- My research interests focus on the issue of the (mathematical and philosophical) infinite. In particular, I'm interes... moreMy research interests focus on the issue of the (mathematical and philosophical) infinite. In particular, I'm interested in set theory, its foundations, and philosophy.
Over the years, I have researched such topics as: the Continuum Hypothesis (history and philosophy of), the new set-theoretic axioms, the set-theoretic multiverse, the problem of justification, the nature of mathematical Platonism, aspects of the history and philosophy of logic.
I have also done research (and I'm still deeply interested) in ancient philosophy (esp., Platonism and the platonic tradition in the Middle Ages).
Currently, I'm a Beatriu de Pinós (MSC COFUND) Post-Doctoral Fellow at the University of Barcelona (UB). Before coming to Barcelona, I was Post-Doctoral Fellow of Theoretical Philosophy at the University of Tartu (2019-2020) and, earlier, at the University of Vienna's KGRC (2013-2015). Between 2015 and 2019, I have been a History and Philosophy teacher in the Italian High Schools (Licei).
I love languages, in particular ancient languages (Latin and Greek), classical music, playing piano, and tennis.edit
In this talk, I address the issue of whether Cantor's conception of abstraction is fully adequate to motivate the acceptance of Hume's Principle (HP). I first address Cantor's two formulations of numerical abstractionism, and show that... more
In this talk, I address the issue of whether Cantor's conception of abstraction is fully adequate to motivate the acceptance of Hume's Principle (HP). I first address Cantor's two formulations of numerical abstractionism, and show that they are compatible with Frege's abstractionism. Then, I proceed to review Gödel's conception of abstraction (which I call GMAA, Gödel's Minimal Account of Abstraction), which represents a reprise and defence of Cantor's conception. I also show that alternative notions of cardinal number, such as those proposed by numerosity theorists, fail to satisfy both Cantor's conception and GMAA. In the talk, I also illustrate the relevance of the topic for the current debate about HP within the neo-logicist programme.
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The talk presents recent work done by myself and Joan Bagaria on Steel's MV theory (multiverse axioms). I present results concerning the existence and definability of the core, its features and its indeterminacy. I also address the... more
The talk presents recent work done by myself and Joan Bagaria on Steel's MV theory (multiverse axioms). I present results concerning the existence and definability of the core, its features and its indeterminacy. I also address the rationale for the acceptance of the MV axioms, in particular, the role played by Large Cardinals in contemporary set theory and the use of forcing to address different *worlds*, and what I call Steel's programme, that is, the use of the MV axioms to establish the optimality of the theory ZFC+V=Ultimate-L.
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Introductory (and rather accessible) lecture on my recent research topics: in particular, the ontology of set theory, pluralism, the meaning of indeterminacy (held at Tartu, 14/11/2019).
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In recent years, the notion of 'set-theoretic multiverse' has emerged and progressively gained prominence in the debate on the foundations of set theory. Several conceptions of the set-theoretic multiverse have been presented so far, all... more
In recent years, the notion of 'set-theoretic multiverse' has emerged and progressively gained prominence in the debate on the foundations of set theory. Several conceptions of the set-theoretic multiverse have been presented so far, all of which have advantages and disadvantages. Hamkins' broad multiverse ([4]), consisting of all models of all collections of set-theoretic axioms, is philosophically robust, but mathematically unattractive, as it may fail to fulfil fundamental foundational requirements of set theory. Steel's set-generic multiverse ([5]) consisting of all Boolean-valued models V B of the axioms ZFC+Large Cardinals, is mathematically very attractive and fertile , but too restrictive. In particular, it cannot capture all possible outer models, focusing only on the set-generic extensions. Finally, Sy Friedman's hyperuniverse conception ([2]), although mathematically versatile and foun-dationally attractive, has the main disadvantage of postulating that V is countable. In this paper, we introduce a new conception of the set-theoretic mul-tiverse, that is, the 'V-logic multiverse', which expands on mathematical work conducted within the Hyperunuverse Programme ([1], [3]), but also draws on features of the set-generic multiverse, in particular, on Steel's proposed axiomatisation of it. V-logic is an infinitary logic (a logic admitting formulas and proofs of infinite length) whose language L κ + ,ω , in addition to symbols already used in first-order logic, consists of κ-many constants a, one for each set a ∈ V , and of a special constant symbol V , which denotes V. In V-logic, one can ensure that the statement asserting the consistency of ZFC+ψ, for some set-theoretic statement ψ, is satisfied by some model M , if and only if M is an outer model of V. By outer model we mean here: models obtained through set-forcing, class-forcing, hyperclass-forcing and, in general, any model-theoretic technique able to produce width extensions of V. Thus, through the choice of suitable consistency statements, we can generate outer models M , endowed with specific features. The V-logic multiverse is precisely the collection of all such outer models of V.
