb’s theorem is considered. Main result is: let κ be an inaccessible cardinal, then ¬Con( ZFC +∃κ) .

This paper presents a model-theoretic semantics for discourse "about" natural numbers, one that captures what I call "the mathematical-object picture", but avoids what I can "the mathematical-object theory".

En el ámbito de la lógica matemática existe un problema sobre la relación lógica entre dos versiones débiles del Axioma de elección (AE) que no se ha podido resolver desde el año 2000 (aproximadamente). Tales versiones están relacionadas con ultrafiltros no principales y con Propiedades Ramsey (Bernstein, Polarizada, Subretículo, Ramsey, Ordinales flotantes, etc). La primera versión débil del AE es la siguiente (A): “Existen ultrafiltros no principales sobre el conjunto de los números naturales (ℕ)”. Y la segunda versión débil del (...) AE es la siguiente (B): “Existen ultraflitros sobre ℕ”. Se sabe que A implica B, pero se desconoce si B implica A. Di Prisco y Henle conjeturan en los artículos ([1], [2]) que esto no ocurre, es decir, conjeturan que B no implica A, en otras palabras, conjeturan que A es más fuerte estrictamente que B, que A es independiente de B, pero esto no se ha podido demostrar todavía aunque se ha intentado hacer desde hace aproximadamente 21 años. Una descripción detallada de este problema abierto puede encontrarse en esta ponencia (dictada en el marco del Día Mundial de la Lógica 14-01-2022) y en el artículo [3]. [1] C. Di Prisco y H. Henle. “Doughnuts, Floating Ordinals, Square Brackets, and Ultraflitters”. Journal of Symbolic Logic 65 (2000) 462-473. [2] C. Di Prisco y H. Henle. “Partitions of the reals and choice”. En “Models, algebras and proofs”. X. Caicedo y C.M. Montenegro. Eds. Lecture Notes in Pure and Appl. Math, 203, Marcel Dekker, 1999. [3] F. Galindo. “Tópicos de ultrafiltros”. Divulgaciones Matemáticas. Vol. 21, No 1-2, 2020. (shrink)

An ancient argument attributed to the philosopher Carneades is presented that raises critical questions about the concept of an all-virtuous Divine being. The argument is based on the premises that virtue involves overcoming pains and dangers, and that only a being that can suffer or be destroyed is one for whom there are pains and dangers. The conclusion is that an all-virtuous Divine (perfect) being cannot exist. After presenting this argument, reconstructed from sources in Sextus Empiricus and Cicero, this paper (...) goes on to model it as a deductively valid sequence of reasoning. The paper also discusses whet her the premises are true. Questions about the possibility and value of proving and disproving the existence of God by logical reasoning are raised, as well as ethical questions about how the cardinal ethical virtues should be defined. (shrink)

This article reintroduces Fr. Zeferino González, OP (1831-1894) to scholars of Church history, philosophy, and cultural heritage. He was an alumnus of the University of Santo Tomás in Manila, a Cardinal, and a champion of the revival of Catholic Philosophy that led to the promulgation of Leo XIII’s encyclical Aeterni Patris. Specifically, this essay presents, firstly, the Cardinal’s biography in the context of his experience as a missionary in the Far East; secondly, the intellectual tradition in Santo Tomás (...) in Manila, which he carried with him until his death; and lastly, some reasons for his once-radiant memory to slip into an undeserved forgetfulness. (shrink)

In this article we derived an important example of the inconsistent countable set in second order ZFC (ZFC_2) with the full second-order semantics. Main results: (i) :~Con(ZFC2_); (ii) let k be an inaccessible cardinal, V is an standard model of ZFC (ZFC_2) and H_k is a set of all sets having hereditary size less then k; then : ~Con(ZFC + E(V)(V = Hk)):.

