We present recent results on the model companions of set theory, placing them in the context of a current debate in the philosophy of mathematics. We start by describing the dependence of the notion of model companionship on the signature, and then we analyze this dependence in the specific case of set theory. We argue that the most natural model companions of set theory describe (as the signature in which we axiomatize set theory varies) theories of $H_{\kappa ^+}$, as $\kappa (...) $ ranges among the infinite cardinals. We also single out $2^{\aleph _0}=\aleph _2$ as the unique solution of the continuum problem which can (and does) belong to some model companion of set theory (enriched with large cardinal axioms). While doing so we bring to light that set theory enriched by large cardinal axioms in the range of supercompactness has as its model companion (with respect to its first order axiomatization in certain natural signatures) the theory of $H_{\aleph _2}$ as given by a strong form of Woodin’s axiom $(*)$ (which holds assuming $\mathsf {MM}^{++}$ ). Finally this model-theoretic approach to set-theoretic validities is explained and justified in terms of a form of maximality inspired by Hilbert’s axiom of completeness. (shrink)

I shall commence with an eminently subjective statement: reading this book was a pleasant experience. Let me explain why, then, hence providing a somewhat more objective set of reasons for the posi...

This is an introductory paper to a series of results linking generic absoluteness results for second and third order number theory to the model theoretic notion of model companionship. Specifically we develop here a general framework linking Woodin’s generic absoluteness results for second order number theory and the theory of universally Baire sets to model companionship and show that (with the required care in details) a $$\Pi _2$$ -property formalized in an appropriate language for second order number theory is forcible (...) from some $$T\supseteq \mathsf {ZFC}+$$ large cardinals if and only if it is consistent with the universal fragment of T if and only if it is realized in the model companion of T. In particular we show that the first order theory of $$H_{\omega _1}$$ is the model companion of the first order theory of the universe of sets assuming the existence of class many Woodin cardinals, and working in a signature with predicates for $$\Delta _0$$ -properties and for all universally Baire sets of reals. We will extend these results also to the theory of $$H_{\aleph _2}$$ in a follow up of this paper. (shrink)

In this article we analyze the method of forcing from a more philosophical perspective. After a brief presentation of this technique we outline some of its philosophical imports in connection with realism. We shall discuss some philosophical reactions to the invention of forcing, concentrating on Mostowski’s proposal of sharpening the notion of generic set. Then we will provide an overview of the notions of multiverse and the related philosophical debate on the foundations of set theory. In conclusion, we connect this (...) modern debate and Mostowski’s proposal, suggesting a way to analyze the notion of genericity within the framework of a multiverse structure. (shrink)

In this article we present a technique for selecting models of set theory that are complete in a model-theoretic sense. Specifically, we will apply Robinson infinite forcing to the collections of models of ZFC obtained by Cohen forcing. This technique will be used to suggest a unified perspective on generic absoluteness principles.

It is prima facie uncontroversial that the justification of an assertion amounts to a collection of other (inferentially related) assertions. In this paper, we point at a class of assertions, i.e. mathematical assertions, that appear to systematically flout this principle. To justify a mathematical assertion (e.g. a theorem) is to provide a proof—and proofs are sequences of directives. The claim is backed up by linguistic data on the use of imperatives in proofs, and by a pragmatic analysis of theorems and (...) their proofs. Proofs, we argue, are sequences of instructions whose performance inevitably gets one to truth. It follows that a felicitous theorem, i.e. a theorem that has been correctly proven, is a persuasive theorem. When it comes to mathematical assertions, there is no sharp distinction between illocutionary and perlocutionary success. (shrink)