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Slides del seminario che ho tenuto all'Università S. Raffaele di Milano il 21 maggio 2019 nell'ambito del corso di Logica filosofica della prof.ssa Francesca Boccuni. Il seminario ha toccato temi di (filosofia della) teoria degli... more
Slides del seminario che ho tenuto all'Università S. Raffaele di Milano il 21 maggio 2019 nell'ambito del corso di Logica filosofica della prof.ssa Francesca Boccuni.
Il seminario ha toccato temi di (filosofia della) teoria degli insiemi, in particolare questioni concernenti l'alternativa fra monismo e pluralismo in teoria degli insiemi (in particolare, la dicotomia universismo/multiversismo) e ha fornito un'analisi di alcune delle più rilevanti teorie del multiverso, e di una possibile interpretazione platonistica del multiverso (radicale).
Il seminario ha toccato temi di (filosofia della) teoria degli insiemi, in particolare questioni concernenti l'alternativa fra monismo e pluralismo in teoria degli insiemi (in particolare, la dicotomia universismo/multiversismo) e ha fornito un'analisi di alcune delle più rilevanti teorie del multiverso, e di una possibile interpretazione platonistica del multiverso (radicale).
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My presentation at FOMUS 2016. The paper is meant to provide a response to Maddy's recent claim that set theory is concerned with V, the universe of sets. I show that a multiverse theory also befits the very Maddian requirements of a... more
My presentation at FOMUS 2016. The paper is meant to provide a response to Maddy's recent claim that set theory is concerned with V, the universe of sets. I show that a multiverse theory also befits the very Maddian requirements of a foundational theory in mathematics and that, consequently, choosing between a universe and a multiverse theory will most likely depend on further requirements of a good foundation.
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Within the Hyperuniverse Programme, 'new' set-theoretic axioms are consequences of higher-order principles describing properties of an *ideal* V in members of the hyperuniverse, which is, in turn, the collection of all c.t.m. of ZFC. We... more
Within the Hyperuniverse Programme, 'new' set-theoretic axioms are consequences of higher-order principles describing properties of an *ideal* V in members of the hyperuniverse, which is, in turn, the collection of all c.t.m. of ZFC. We present some arguments that new axioms conceived in this manner can be viewed as intrinsically justified truths of set theory.
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We review some conceptions of the set-theoretic multiverse, their mathematical and philosophical features, and their tenability. Afterwards, we propose our own conception of the multiverse, the hyperuniverse, which is currently... more
We review some conceptions of the set-theoretic multiverse, their mathematical and philosophical features, and their tenability. Afterwards, we propose our own conception of the multiverse, the hyperuniverse, which is currently investigated in the context of an ongoing programme on the foundations of set theory and the search for new axioms at the KGRC in Vienna.
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I present some strands of Gödel's thought in the light of Cantor's philosophy of mathematics. Gödel's Cantorianism is transparent in three areas: 1) Cantor's distinction between immanent/transient mathematical existence, and the... more
I present some strands of Gödel's thought in the light of Cantor's philosophy of mathematics. Gödel's Cantorianism is transparent in three areas: 1) Cantor's distinction between immanent/transient mathematical existence, and the requirement that the mathematician
be only concerned with the immanent one, is reflected by Gödel's belief in the objectivity of concepts (conceptual realism), whereas belief in their trans-subjective existence is also a constituent of Gödel's Platonism. 2) Cantor's suggestion that mathematical concepts can be non-arbitrarily expanded is resumed by Gödel and re-cast in a phenomenological fashion. In particular, Gödel seems to think that there is an objective conceptual development of set theory and, thereby, a unique notion of set in the same way as Cantor thought that there was only one correct realisation of the notion of actual in finite. 3) Cantor also defined extrinsic criteria for introducing new axioms: consistency, success, fruitfulness. Gödel describes analogous criteria while discussing his programme for fi nding new set-theoretic axioms.""
be only concerned with the immanent one, is reflected by Gödel's belief in the objectivity of concepts (conceptual realism), whereas belief in their trans-subjective existence is also a constituent of Gödel's Platonism. 2) Cantor's suggestion that mathematical concepts can be non-arbitrarily expanded is resumed by Gödel and re-cast in a phenomenological fashion. In particular, Gödel seems to think that there is an objective conceptual development of set theory and, thereby, a unique notion of set in the same way as Cantor thought that there was only one correct realisation of the notion of actual in finite. 3) Cantor also defined extrinsic criteria for introducing new axioms: consistency, success, fruitfulness. Gödel describes analogous criteria while discussing his programme for fi nding new set-theoretic axioms.""
