DOCSLIB.ORG

Structural Reflection and the Ultimate L As the True, Noumenal Universe Of

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Structural Reflection and the Ultimate L As the True, Noumenal Universe Of Scuola Normale Superiore di Pisa CLASSE DI LETTERE Corso di Perfezionamento in Filosofia PHD degree in Philosophy Structural Reflection and the Ultimate L as the true, noumenal universe of mathematics Candidato: Relatore: Emanuele Gambetta Prof.- Massimo Mugnai anno accademico 2015-2016 Structural Reflection and the Ultimate L as the true noumenal universe of mathematics Emanuele Gambetta Contents 1. Introduction 7 Chapter 1. The Dream of Completeness 19 0.1. Preliminaries to this chapter 19 1. G¨odel'stheorems 20 1.1. Prerequisites to this section 20 1.2. Preliminaries to this section 21 1.3. Brief introduction to unprovable truths that are mathematically interesting 22 1.4. Unprovable mathematical statements that are mathematically interesting 27 1.5. Notions of computability, Turing's universe and Intuitionism 32 1.6. G¨odel'ssentences undecidable within PA 45 2. Transfinite Progressions 54 2.1. Preliminaries to this section 54 2.2. Gottlob Frege's definite descriptions and completeness 55 2.3. Transfinite progressions 59 3. Set theory 65 3.1. Preliminaries to this section 65 3.2. Prerequisites: ZFC axioms, ordinal and cardinal numbers 67 3.3. Reduction of all systems of numbers to the notion of set 71 3.4. The first large cardinal numbers and the Constructible universe L 76 3.5. Descriptive set theory, the axioms of determinacy and Luzin's problem formulated in second-order arithmetic 84 3 4 CONTENTS 3.6. The method of forcing and Paul Cohen's independence proof 95 3.7. Forcing Axioms, BPFA assumed as a phenomenal solution to the continuum hypothesis and a Kantian metaphysical distinction 103 3.8. Woodin's program applied to third-order arithmetic, Woodin's Maximum and a comparison between Turing's completeness and Woodin's completeness 111 Chapter 2. Reflection 127 0.1. Preliminaries to this chapter 127 1. The Reflection Principle 130 2. Indescribability 133 3. Koellner's limitative results about Tait's reflection principles 137 4. Welch's Global Reflection 140 Chapter 3. Structural reflection 143 0.1. Preliminaries to this chapter 143 1. Structural reflection and the philosophical concept of richness 144 2. Beginning inner model theory 149 3. Relativising structural reflection to inner models 153 4. The Core Model and Structural Reflection 157 5. The theory of 0y 161 6. The theory of L[E∗], extender models 167 7. The HOD conjecture and the Wholeness axioms 171 8. The Ultimate L model 175 9. The philosophy of mathematics that I sustain 187 Chapter 4. Philosophical Aspects 197 0.1. Preliminaries to this chapter 197 CONTENTS 5 1. Philosophy of the Infinite: comparison between Cantor's absolute infinite and the Absolute principle of Damascius and Plotinus. 198 2. Stressing philosophical aspects: the Kantian distinction again within set theory and the problematic nature of real numbers 207 3. The plausibility of a new axiom, namely ZFC + 0] exists 215 4. Melissus of Samo and Georg Cantor 222 5. John Duns Scotus, the infinite and philosophy of mind 225 6. Reinhardt Cardinals and Anselm's argument 230 7. Paradoxes and the Curry-Liar paradox 234 Chapter 5. Appendix 239 1. Silver indiscernibles 239 Bibliography 247 Bibliography 247 1. INTRODUCTION 7 1. Introduction In this dissertation I am going to discuss structural reflection and the philosophy of set theory. We can find many different kinds of reflection principles in set theory, but I found structural reflection, as conceived by Joan Bagaria, an interesting and powerful method to characterize large cardinals in terms of reflection. In fact, structural reflection can produce a proper class of supercompact cardinals and a proper class of extendible cardinals (by using different conditions). Thus, structural reflection is important because it provides an intrinsic philosophical justification of large cardinals. In fact, the large cardinals, that we are able to interpret as principles of structural reflection are fundamental for Ω-logic and second-order arithmetic. By adopting structural reflection, we reflect an internal structural property of the membership relation. We can try to clarify immediately what one may mean by reflecting an internal structural property of the membership relation by following Bagaria's thought. We could answer that it is a property of some structure of the form (X; 2; (Ri)i2I ), where X is a set or a proper class and (Ri)i2I is a family of relations on X, and where I is a set that may be empty. So, an internal structural property of 2 would be formally given by a formula φ(x), possibly with parameters, that defines a class of structures of the form (X; 2; (Ri)i2I ). We might interpret this fact by saying that there exists an ordinal α that reflects φ and such that for every structure A in the class (that is, for every structure A that satisfies φ) there exists a