En esta entrevista se resolvieron las interrogantes formuladas y orientadas a la preservación de la lengua castellana en Hispanoamérica. En principio, se abordaron tres temas fundamentales derivados de esta propuesta: la escritura, la investigación y la publicidad. Con el primer tópico, se discernió la función de las normativas, los diccionarios y los manuales elaborados por la Real Academia Española. Su intervención es enjundiosa, ya que direcciona al lector y académico al perfeccionamiento de su lenguaje, con la finalidad de que no (...) reincida en errores gramaticales. El segundo rubro abarca la investigación. Esta fue destacada porque facilita la interacción de los hablantes de la lengua castellana con las metodologías que se instauran para configurar un lenguaje totémico. Además, se asume que la única manera de difundir y contribuir a ese conocimiento sería por medio de las publicaciones científicas. La tercera variante consiste en cómo se inserta esta inquietud en torno a la lengua castellana en los ámbitos publicitarios. Esto es neurálgico por su audiencia limitada y el desinterés de respaldo por los Gobiernos. Frente a ello, consideré que el Dr. Ignacio Bosque Muñoz era la persona idónea para confrontar esta realidad sobre el castellano. Su rol importante en la RAE y la Asociación de Academias de la Lengua Española (ASALE) lo amerita, así como sus publicaciones orientadas al estudio del idioma con enfoques gramaticales. Con su interacción, la volición de este trabajo se evidenciará con la explicación del panorama vigente de la lengua y su preservación, como cuando uno desea comprender el Quijote, entender las contribuciones gramaticales y lexicales de la RAE y la ASALE o cuestionar la proliferación del idioma desde los medios de comunicación. La auscultación de esas oscilaciones será el aporte de esta entrevista. (shrink)

La entrevista al doctor Darío Villanueva es sobre el panorama literario del siglo XXI. A partir de cuatro tópicos fundamentales y reincidentes: los libros, los escritores, las editoriales y la realidad. Estos han sido incorporados en las preguntas para desentrañar el sistema literario que se ha originado en los últimos años. Frente a estas interrogantes, se notará que existen algunos obstáculos que han tergiversado y entorpecido la labor de la escritura, así como el canon literario, tal como la cultura de (...) masas y la diversificación que se aprecia desde la posmodernidad. (shrink)

An analysis of Cardinal Joseph Ratzinger's statements regarding relativism in his 2005 homily to the conclave meeting to elect the new pope in the context of the charge of "relativism" in 20th-century philosophy. Parts of this essay are adapted from Barbara Herrnstein Smith,"Pre-Post-Modern Relativism," in *Scandalous Knowledge: Science, Truth and the Human* (Edinburgh: Edinburgh University Press, 2005; Durham, NC: Duke University Press, 2006), 18 – 45.

This paper presents and defends an argument that the continuum hypothesis is false, based on considerations about objective chance and an old theorem due to Banach and Kuratowski. More specifically, I argue that the probabilistic inductive methods standardly used in science presuppose that every proposition about the outcome of a chancy process has a certain chance between 0 and 1. I also argue in favour of the standard view that chances are countably additive. Since it is possible to randomly pick (...) out a point on a continuum, for instance using a roulette wheel or by flipping a countable infinity of fair coins, it follows, given the axioms of ZFC, that there are many different cardinalities between countable infinity and the cardinality of the continuum. (shrink)

According to the so-called strong variant of Composition as Identity (CAI), the Principle of Indiscernibility of Identicals can be extended to composition, by resorting to broadly Fregean relativizations of cardinality ascriptions. In this paper we analyze various ways in which this relativization could be achieved. According to one broad variety of relativization, cardinality ascriptions are about objects, while concepts occupy an additional argument place. It should be possible to paraphrase the cardinality ascriptions in plural logic and, as a consequence, relative (...) counting requires the relativization either of quantifiers, or of identity, or of the is one of relation. However, some of these relativizations do not deliver the expected results, and others rely on problematic assumptions. In another broad variety of relativization, cardinality ascriptions are about concepts or sets. The most promising development of this approach is prima facie connected with a violation of the so-called Coreferentiality Constraint, according to which an identity statement is true only if its terms have the same referent. Moreover - even provided that the problem with coreferentiality can be fixed - the resulting analysis of cardinality ascriptions meets several difficulties. (shrink)