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Are there sufficient grounds for the Gödelian platonist's claiming that we have a definite conception of the universe of sets whereby truth-value determinacy of ZFC undecidable statements necessarily follows? I argue that set theory is a... more
Are there sufficient grounds for the Gödelian platonist's claiming that we have a definite conception of the universe of sets whereby truth-value determinacy of ZFC undecidable statements necessarily follows? I argue that set theory is a perfect case study to test this conception and I conclude that evidence can be brought forward that Gödelian platonism may be untenable.
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In this work, I will be looking at the issues raised by set-theoretic indeterminacy for a Gődelian platonist, who holds that there is a universe of independently existing math- ematical objects and that there are objective unique... more
In this work, I will be looking at the issues raised by set-theoretic indeterminacy for a Gődelian platonist, who holds that there is a universe of independently existing math- ematical objects and that there are objective unique truth-values for any set-theoretic statement. After careful consideration of the philosophical and mathematical issues involved, I claim that Gődelian platonism is untenable. In Chapter 1, I examine dif- ferent forms of mathematical platonism and I elucidate their features. In particular, I distinguish between a substantive form (Gődel's platonism) and an operational form (anti-constructivism). I also make it clear that I will be concerned with set-theoretic Gődelian platonism. In Chapter 2, I examine the indeterminacy phenomenon in set theory through a detailed analysis of the most famous open conjecture, the Continuum Hypothesis (CH). In Chapter 3, I move on to describe the main philosophical orien- tations with regard to the indeterminacy phenomenon ...
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Richard Kimberly Heck and Paolo Mancosu have claimed that the possibility of non-Cantorian assignments of cardinalities to infinite concepts shows that Hume's Principle (HP) is not implicit in the ...
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Gödel’s philosophical conceptions bear striking similarities to Cantor’s. Although there is no conclusive evidence that Gödel deliberately used or adhered to Cantor’s views, one can successfully reconstruct and see his “Cantorianism” at... more
Gödel’s philosophical conceptions bear striking similarities to Cantor’s. Although there is no conclusive evidence that Gödel deliberately used or adhered to Cantor’s views, one can successfully reconstruct and see his “Cantorianism” at work in many parts of his thought. In this paper, I aim to describe the most prominent conceptual intersections between Cantor’s and Gödel’s thought, particularly on such matters as the nature and existence of mathematical entities (sets), concepts, Platonism, the Absolute Infinite, the progress and inexhaustibility of mathematics.
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Penelope Maddy has recently addressed the set-theoretic multiverse, and expressed reservations on its status and merits (Maddy, Set-theoretic foundations. In: Caicedo et al (eds) Foundations of mathematics. Essays in honor of W. Hugh... more
Penelope Maddy has recently addressed the set-theoretic multiverse, and expressed reservations on its status and merits (Maddy, Set-theoretic foundations. In: Caicedo et al (eds) Foundations of mathematics. Essays in honor of W. Hugh Woodin’s 60th birthday. Contemporary mathematics. American Mathematical Society, Providence, pp. 289–322, 2017). The purpose of the paper is to examine her concerns, by using the interpretative framework of set-theoretic naturalism. I first distinguish three main forms of ‘multiversism’, and then I proceed to analyse Maddy’s concerns. Among other things, I take into account salient aspects of multiverse-related mathematics, in particular, research programmes in set theory for which the use of the multiverse seems to be crucial, and show how one may provide responses to Maddy’s concerns based on a careful analysis of ‘multiverse practice’.
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In the contemporary philosophy of set theory, discussion of new axioms that purport to resolve independence necessitates an explanation of how they come to be justified. Ordinarily, justification is divided into two broad kinds: intrinsic... more
In the contemporary philosophy of set theory, discussion of new axioms that purport to resolve independence necessitates an explanation of how they come to be justified. Ordinarily, justification is divided into two broad kinds: intrinsic justification relates to how ‘intuitively plausible’ an axiom is, whereas extrinsic justification supports an axiom by identifying certain ‘desirable’ consequences. This paper puts pressure on how this distinction is formulated and construed. In particular, we argue that the distinction as often presented is neither well-demarcated nor sufficiently precise. Instead, we suggest that the process of justification in set theory should not be thought of as neatly divisible in this way, but should rather be understood as a conceptually indivisible notion linked to the goal of explanation.