structure B also in the class which belongs to Vα and is like A. Since, in general, A may be much larger than any B in Vα, the closest resemblance of B to A will be attained in the case that B can be elementarily embedded into A. Thus we can now formulate the principle of structural reflection as follows: Definition 1. (Bagaria) (Structural reflection, SR) For every definable (in the first order language of set theory, with parameters) class of structures C of the form (X; 2 8 CONTENTS Vα ; (Ri)i2I ), there exists α such that α reflects C, i.e. C = C \ Vα and for every A in C there exists B in C \ Vα and an elementary embedding from B into A. From a mathematical perspective, the main objective of this dissertation is the ap- plication of structural reflection to the canonical inner model for a measurable cardinal, namely L[U]. following Bagaria's thought and his results [Bagaria 13], I will prove that structural reflection for Π1 classes of structures definable in V relativized to this canonical inner model is equivalent to the existence of 0y. Then, I will prove that structural reflection for classes of structures (whatever complexity) definable within L[U] is implied by the exis- y tence of 0 . The mathematical result concerning classes of structures Π1 definable in V will support my philosophical thesis that considers Woodin's Ultimate L as the true, noumenal universe of mathematics very close to V. In fact, I will prove that if we apply structural reflection with (Π1 definable in V) classes of structures to a weak extender model for a supercompact cardinal, we do not get transcendence over this inner model. At the same time, I will conjecture that if we apply structural reflection to a canonical inner model for a strong cardinal, we obtain 0{. Bagaria [Bagaria 13] proved that 0] existence is equivalent to structural reflection (with Π1 definable classes of structures in V) relativized to G¨odel's constructible universe L. Therefore, on one side, if we relativize structural reflection with Π1 definable classes of structures in V to G¨odel'sconstructible universe, inner model of iterated sharps, inner model of measurability, inner model of iterated daggers and inner model for a strong cardinal we obtain a sharp for these specific inner models. On the other side, if we relativize structural reflection for Π1 classes of structures definable in V to a weak extender model for a supercompact cardinal, we do not get transcendence. So, we may assert that structural reflection supports the philosophical thesis claiming that Woodin's Ultimate L (not yet constructed) is very close to V and it can be considered as the true, noumenal (I will clarify immediately the philosophical meaning of this word) uni- verse of mathematics where undecided mathematical statements, such as the Continuum 1. INTRODUCTION 9 Hypothesis, are settled. There are many aspects that justify structural reflection itself. First of all, Σ1 structural reflection can be proved from the axioms of ZFC. So, structural reflection is a feature of the universe of sets. Secondly, we have the concept of richness. When we relativize structural reflection to inner models, we are able to transcend the specific inner model and obtain a bigger, richer universe of sets. Thus, the philosophical concept of richness justifies structural reflection. On the one side, structural reflection pro- vides intrinsic philosophical justifications for large cardinals. On the other side, we have philosophical reasons that support structural reflection and render structural reflection a powerful method to interpret large cardinals as principles of reflection. Therefore, from a philosophical perspective, structural reflection is able to justify intrinsically large cardinal notions such as infinitely-many Woodin cardinals and a proper class of Woodin cardinals, that are fundamental for second-order arithmetic and Ω-logic. The precedent two cardinal notions reduce the phenomenon of incompleteness which arises within the universe of sets. Surely, infinitely-many Woodin cardinals and a proper class of Woodin are justified also extrinsically (fruitfullness of the results, i. e., what they are able to prove). Structural reflection is fundamental from a philosophical perspective because it gives us an intrinsic philosophical justification of large cardinals. Intrinsic philosophical justifications are based on the conceptual analysis of the sets themselves. In fact, as we will see, reflec- tion is an essential feature of the universe of sets. Bill Tait's reflection [Koellner 09] could not overcome the barrier represented by G¨odel'sconstructible universe and Philip Welch's [Welch 10] global reflection implies embeddings of proper classes which can be seen as problematic mathematical objects. Therefore, structural reflection seems to be more nat- ural and a more powerful method to characterize very large cardinal numbers. The best feature of this kind of reflection is that it seems to improve on Tait's and Welch's methods of reflection.