On croit connaître le Prince. En fait, on colporte parfois à son propos une vulgate interprétative fondée sur des approximations ou des partis pris de traduction dans des éditions françaises qui se répètent et s'engendrent l'une l'autre. Le texte paraît si " connu ", voire si " évident ", que l'on pourrait oublier ainsi de s'interroger sur le rapport des traductions avec l'original italien ; c'est une des raisons pour lesquelles le texte original (récemment établi par Giorgio Inglese sous (...) le titre de De principatibus) est publié, dans cette édition, en regard d'une traduction entièrement inédite. Le travail des traducteurs repose ici sur la reconstitution de la logique propre de la syntaxe, sans en gommer les condensations étranges voire les anacoluthes, sur le respect de la cohérence du lexique, sans sacrifier les multiples échos qui balisent l'ouvrage, mais aussi sur la tentative de retrouver le rythme si particulier de la prose machiavélienne qui a tant fait pour le succès de ce que Machiavel nomme " son opuscule ". Cette édition est accompagnée d'une présentation qui inscrit le texte dans son temps et dans le réseau complexe de ses enjeux politiques et théoriques. Elle comporte également un grand commentaire linéaire du Prince qui engage une nouvelle interprétation de l'œuvre. La postface permet de voir comment se forge et naît, dans l'écriture du Prince, une langue capable de rendre compte des " choses nouvelles " de la politique. (shrink)

Se analiza lo que parece ser una crisis perpetua del pensamiento humano, cuando resurge de tiempo en tiempo el error fundamental de atribuir al mundo un significado material, excluyendo el significado moral. La historia del pensamiento humano nos muestra hasta qué punto ideas que consideramos nuevas, tuvieron en realidad expresiones muy remotas, de ahí la importancia de estudiar el pasado. Del Vecchio defiende el valor de la filosofía escolástica, frente a la imprecisión de la moderna filosofía del derecho y del (...) positivismo jurídico. Al tiempo que rechaza como error probado, el materialismo histórico y dialéctico, teorizado por Marx. El autor del artículo considera que las cosas materiales entran en las relaciones sociales como exigencias, inclinaciones o aspiraciones, pero no determinan su surgir, pues los hechos y las relaciones de la vida social y política tienen su raíz en el espíritu humano. (shrink)

ABSTRACT We present and discuss a change in the introduction of Hilbert’s Grundlagen der Geometrie between the first and the subsequent editions: the disappearance of the reference to the independence of the axioms. We briefly outline the theoretical relevance of the notion of independence in Hilbert’s work and we suggest that a possible reason for this disappearance is the discovery that Hilbert’s axioms were not, in fact, independent. In the end we show how this change gives textual evidence for the (...) connection between the notions of independence and simplicity. (shrink)

In this article I propose to look at set theory not only as a founda­tion of mathematics in a traditional sense, but as a foundation for mathemat­ical practice. For this purpose I distinguish between a standard, ontological, set theoretical foundation that aims to find a set theoretical surrogate to every mathematical object, and a practical one that tries to explain mathematical phenomena, giving necessary and sufficient conditions for the proof of mathematical propositions. I will present some example of this use (...) of set theoretical methods, in the context of mainstream mathematics, in terms of independence proofs, equiconsistency results and discussing some recent results that show how it is possible to “complete” the structures H(ℵ1) and H(ℵ2). Then I will argue that a set theoretical foundation of mathematics can be relevant also for the philosophy of mathematical practice, as long as some axioms of set theory can be seen as explanations of mathematical phenomena. In the end I will propose a more general distinction between two different kinds of foundation: a practical one and a theoretical one, drawing some examples from the history of the foundation of mathematics. (shrink)

In this article I propose to look at set theory not only as a founda­tion of mathematics in a traditional sense, but as a foundation for mathemat­ical practice. For this purpose I distinguish between a standard, ontological, set theoretical foundation that aims to find a set theoretical surrogate to every mathematical object, and a practical one that tries to explain mathematical phenomena, giving necessary and sufficient conditions for the proof of mathematical propositions. I will present some example of this use (...) of set theoretical methods, in the context of mainstream mathematics, in terms of independence proofs, equiconsistency results and discussing some recent results that show how it is possible to “complete” the structures H(ℵ1) and H(ℵ2). Then I will argue that a set theoretical foundation of mathematics can be relevant also for the philosophy of mathematical practice, as long as some axioms of set theory can be seen as explanations of mathematical phenomena. In the end I will propose a more general distinction between two different kinds of foundation: a practical one and a theoretical one, drawing some examples from the history of the foundation of mathematics. (shrink)