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We address Steel’s Programme to identify a ‘preferred’ universe of set theory and the best axioms extending $\mathsf {ZFC}$ by using his multiverse axioms $\mathsf {MV}$ and the ‘core hypothesis’. In the first part, we examine the... more
We address Steel’s Programme to identify a ‘preferred’ universe of set theory and the best axioms extending $\mathsf {ZFC}$ by using his multiverse axioms $\mathsf {MV}$ and the ‘core hypothesis’. In the first part, we examine the evidential framework for $\mathsf {MV}$ , in particular the use of large cardinals and of ‘worlds’ obtained through forcing to ‘represent’ alternative extensions of $\mathsf {ZFC}$ . In the second part, we address the existence and the possible features of the core of $\mathsf {MV}_T$ (where T is $\mathsf {ZFC}$ +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 $\mathsf {ZFC}$ . To this end, we take into account several strategies, and assess their prospects in the light of $\mathsf {MV}$ ’s evidential framework.
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Godel’s philosophical conceptions bear striking similarities to Cantor’s. Although there is no conclusive evidence that Godel deliberately used or adhered to Cantor’s views, one can successfully reconstruct and see his “Cantorianism” at... more
Godel’s philosophical conceptions bear striking similarities to Cantor’s. Although there is no conclusive evidence that Godel deliberately used or adhered to Cantor’s views, one can successfully reconstruct and see his “Cantorianism” at work in many parts of his thought. In this paper, I aim to describe the most prominent conceptual intersections between Cantor’s and Godel’s thought, particularly on such matters as the nature and existence of mathematical entities (sets), concepts, Platonism, the Absolute Infinite, the progress and inexhaustibility of mathematics.
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We review some conceptions of the set-theoretic multiverse and evaluate their strength. In §1, we introduce the universe/multiverse dichotomy and discuss its significance. In §2, we discuss three alternative conceptions. Finally, in §3,... more
We review some conceptions of the set-theoretic multiverse and evaluate their strength. In §1, we introduce the universe/multiverse dichotomy and discuss its significance. In §2, we discuss three alternative conceptions. Finally, in §3, we present our own conception as integral to the Hyperuniverse Programme launched by Friedman and Arrigoni in [10]. We believe that ours strongly differentiates itself from those examined in §2 (and, more generally, from any other merely descriptive multiverse conception), insofar as it is primarily concerned with the search for new axioms. 1. The Set-theoretic Multiverse The current situation in the foundations of set theory sees the multiverse view and the universe view in confrontation with each other. The latter conception is easily characterised: its supporters think that there is a definite and unique set-theoretic structure which captures all true properties of sets. They are aware that the first-order axioms of sets do not have a fixed refere...
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The (maximal) iterative concept of set is standardly taken to justify ZFC and some of its extensions. In this paper, we show that the maximal iterative concept also lies behind a class of further maximality principles expressing the... more
The (maximal) iterative concept of set is standardly taken to justify ZFC and some of its extensions. In this paper, we show that the maximal iterative concept also lies behind a class of further maximality principles expressing the maximality of the universe of sets V in height and width. These principles have been heavily investigated by the first author and his collaborators within the Hyperuniverse Programme. The programme is based on two essential tools: the hyperuniverse, consisting of all countable transitive models of ZFC, and V -logic, both of which are also fully discussed in the paper.