Load more

Recommended publications
  • A Cardinal Sin: the Infinite in Spinoza's Philosophy
    Macalester College DigitalCommons@Macalester College Philosophy Honors Projects Philosophy Department Spring 2014 A Cardinal Sin: The nfinitI e in Spinoza's Philosophy Samuel H. Eklund Macalester College, [email protected] Follow this and additional works at: http://digitalcommons.macalester.edu/phil_honors Part of the Philosophy Commons Recommended Citation Eklund, Samuel H., "A Cardinal Sin: The nfinitI e in Spinoza's Philosophy" (2014). Philosophy Honors Projects. Paper 7. http://digitalcommons.macalester.edu/phil_honors/7 This Honors Project is brought to you for free and open access by the Philosophy Department at DigitalCommons@Macalester College. It has been accepted for inclusion in Philosophy Honors Projects by an authorized administrator of DigitalCommons@Macalester College. For more information, please contact [email protected]. A Cardinal Sin: The Infinite in Spinoza’s Philosophy By: Samuel Eklund Macalester College Philosophy Department Honors Advisor: Geoffrey Gorham Acknowledgements This thesis would not have been possible without my advisor, Professor Geoffrey Gorham. Through a collaborative summer research grant, I was able to work with him in improving a vague idea about writing on Spinoza’s views on existence and time into a concrete analysis of Spinoza and infinity. Without his help during the summer and feedback during the past academic year, my views on Spinoza wouldn’t have been as developed as they currently are. Additionally, I would like to acknowledge the hard work done by the other two members of my honors committee: Professor Janet Folina and Professor Andrew Beveridge. Their questions during the oral defense and written feedback were incredibly helpful in producing the final draft of this project.
    [Show full text]
  • On the Infinite in Leibniz's Philosophy
    On the Infinite in Leibniz's Philosophy Elad Lison Interdisciplinary Studies Unit Science, Technology and Society Ph.D. Thesis Submitted to the Senate of Bar-Ilan University Ramat-Gan, Israel August 2010 This work was carried out under the supervision of Dr. Ohad Nachtomy (Department of Philosophy), Bar-Ilan University. Contents א.……………………………….…………………………………………Hebrew Abstract Prologue…………………………………………………………...………………………1 Part A: Historic Survey Methodological Introduction…………………………………………………………..15 1. Aristotle: Potential Infinite………………………………………………………….16 2. Thomas Aquinas: God and the Infinite………………………………………..…….27 3. William of Ockham: Syncategorematic and Actual Infinite……………………..….32 4. Rabbi Abraham Cohen Herrera: Between Absolute Unity and Unbounded Multitude………………………………………………………………………..….42 5. Galileo Galilei: Continuum Constructed from Infinite Zero's………………………49 6. René Descartes: Infinite as Indefinite…………………………………………….…58 7. Pierre Gassendi: Rejection of the Infinite…………………………………………...69 8. Baruch Spinoza: Infinite Unity…………………………………………………...…73 9. General Background: Leibniz and the History of the Infinite……………………....81 Summary…………………………………………………………………………….…94 Part B: Mathematics Introduction…………………………………………………………………………….99 1. 'De Arte Combinatoria' as a Formal Basis for Thought: Retrospective on Leibniz's 1666 Dissertation………………………………………………………………....102 2. Leibniz and the Infinitesimal Calculus……………………………………….……111 2.1. Mathematical Background: Mathematical Works in 16th-17th Centuries…..111 2.2. Leibniz's Mathematical Development…………………………………….…127
    [Show full text]
  • Cantor on Infinity in Nature, Number, and the Divine Mind
    Cantor on Infinity in Nature, Number, and the Divine Mind Anne Newstead Abstract. The mathematician Georg Cantor strongly believed in the existence of actually infinite numbers and sets. Cantor’s “actualism” went against the Aristote- lian tradition in metaphysics and mathematics. Under the pressures to defend his theory, his metaphysics changed from Spinozistic monism to Leibnizian volunta- rist dualism. The factor motivating this change was two-fold: the desire to avoid antinomies associated with the notion of a universal collection and the desire to avoid the heresy of necessitarian pantheism. We document the changes in Can- tor’s thought with reference to his main philosophical-mathematical treatise, the Grundlagen (1883) as well as with reference to his article, “Über die verschiedenen Standpunkte