: In this article we analyze the key concept of Hilbert's axiomatic method, namely that of axiom. We will find two different concepts: the first one from the period of Hilbert's foundation of geometry and the second one at the time of the development of his proof theory. Both conceptions are linked to two different notions of intuition and show how Hilbert's ideas are far from a purely formalist conception of mathematics. The principal thesis of this article is that one (...) of the main problems that Hilbert encountered in his foundational studies consisted in securing a link between formalization and intuition. We will also analyze a related problem, that we will call "Frege's Problem", form the time of the foundation of geometry and investigate the role of the Axiom of Completeness in its solution. (shrink)

The work of Giorgio Agamben, one of Italy's most important and original philosophers, has been based on an uncommon erudition in classical traditions of philosophy and rhetoric, the grammarians of late antiquity, Christian theology, and modern philosophy. Recently, Agamben has begun to direct his thinking to the constitution of the social and to some concrete, ethico-political conclusions concerning the state of society today, and the place of the individual within it. In Homo Sacer, Agamben aims to connect the problem (...) of pure possibility, potentiality, and power with the problem of political and social ethics in a context where the latter has lost its previous religious, metaphysical, and cultural grounding. Taking his cue from Foucault's fragmentary analysis of biopolitics, Agamben probes with great breadth, intensity, and acuteness the covert or implicit presence of an idea of biopolitics in the history of traditional political theory. He argues that from the earliest treatises of political theory, notably in Aristotle's notion of man as a political animal, and throughout the history of Western thinking about sovereignty, a notion of sovereignty as power over "life" is implicit. The reason it remains merely implicit has to do, according to Agamben, with the way the sacred, or the idea of sacrality, becomes indissociable from the idea of sovereignty. Drawing upon Carl Schmitt's idea of the sovereign's status as the exception to the rules he safeguards, and on anthropological research that reveals the close interlinking of the sacred and the taboo, Agamben defines the sacred person as one who can be killed and yet not sacrificed—a paradox he sees as operative in the status of the modern individual living in a system that exerts control over the collective "naked life" of all individuals. (shrink)

This volume constitutes the largest collection of writings by the Italian philosopher Giorgio Agamben hitherto published in any language and all but one appear in English for the first time. The essays consider figures in the history of philosophy (Plato, Plotinus, Spinoza, Hegel) and twentieth-century thought (Walter Benjamin, Heidegger, Derrida, Deleuze, the historian Aby Warburg, and the linguist J.-C. Milner). They also examine several central concerns of Agamben: the relation of linguistic and metaphysical categories; messianism in Islamic, Jewish, and (...) Christian theology; and the state and future of contemporary politics. Despite the diversity of the texts collected here, they show a consistent concern for a set of overriding philosophical themes concerning language, history, and potentiality. (shrink)

The end of human history is an event that has been foreseen or announced by both messianics and dialecticians. But who is the protagonist of that history that is coming—or has come—to a close? What is man? How did he come on the scene? And how has he maintained his privileged place as the master of, or first among, the animals? In The Open, contemporary Italian philosopher Giorgio Agamben considers the ways in which the “human” has been thought of (...) as either a distinct and superior type of animal, or a kind of being that is essentially different from animal altogether. In an argument that ranges from ancient Greek, Christian, and Jewish texts to twentieth-century thinkers such as Heidegger, Benjamin, and Kojève, Agamben examines the ways in which the distinction between man and animal has been manufactured by the logical presuppositions of Western thought, and he investigates the profound implications that the man/animal distinction has had for disciplines as seemingly disparate as philosophy, law, anthropology, medicine, and politics. (shrink)

Giorgio Agamben's Homo Sacer was one of the seminal works of political philosophy in recent decades. It was also the beginning of a series of interconnected investigations of staggering ambition and scope, investigating the deepest foundations of Western politics and thought. The Use of Bodies represents the ninth and final volume in this twenty-year undertaking, breaking considerable new ground while clarifying the stakes and implications of the project as a whole. It comprises three major sections. The first uses Aristotle's (...) discussion of slavery as a starting point for radically rethinking notions of selfhood; the second calls for a complete reworking of Western ontology; and the third explores the enigmatic concept of "form-of-life," which is in many ways the motivating force behind the entire Homo Sacer project. Interwoven between these major sections are shorter reflections on individual thinkers, while the epilogue pushes toward a new approach to political life that breaks with the destructive deadlocks of Western thought. The Use of Bodies represents a true masterwork by one of our greatest living philosophers. (shrink)