(ENG) The Continuum Hypothesis, formulated by Cantor in 1878, is one of the most renowned open conjectures in set theory. Already in 1900, the Continuum Problem appeared in Hilbert’s famous list of the most important unsolved problems in... more
(ENG) The Continuum Hypothesis, formulated by Cantor in 1878, is one of the most renowned open conjectures in set theory. Already in 1900, the Continuum Problem appeared in Hilbert’s famous list of the most important unsolved problems in mathematics. As a consequence of the proof of the independence of the Continuum Hypothesis from ZFC, the current status of the Continuum Problem is controversial. In more recent times, the search for a solution to the problem has been one of the main thrusts of the search for new axioms in mathematics. The article provides a description of the most fundamental mathematical results, as well as an examination of the philosophical issues related to the Continuum Problem. (ITA) L’Ipotesi del Continuo (IC), formulata da Cantor nel 1878, è una delle congetture più note e controverse della matematica. Il Problema del Continuo (PC), che vi è collegato, fu collocato da Hilbert, nel 1900, fra i principali problemi “insoluti” della matematica. A seguito della dimostrazione di indipendenza di IC da ZFC, lo status attuale del problema è controverso. In anni più recenti, la ricerca di una soluzione di PC è stata anche una delle ragioni fondamentali per la ricerca di nuovi assiomi in matematica. L’articolo fornisce un quadro generale dei risultati matematici fondamentali, e una breve analisi di alcune delle questioni filosofiche connesse a PC.
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This collection documents the work of the Hyperuniverse Project which is a new approach to set-theoretic truth based on justifiable principles and which leads to the resolution of many questions independent from ZFC. The contributions... more
This collection documents the work of the Hyperuniverse Project which is a new approach to set-theoretic truth based on justifiable principles and which leads to the resolution of many questions independent from ZFC. The contributions give an overview of the program, illustrate its mathematical content and implications, and also discuss its philosophical assumptions. It will thus be of wide appeal among mathematicians and philosophers with an interest in the foundations of set theory. The Hyperuniverse Project was supported by the John Templeton Foundation from January 2013 until September 2015
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FOMUS: Foundations of Mathematics Univalent Foundations and Set Theory was held at the Center for Interdisciplinary Research (ZiF) in Bielefeld, Germany from July 18–23, 2016. This interdisciplinary workshop, designed as a hybrid between... more
FOMUS: Foundations of Mathematics Univalent Foundations and Set Theory was held at the Center for Interdisciplinary Research (ZiF) in Bielefeld, Germany from July 18–23, 2016. This interdisciplinary workshop, designed as a hybrid between a summer school and research conference, was aimed at graduate students, junior researchers, and leading experts from the fields of mathematics, philosophy and computer science. Within this framework, students and researchers from all over the world gathered to investigate and discuss suitable foundations for mathematics and their qualifying criteria, with an emphasis on homotopy type theory/univalent foundations and set theory. The workshop was generously funded by the Association for Symbolic Logic (ASL), the Association of German Mathematicians (DMV), the Berlin Mathematical School (BMS), the Center of Interdisciplinary Research (ZiF), the Deutsche Vereinigung fürMathematische Logik und für Grundlagenforschung der Exakten Wissenschaften (DVMLG), the German Academic Merit Foundation (Stipendiaten machen Programm), the Fachbereich Grundlagen der Informatik of theGerman Informatics Society (GI), and theGerman Society for Analytic Philosophy (GAP). The organizers of the conference were Balthasar Grabmayr (Berlin), Deborah Kant (Berlin), Lukas Kühne (Bonn), and Deniz Sarikaya (Hamburg). The meeting opened with a public plenary talk by Vladimir Voevodsky (Institute for Advanced Study, Princeton) on Multiple concepts of equality in the New Foundations of Mathematics. The other invited talks included:
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We review different conceptions of the set-theoretic multiverse and evaluate their features and strengths. In Sect. 1, we set the stage by briefly discussing the opposition between the ‘universe view’ and the ‘multiverse view’.... more
We review different conceptions of the set-theoretic multiverse and evaluate their features and strengths. In Sect. 1, we set the stage by briefly discussing the opposition between the ‘universe view’ and the ‘multiverse view’. Furthermore, we propose to classify multiverse conceptions in terms of their adherence to some form of mathematical realism. In Sect. 2, we use this classification to review four major conceptions. Finally, in Sect. 3, we focus on the distinction between actualism and potentialism with regard to the universe of sets, then we discuss the Zermelian view, featuring a ‘vertical’ multiverse, and give special attention to this multiverse conception in light of the hyperuniverse programme introduced in Arrigoni and Friedman (Bull Symb Logic 19(1):77–96, 2013). We argue that the distinctive feature of the multiverse conception chosen for the hyperuniverse programme is its utility for finding new candidates for axioms of set theory.
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In recent years, one of the main thrusts of set-theoretic research has been the investigation of maximality principles for V, the universe of sets. The Hyperuniverse Programme (HP) has formulated several maximality principles, which... more
In recent years, one of the main thrusts of set-theoretic research has been the investigation of maximality principles for V, the universe of sets. The Hyperuniverse Programme (HP) has formulated several maximality principles, which express the maximality of V both in height and width. The paper provides an overview of the principles which have been investigated so far in the programme, as well as of the logical and model-theoretic tools which are needed to formulate them mathematically, and also briefly shows how optimal principles, among those available, may be selected in a justifiable way.