in bezug auf das aktuelle Unendliche” (“Concerning Various Perspec- tives on the Actual Infinite”) (1885). I. he Philosophical Reception of Cantor’s Ideas. Georg Cantor’s dis- covery of transfinite numbers was revolutionary. Bertrand Russell Tdescribed it thus: The mathematical theory of infinity may almost be said to begin with Cantor. The infinitesimal Calculus, though it cannot wholly dispense with infinity, has as few dealings with it as possible, and contrives to hide it away before facing the world Cantor has abandoned this cowardly policy, and has brought the skeleton out of its cupboard. He has been emboldened on this course by denying that it is a skeleton. Indeed, like many other skeletons, it was wholly dependent on its cupboard, and vanished in the light of day.1 1Bertrand Russell, The Principles of Mathematics (London: Routledge, 1992 [1903]), 304.
    [Show full text]
  • The Journal of Symbolic Logic a Definable Nonstandard Model of the Reals
    Sh:825 The Journal of Symbolic Logic http://journals.cambridge.org/JSL Additional services for The Journal of Symbolic Logic: Email alerts: Click here Subscriptions: Click here Commercial reprints: Click here Terms of use : Click here A denable nonstandard model of the reals Vladimir Kanovei and Saharon Shelah The Journal of Symbolic Logic / Volume 69 / Issue 01 / March 2004, pp 159 - 164 DOI: 10.2178/jsl/1080938834, Published online: 12 March 2014 Link to this article: http://journals.cambridge.org/abstract_S0022481200008094 How to cite this article: Vladimir Kanovei and Saharon Shelah (2004). A denable nonstandard model of the reals . The Journal of Symbolic Logic, 69, pp 159-164 doi:10.2178/jsl/1080938834 Request Permissions : Click here Downloaded from http://journals.cambridge.org/JSL, IP address: 128.210.126.199 on 19 May 2015 Sh:825 THE JOURNAL OF SYMBOLIC LOGIC Volume 69, Number 1. March 2004 A DEFINABLE NONSTANDARD MODEL OF THE REALS VLADIMIR KANOVEIf AND SAHARON SHELAH * Abstract. We prove, in ZFC, the existence of a definable, countably saturated elementary extension of the reals. §1. Introduction. It seems that it has been taken for granted that there is no distinguished, definable nonstandard model of the reals. (This means a countably saturated elementary extension of the reals.) Of course if V = L then there is such an extension (just take the first one in the sense of the canonical well-ordering of L), but we mean the existence provably in ZFC. There were good reasons for this: without Choice we cannot prove the existence of any elementary extension of the reals containing an infinitely large integer.' 2 Still there is one.
    [Show full text]
  • Cantor-Von Neumann Set-Theory Fa Muller
    Logique & Analyse 213 (2011), x–x CANTOR-VON NEUMANN SET-THEORY F.A. MULLER Abstract In this elementary paper we establish a few novel results in set the- ory; their interest is wholly foundational-philosophical in motiva- tion. We show that in Cantor-Von Neumann Set-Theory, which is a reformulation of Von Neumann's original theory of functions and things that does not introduce `classes' (let alone `proper classes'), developed in the 1920ies, both the Pairing Axiom and `half' the Axiom of Limitation are redundant — the last result is novel. Fur- ther we show, in contrast to how things are usually done, that some theorems, notably the Pairing Axiom, can be proved without invok- ing the Replacement Schema (F) and the Power-Set Axiom. Also the Axiom of Choice is redundant in CVN, because it a theorem of CVN. The philosophical interest of Cantor-Von Neumann Set- Theory, which is very succinctly indicated, lies in the fact that it is far better suited than Zermelo-Fraenkel Set-Theory as an axioma- tisation of what Hilbert famously called Cantor's Paradise. From Cantor one needs to jump to Von Neumann, over the heads of Zer- melo and Fraenkel, and then reformulate. 0. Introduction In 1928, Von Neumann published his grand axiomatisation of Cantorian Set- Theory [1925; 1928]. Although Von Neumann's motivation was thoroughly Cantorian, he did not take the concept of a set and the membership-relation as primitive notions, but the concepts of a thing and a function — for rea- sons we do not go into here. This, and Von Neumann's cumbersome nota- tion and terminology (II-things, II.I-things) are the main reasons why ini- tially his theory remained comparatively obscure.