We address Steel’s Programme to identify a ‘preferred’ universe of set theory and the best axioms extending ZFC by using his multiverse axioms MV and the ‘core hypothesis’. In the first part, we examine the evidential framework for MV, in particular the use of large cardinals and of ‘worlds’ obtained through forcing to ‘represent’ alternative extensions of ZFC. In the second part, we address the existence and the possible features of the core of MV_T (where T is ZFC+Large Cardinals). In (...) the last part, we discuss the hypothesis that the core is Ultimate-L, and examine whether and how, based on this fact, the Core Universist can justify V=Ultimate-L as the best (and ultimate) extension of ZFC. To this end, we take into account several strategies, and assess their prospects in the light of MV’s evidential framework. (shrink)

This paper analyses the hidden use of new axioms in set-theoretic practice with a focus on large cardinal axioms and presents a general overview of set-theoretic practices using large cardinal axioms. The hidden use of a new axiom provides extrinsic reasons in support of this axiom via the idea of verifiable consequences, which is especially relevant for set-theoretic practitioners with an absolutist view. Besides that, the hidden use has pragmatic significance for further important sub-groups of the set-theoretic community---set-theoretic (...) practitioners with a pluralist view and set-theoretic practitioners who aim for ZFC-proofs. By describing this, the paper gives a more complete picture of new axioms in set-theoretic practice. These observations, for instance, show that set-theoretic practitioners interested in ZFC-proofs use tools that go beyond ZFC. The analysis is based on empirical data that was collected in an extensive interview study with set-theoretic practitioners. (shrink)

I consider the first-order modal logic which counts as valid those sentences which are true on every interpretation of the non-logical constants. Based on the assumptions that it is necessary what individuals there are and that it is necessary which propositions are necessary, Timothy Williamson has tentatively suggested an argument for the claim that this logic is determined by a possible world structure consisting of an infinite set of individuals and an infinite set of worlds. He notes that only the (...) cardinalities of these sets matters, and that not all pairs of infinite sets determine the same logic. I use so-called two-cardinal theorems from model theory to investigate the space of logics and consequence relations determined by pairs of infinite sets, and show how to eliminate the assumption that worlds are individuals from Williamson’s argument. (shrink)

This paper explores the value of benevolence as a cardinal virtue by analyzing the evolving history of virtue ethics from ancient Greek tradition to emotivism and contemporary thoughts. First, I would like to start with a brief idea of virtue ethics. Greek virtue theorists recognize four qualities of moral character, namely, wisdom, temperance, courage, and justice. Christianity recognizes unconditional love as the essence of its theology. Here I will analyze the transition within the doctrine of virtue ethics in the (...) Christian era and afterward since the eighteenth-century thinkers are immensely inspired by this Christian notion of love consider universal benevolence as the cardinal virtue. Later, Hume introduces an emotivist turn by considering the moral worth of sympathetic emotions in his ethical doctrine. In this paper, I aim to discover the cardinality of the virtue of benevolence following the evolutionary history of virtue ethics. (shrink)

The Overgeneration Argument is a prominent objection against the model-theoretic account of logical consequence for second-order languages. In previous work we have offered a reconstruction of this argument which locates its source in the conflict between the neutrality of second-order logic and its alleged entanglement with mathematics. Some cases of this conflict concern small large cardinals. In this article, we show that in these cases the conflict can be resolved by moving from a set-theoretic implementation of the model-theoretic account to (...) one which uses higher-order resources. (shrink)

The Overgeneration Argument is a prominent objection against the model-theoretic account of logical consequence for second-order languages. In previous work we have offered a reconstruction of this argument which locates its source in the conflict between the neutrality of second-order logic and its alleged entanglement with mathematics. Some cases of this conflict concern small large cardinals. In this article, we show that in these cases the conflict can be resolved by moving from a set-theoretic implementation of the model-theoretic account to (...) one which uses higher-order resources. (shrink)

In this paper I present two new arguments against the possibility of an omniscient being. My new arguments invoke considerations of cardinality and resemble several arguments originally presented by Patrick Grim. Like Grim, I give reasons to believe that there must be more objects in the universe than there are beliefs. However, my arguments will rely on certain mereological claims, namely that Classical Extensional Mereology is necessarily true of the part-whole relation. My first argument is an instance of a problem (...) first noted by Gideon Rosen and requires an additional assumption about the mereological structure of certain beliefs. That assumption is that an omniscient being’s beliefs are mereological simples. However, this assumption is dropped when I present my second argument. Thus, I hope to show that if Classical Extensional Mereology is true of the part-whole relation, there cannot be an omniscient being. (shrink)