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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,... more
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.
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In the contemporary philosophy of set theory, discussion of new axioms that purport to resolve independence necessitates an explanation of how they come to be justified. Ordinarily, justification is divided into two broad kinds: intrinsic... more
In the contemporary philosophy of set theory, discussion of new axioms that purport to resolve independence necessitates an explanation of how they come to be justified. Ordinarily, justification is divided into two broad kinds: intrinsic justification relates to how 'intuitively plausible' an axiom is, whereas extrinsic justification supports an axiom by identifying certain 'desirable' consequences. This paper puts pressure on how this distinction is formulated and construed. In particular, we argue that the distinction as often presented is neither well-demarcated nor sufficiently precise. Instead, we suggest that the process of justification in set theory should not be thought of as neatly divisible in this way, but should rather be understood as a conceptually indivisible notion linked to the goal of explanation.
Research Interests:
In recent years, one of the main thrusts of set-theoretic research has been the investigation of maximality principles for V, the universe of sets. The Hyperuniverse Programme (HP) has formulated several maximality principles, which... more
In recent years, one of the main thrusts of set-theoretic research has been the investigation of maximality principles for V, the universe of sets. The Hyperuniverse Programme (HP) has formulated several maximality principles, which express the maximality of V both in height and width. The paper provides an overview of the principles which have been investigated so far in the programme, as well as of the logical and model-theoretic tools which are needed to formulate them mathematically, and also briefly shows how optimal principles, among those available, may be selected in a justifiable way.
Research Interests:
Penelope Maddy has recently addressed the set-theoretic multiverse, and expressed reservations on its status and merits ([Maddy, 2017]). The purpose of the paper is to examine her concerns, by using the interpretative framework of... more
Penelope Maddy has recently addressed the set-theoretic multiverse, and expressed reservations on its status and merits ([Maddy, 2017]). The purpose of the paper is to examine her concerns, by using the interpretative framework of set-theoretic naturalism. I first distinguish three main forms of 'multiversism', and then I proceed to analyse Maddy's concerns. Among other things, I take into account salient aspects of multiverse-related mathematics , in particular, research programmes in set theory for which the use of the multiverse seems to be crucial, and show how one may provide responses to Maddy's concerns based on a careful analysis of 'multiverse practice'.
(ENG) The Continuum Hypothesis, formulated by Cantor in 1878, is one of the most renowned open conjectures in set theory. Already in 1900, the Continuum Problem appeared in Hilbert’s famous list of the most important unsolved problems in... more
(ENG) The Continuum Hypothesis, formulated by Cantor in 1878, is one of the most renowned open conjectures in set theory. Already in 1900, the Continuum Problem appeared in Hilbert’s famous list of the most important unsolved problems in mathematics. As a consequence of the proof of the independence of the Continuum Hypothesis from ZFC, the current status of the Continuum Problem is controversial. In more recent times, the search for a solution to the problem has been one of the main thrusts of the search for new axioms in mathematics. The article provides a description of the most fundamental mathematical results, as well as an examination of the philosophical issues related to the Continuum Problem.
(ITA) L’Ipotesi del Continuo (IC), formulata da Cantor nel 1878, è una delle congetture più note e controverse della matematica. Il Problema del Continuo (PC), che vi è collegato, fu collocato da Hilbert, nel 1900, fra i principali problemi “insoluti” della matematica. A seguito della dimostrazione di indipendenza di IC da ZFC, lo status attuale del problema è controverso. In anni più recenti, la ricerca di una soluzione di PC è stata anche una delle ragioni fondamentali per la ricerca di nuovi assiomi in matematica. L’articolo fornisce un quadro generale dei risultati matematici fondamentali, e una breve analisi di alcune delle questioni filosofiche connesse a PC.
(ITA) L’Ipotesi del Continuo (IC), formulata da Cantor nel 1878, è una delle congetture più note e controverse della matematica. Il Problema del Continuo (PC), che vi è collegato, fu collocato da Hilbert, nel 1900, fra i principali problemi “insoluti” della matematica. A seguito della dimostrazione di indipendenza di IC da ZFC, lo status attuale del problema è controverso. In anni più recenti, la ricerca di una soluzione di PC è stata anche una delle ragioni fondamentali per la ricerca di nuovi assiomi in matematica. L’articolo fornisce un quadro generale dei risultati matematici fondamentali, e una breve analisi di alcune delle questioni filosofiche connesse a PC.