    [Show full text]
  • Extending Modal Logic
    Extending Modal Logic Maarten de Rijke Extending Modal Logic ILLC Dissertation Series 1993-4 institute [brlogic, language andcomputation For further information about ILLC-publications, please contact: Institute for Logic, Language and Computation Universiteit van Amsterdam Plantage Muidergracht 24 1018 TV Amsterdam phone: +31-20-5256090 fax: +31—20-5255101 e-mail: [email protected] Extending Modal Logic Academisch Proefschrift ter verkrijging van de graad van doctor aan de Universiteit van Amsterdam, op gezag van de Rector Magnificus Prof. Dr. P.W.M. de Meijer in het openbaar te verdedigen in de Aula der Universiteit (Oude Lutherse Kerk, ingang Singel 411, hoek Spui) op vrijdag 3 december 1993 te 15.00 uur door Maarten de Rijke geboren te Vlissingen. Promotor: Prof.dr. J.F.A.K. van Benthem Faculteit Wiskunde en Informatica Universiteit van Amsterdam Plantage Muidergracht 24 1018 TV Amsterdam The investigations were supported by the Philosophy Research Foundation (SWON), which is subsidized by the Netherlands Organization for Scientific Research (NWO). Copyright © 1993 by ‘.\/Iaarten de Rijke Cover design by Han Koster. Typeset in LATEXbythe author. Camera-ready copy produced at the Faculteit der Wiskunde en Informatica, Vrije Universiteit Amsterdam. Printed and bound by Haveka B.V., Alblasserdam. ISBN: 90-800 769—9—6 Contents Acknowledgments vii I Introduction 1 1 What this dissertation is about 3 1.1 Modal logic . 3 1.2 Extending modal logic . 3 1.3 A look ahead . 4 2 What is Modal Logic? 6 2.1 Introduction . 6 2.2 A framework for modal logic 7 2.3 Examples . 10 2.4 Questions and comments .
    [Show full text]
  • Relation 2016 08 27.Pdf
    For updated version of this document see http://imechanica.org/node/19709 Zhigang Suo RELATION Binary Relation Relation. A binary relation is a collection of ordered pairs. If there is no danger for confusion, we will simply call a binary relation a relation. Thus, a relation is a set, each element of which is an ordered pair. If an ordered pair x,y belongs to a relation R, we write ( ) x,y ∈R . ( ) We say that the items x and y stand in the relation R. Specify a relation by a collection of ordered pairs. We specify a relation in the same way as we specify any set, by specifying the elements of the set. Here is a collection of three ordered pairs: red, blue , conservative, libral , elephant, donkey . {( ) ( ) ( )} Do we call this collection a relation? Yes, any collection of any ordered pairs defines a relation. This set may or may not be a useful relation, but it surely fits the definition of a relation. We can, of course, try to discover the “meaning” of this relation, or the property that specifies this relation. But we do not have to do anything to call the set a relation. Specify a relation by a property. This said, we do want to know the “meaning” of a relation. Like a useful set, a useful relation is often specified by a property, a rule that picks the ordered pairs in the relation. Consider a collection of two ordered pairs: Lisa, Eric , Daniel, Michael . {( ) ( )} By talking to the people, we learn that this relation is specified by a property: each ordered pair is a pair of siblings.