It is widely held that under ordinal utility, utility differences are ill-defined. Allegedly, for these to be well-defined (without turning to choice under risk or the like), one should adopt as a new kind of primitive quaternary relations, instead of the traditional binary relations underlying ordinal utility functions. Correlatively, it is also widely held that the key structural properties of quaternary relations are entirely arbitrary from an ordinal point of view. These properties would be, in a nutshell, the hallmark of (...) cardinal utility. While much is obviously true in these two tenets, this note explains why, as stated, they should be abandoned. Any ordinal utility function induces a rich quaternary relation. There is such a thing as ordinal utility differences. Furthermore, this induced quaternary relation respects, apart from completeness, the most standard structural properties of quaternary relations. These properties are, from an ordinal point of view, anything but arbitrary; from a quaternary perspective only completeness should be considered the hallmark—if any—of cardinal utility. These facts are explained to be especially relevant to the critical appreciation of the ordinalist methodology. (shrink)

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

According to Cantor (Mathematische Annalen 21:545–586, 1883 ; Cantor’s letter to Dedekind, 1899 ) a set is any multitude which can be thought of as one (“jedes Viele, welches sich als Eines denken läßt”) without contradiction—a consistent multitude. Other multitudes are inconsistent or paradoxical. Set theoretical paradoxes have common root—lack of understanding why some multitudes are not sets. Why some multitudes of objects of thought cannot themselves be objects of thought? Moreover, it is a logical truth that such multitudes do (...) exist. However we do not understand this logical truth so well as we understand, for example, the logical truth $${\forall x \, x = x}$$ . In this paper we formulate a logical truth which we call the productivity principle. Rusell (Proc Lond Math Soc 4(2):29–53, 1906 ) was the first one to formulate this principle, but in a restricted form and with a different purpose. The principle explicates a logical mechanism that lies behind paradoxical multitudes, and is understandable as well as any simple logical truth. However, it does not explain the concept of set. It only sets logical bounds of the concept within the framework of the classical two valued $${\in}$$ -language. The principle behaves as a logical regulator of any theory we formulate to explain and describe sets. It provides tools to identify paradoxical classes inside the theory. We show how the known paradoxical classes follow from the productivity principle and how the principle gives us a uniform way to generate new paradoxical classes. In the case of ZFC set theory the productivity principle shows that the limitation of size principles are of a restrictive nature and that they do not explain which classes are sets. The productivity principle, as a logical regulator, can have a definite heuristic role in the development of a consistent set theory. We sketch such a theory—the cumulative cardinal theory of sets. The theory is based on the idea of cardinality of collecting objects into sets. Its development is guided by means of the productivity principle in such a way that its consistency seems plausible. Moreover, the theory inherits good properties from cardinal conception and from cumulative conception of sets. Because of the cardinality principle it can easily justify the replacement axiom, and because of the cumulative property it can easily justify the power set axiom and the union axiom. It would be possible to prove that the cumulative cardinal theory of sets is equivalent to the Morse–Kelley set theory. In this way we provide a natural and plausibly consistent axiomatization for the Morse–Kelley set theory. (shrink)

Este número de acceso abierto tiene como objetivo resaltar los puntos de vista de los países latinoamericanos sobre la justicia en un contexto de pandemia y contribuir al diálogo entre estos y con la comunidad científica global. Explora los desafíos globales de la pandemia de COVID-19, las diferencias relevantes entre las medidas de salud pública y su impacto en los países de ingresos altos versus los países de ingresos bajos o medios, y cómo la injusticia global se profundizó debido (...) a la pandemia de COVID-19. También llama la atención sobre las experiencias, los resultados y las respuestas del norte global a la pandemia con respecto a poblaciones vulnerables y trayectorias de exposición que se reproducen en América Latina. (shrink)

We show that the statement "I only know that I know nothing," attributed to the Greek philosopher Socrates, contains, at its core, Zermelo's Axiom of Selection and the arithmetic of the infinite cardinal aleph-0.