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(ITA) In questo lavoro esamino la ragioni dell’identificazione, da parte di Guglielmo di Conches e di Bernardo di Chartres, della iusticia naturalis come argomento del Timeo. Attraverso un’analisi delle fonti primarie (Platone stesso) e... more
(ITA) In questo lavoro esamino la ragioni dell’identificazione, da parte di Guglielmo di Conches e di Bernardo di Chartres, della iusticia naturalis come argomento del Timeo. Attraverso un’analisi delle fonti primarie (Platone stesso) e secondarie (la tradizione neoplatonica e, in particolare, il commentarius di Calcidio, che tradusse il Timeo in latino fra il IV e il V sec.), ricostruisco brevemente il processo storico (e logico) in seguito al quale la densa trattazione di argomenti naturali del Timeo poté essere
inquadrata e definita dai due attraverso il riferimento, per noi oscuro, alla iusticia naturalis e, nello stesso tempo, mostro come la nozione stessa di iusticia dei due autori abbia subito una risemantizzazione specifica, legata a considerazioni teologiche ed ermeneutiche.
inquadrata e definita dai due attraverso il riferimento, per noi oscuro, alla iusticia naturalis e, nello stesso tempo, mostro come la nozione stessa di iusticia dei due autori abbia subito una risemantizzazione specifica, legata a considerazioni teologiche ed ermeneutiche.
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The Hyperuniverse Programme, introduced in Arrigoni-Friedman (2013), fosters the search for new set-theoretic axioms. In this paper, we present the procedure envisaged by the programme to find new axioms and the conceptual framework... more
The Hyperuniverse Programme, introduced in Arrigoni-Friedman (2013), fosters the search for new set-theoretic axioms. In this paper, we present the procedure envisaged by the programme to find new axioms and the conceptual framework behind it. The procedure comes in several steps. Intrinsically motivated axioms are those statements which are suggested by the standard concept of set, that is the `maximal iterative concept', and the programme identifies higher-order statements motivated by the maximal iterative concept. The satisfaction of these statements in countable transitive models, the collection of which constitutes the `hyperuniverse', has remarkable consequences, which should be viewed as new intrinsically motivated set-theoretic axioms (H-axioms).
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Gödel’s philosophical conceptions bear striking similarities to Cantor’s. Although there is no conclusive evidence that Gödel deliberately used or adhered to Cantor’s views, one can successfully reconstruct and see his “Cantorianism” at... more
Gödel’s philosophical conceptions bear striking similarities to Cantor’s. Although there is no conclusive evidence that Gödel deliberately used or adhered to Cantor’s views, one can successfully reconstruct and see his “Cantorianism” at work in many parts of his thought. In this paper, I aim to describe the most prominent conceptual intersections between Cantor’s and Gödel’s thought, particularly on such matters as the nature and existence of mathematical entities (sets), concepts, Platonism, the Absolute Infinite, the progress and inexhaustibility of mathematics.
We review different conceptions of the set-theoretic multiverse and evaluate their features and strengths. In Sect. 1, we set the stage by briefly discussing the opposition between the ‘universe view’ and the ‘multiverse view’.... more
We review different conceptions of the set-theoretic multiverse and evaluate their features and strengths. In Sect. 1, we set the stage by briefly discussing the opposition between the ‘universe view’ and the ‘multiverse view’. Furthermore, we
propose to classify multiverse conceptions in terms of their adherence to some form of mathematical realism. In Sect. 2, we use this classification to review four major conceptions. Finally, in Sect. 3, we focus on the distinction between actualism and potentialism with regard to the universe of sets, then we discuss the Zermelian view, featuring a ‘vertical’ multiverse, and give special attention to this multiverse conception in light of the hyperuniverse programme introduced in Arrigoni and Friedman (Bull Symb Logic 19(1):77–96, 2013). We argue that the distinctive feature of the multiverse conception chosen for the hyperuniverse programme is its utility for finding new candidates for axioms of set theory.
propose to classify multiverse conceptions in terms of their adherence to some form of mathematical realism. In Sect. 2, we use this classification to review four major conceptions. Finally, in Sect. 3, we focus on the distinction between actualism and potentialism with regard to the universe of sets, then we discuss the Zermelian view, featuring a ‘vertical’ multiverse, and give special attention to this multiverse conception in light of the hyperuniverse programme introduced in Arrigoni and Friedman (Bull Symb Logic 19(1):77–96, 2013). We argue that the distinctive feature of the multiverse conception chosen for the hyperuniverse programme is its utility for finding new candidates for axioms of set theory.