    [Show full text]
  • Lost in the Labyrinth: Spinoza, Leibniz and the Continuum Lost in the Labyrinth: Spinoza, Leibniz and the Continuum
    LOST IN THE LABYRINTH: SPINOZA, LEIBNIZ AND THE CONTINUUM LOST IN THE LABYRINTH: SPINOZA, LEIBNIZ AND THE CONTINUUM By PATRICK RIESTERER, B.A. A Thesis Submitted to the School of Graduate Studies In Partial Fulfillment ofthe Requirements For the Degree Master ofArts McMaster University © Copyright by Patrick Riesterer, August 2006 MASTER OF ARTS (2006) McMaster University (Philosophy) Hamilton, Ontario TITLE: Lost in the Labyrinth: Spinoza, Leibniz and the Continuum AUTHOR: Patrick Riesterer, B.A. (Trinity Western University) SUPERVISOR: Professor Richard Arthur NUMBER OF PAGES: vi, 110 ii Abstract In this thesis, I address the extent ofSpinoza's influence on the development of Leibniz's response to the continuum problem, with particular emphasis on his relational philosophy oftime and space. I expend the first chapter carefully reconstructing Spinoza's position on infinity and time. We see that Spinoza developed a threefold definition ofinfinity to explain the difference between active substance and its passive modes. Spinoza advances a syncategorematic interpretation ofinfinity, and founds a causal theory oftime directly on this conception ofinfinity. In the second chapter, I examine the changes Leibniz's understanding ofthe continuum problem underwent during 1676 and immediately thereafter. During this period, Leibniz's interacted extensively with Spinoza's ideas. We see that several fundamental features ofLeibniz's philosophy oftime take shape at this time. Leibniz adopts a Spinozistic definition ofdivine eternity and immensity, he reevaluates several analogies in an attempt to understand how the attributes ofa substance interrelate, and he develops the notion ofthe law of the series that will become an essential feature ofmonadic appetition. Leibniz synthesizes several ofthese discoveries into a first philosophy ofmotion.
    [Show full text]
  • A Mathematical Model of Divine Infinity
    A Mathematical Model of Divine Infinity Prof. Eric Steinhart, Department of Philosophy, William Paterson University, Wayne NJ 07470. Email: <[email protected]>, <[email protected]>. Published in 2009 in Theology and Science 7 (3), 261 – 274. ABSTRACT: Mathematics is obviously important in the sciences. And so it is likely to be equally important in any effort that aims to understand God in a scientifically significant way or that aims to clarify the relations between science and theology. The degree to which God has any perfection is absolutely infinite. We use contemporary mathematics to precisely define that absolute infinity. For any perfection, we use transfinite recursion to define an endlessly ascending series of degrees of that perfection. That series rises to an absolutely infinite degree of that perfection. God has that absolutely infinite degree. We focus on the perfections of knowledge, power, and benevolence. Our model of divine infinity thus builds a bridge between mathematics and theology. KEYWORDS: God; mathematics; perfection; infinity; Cantor; transfinite recursion. 1. Introduction All will agree that science makes extensive use of mathematics. At least in physics, scientific progress seems to go hand in hand with increasing formalization. More sophisticated physical theories are also more highly mathematical. And the correlation of progress with increasing formalization also appears sciences like chemistry and biology as well. Mathematics is obviously important in the sciences. And so it is likely to be equally important in any effort that aims to understand God in a scientifically significant way or that aims to clarify the relations between science and theology. Nevertheless, mathematics has seen little application in theology.
    [Show full text]
  • A Characterization of Finitary Bisimulation Basic Research in Computer Science
    BRICS Basic Research in Computer Science BRICS RS-97-26 Aceto & Ing A Characterization of Finitary Bisimulation olfsd ´ ottir: A Characterization of Finitary Bisimulation ´ Luca Aceto Anna Ingolfsd´ ottir´ BRICS Report Series RS-97-26 ISSN 0909-0878 October 1997 Copyright c 1997, BRICS, Department of Computer Science University of Aarhus. All rights reserved. Reproduction of all or part of this work is permitted for educational or research use on condition that this copyright notice is included in any copy. See back inner page for a list of recent BRICS Report Series publications. Copies may be obtained by contacting: BRICS Department of Computer Science University of Aarhus Ny Munkegade, building 540 DK–8000 Aarhus C Denmark Telephone: +45 8942 3360 Telefax: +45 8942 3255 Internet: [email protected] BRICS publications are in general accessible through the World Wide Web and anonymous FTP through these URLs: http://www.brics.dk ftp://ftp.brics.dk This document in subdirectory RS/97/26/ A Characterization of Finitary Bisimulation Luca Aceto∗ † BRICS, Department of Computer Science, Aalborg University Fredrik Bajersvej 7E, 9220 Aalborg Ø, Denmark Email: [email protected] Anna Ing´olfsd´ottir‡ Dipartimento di Sistemi ed Informatica, Universit`a di Firenze Via Lombroso 6/17, 50134 Firenze, Italy Email: [email protected] Keywords and Phrases: Concurrency, labelled transition system with di- vergence, bisimulation preorder, finitary relation. 1 Introduction Following a paradigm put forward by Milner and Plotkin, a primary criterion to judge the appropriateness of denotational models for programming and specifi- cation languages is that they be in agreement with operational intuition about program behaviour.