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We address fallacious analogical reasoning and the Metaphoric Fallacy to a Deductive Inference (MFDI), recently discussed by B. Lightbody and M. Berman (2010). We claim that the authors’ proposal to introduce a new fallacy is only partly... more
We address fallacious analogical reasoning and the Metaphoric Fallacy to a Deductive Inference (MFDI), recently discussed by B. Lightbody and M. Berman (2010). We claim that the authors’ proposal to introduce a new fallacy is only partly justified. We also argue that, in some relevant cases, fallacious analogical reasoning involving metaphors is only affected by the use of quaternio terminorum.
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There are multiple connections between Thucydides' and Plato's political thought, especially with regard to their conceptions of history and human evolution. I explore some of these connections, laying particular emphasis on the fact that... more
There are multiple connections between Thucydides' and Plato's political thought, especially with regard to their conceptions of history and human evolution. I explore some of these connections, laying particular emphasis on the fact that both authors are interested in structural properties and general laws of human and cosmic history. The language of the paper is Italian.
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Esistono infiniti di grandezza diversa e quantità infinitesime? Lo spazio, il tempo e la materia sono infiniti? Perché l’infinito è indispensabile per la comprensione della realtà? Il libro affronta queste domande fornendo una panoramica... more
Esistono infiniti di grandezza diversa e quantità infinitesime? Lo spazio, il tempo e la materia sono infiniti? Perché l’infinito è indispensabile per la comprensione della realtà? Il libro affronta queste domande fornendo una panoramica avvincente e aggiornata del problema in ambiti diversi. Dopo un’immersione nella storia e nella filosofia, illustra i risultati della teoria degli insiemi, dell’analisi matematica e della fisica contemporanea. Attraverso queste discipline, gli autori guidano il lettore nei meandri di un interrogativo che, come ha detto il matematico tedesco David Hilbert, «ha eccitato, più di ogni altro, l’animo umano».
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This collection documents the work of the Hyperuniverse Project which is a new approach to set-theoretic truth based on justifiable principles and which leads to the resolution of many questions independent from ZFC. The contributions... more
This collection documents the work of the Hyperuniverse Project which is a new approach to set-theoretic truth based on justifiable principles and which leads to the resolution of many questions independent from ZFC.
The contributions give an overview of the program, illustrate its mathematical content and implications, and also discuss its philosophical assumptions. It will thus be of wide appeal among mathematicians and philosophers with an interest in the foundations of set theory.
The Hyperuniverse Project was supported by the John Templeton Foundation from January 2013 until September 2015
The contributions give an overview of the program, illustrate its mathematical content and implications, and also discuss its philosophical assumptions. It will thus be of wide appeal among mathematicians and philosophers with an interest in the foundations of set theory.
The Hyperuniverse Project was supported by the John Templeton Foundation from January 2013 until September 2015
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Historically, mathematics has often dealt with the ‘expansion’ of previously accepted concepts and notions. In recent years, Buzaglo (in [1]) has provided a formalisation of concept expansion based on forcing. In this paper, I briefly... more
Historically, mathematics has often dealt with the ‘expansion’ of previously
accepted concepts and notions. In recent years, Buzaglo (in [1]) has provided a formalisation
of concept expansion based on forcing. In this paper, I briefly review Buzaglo’s logic of
concept expansion and I apply it to Cantor’s ‘creation’ of the transfinite. I argue that, while
Buzaglo’s epistemological considerations fit well into Cantor’s conceptions, Buzaglo’s logic
of concept expansion might be unsuitable to justify the creation of the transfinite in terms
of a logically rigorous derivation of concepts.
accepted concepts and notions. In recent years, Buzaglo (in [1]) has provided a formalisation
of concept expansion based on forcing. In this paper, I briefly review Buzaglo’s logic of
concept expansion and I apply it to Cantor’s ‘creation’ of the transfinite. I argue that, while
Buzaglo’s epistemological considerations fit well into Cantor’s conceptions, Buzaglo’s logic
of concept expansion might be unsuitable to justify the creation of the transfinite in terms
of a logically rigorous derivation of concepts.