    [Show full text]
  • Outline of Lecture 2 First Order Logic and Second Order Logic Basic Summary and Toolbox
    LogicalMethodsinCombinatorics,236605-2009/10 Lecture 2 Outline of Lecture 2 First Order Logic and Second Order Logic Basic summary and Toolbox • Vocabularies and structures • Isomorphisms and substructures • First Order Logic FOL, a reminder. • Completeness and Compactness • Second Order Logic SOL and SOLn and Monadic Second Order Logic MSOL. • Definability 1 LogicalMethodsinCombinatorics,236605-2009/10 Lecture 2 Vocabularies We deal with (possibly many-sorted) relational structures. Sort symbols are Uα : α ∈ IN Relation symbols are Ri,α : i ∈ Ar, α ∈ IN where Ar is a set of arities, i.e. of finite sequences of sort symbols. In the case of one-sorted vocabularies, the arity is just of the form hU,U,...n ...,Ui which will denoted by n. A vocabulary is a finite set of finitary relation symbols, usually denoted by τ, τi or σ. 2 LogicalMethodsinCombinatorics,236605-2009/10 Lecture 2 τ-structures Structures are interpretations of vocabularies, of the form A = hA(Uα), A(Ri,α): Uα, Ri,α ∈ τi with A(Uα)= Aα sets, and, for i =(Uj1 ,...,Ujr ) A(Ri,α) ⊆ Aj1 × ... × Ajr . Graphs: hV ; Ei with vertices as domain and edges as relation. hV ⊔ E, RGi with two sorted domain of vertices and edges and incidence relation. Labeled Graphs: As graphs but with unary predicates for vertex labels and edge labels depending whether edges are elements or tuples. Binary Words: hV ; R<,P0i with domain lineraly ordered by R< and colored by P0, marking the zero’s. τ-structures: General relational structures. 3 LogicalMethodsinCombinatorics,236605-2009/10 Lecture 2 τ-Substructures Let A, B be two τ-structures.
    [Show full text]
  • Two Pictures of the Iterative Hierarchy
    TWO PICTURES OF THE ITERATIVE HIERARCHY by Ida Marie Myrstad Dahl Thesis for the degree of Master in Philosophy Supervised by Professor Øystein Linnebo Fall 2014 Department of Philosophy, Classics, History of Arts and Ideas University of Oslo Two pictures of the iterative hierarchy ©2014 Two pictures of the iterative hierarchy Ida Marie Myrstad Dahl http://www.duo.uio.no Print: OKPrintShop iii Contents Acknowledgements . v Abstract . vi Introduction 1 1 The iterative conception of set 4 1.1 What it is . 4 1.2 Why the iterative conception? . 8 2 Actualism and potentialism on the iterative conception 15 2.1 The actualist picture . 16 2.2 The potentialist picture . 19 3 From the ancient to the contemporary concept of infinity 24 3.1 Aristotle on infinity . 24 3.2 Three important developments . 30 3.3 Cantor’s theory of the infinite . 31 3.4 A tension between actualism and potentialism . 34 4 Two tenable pictures? 40 4.1 (1) An actual conception . 41 4.2 (2) An intuitive conception . 47 4.3 (3) Explaining paradox . 50 4.4 Two tenable interpretations . 53 Conclusion 56 iv Acknowledgements I owe special thanks to Øystein Linnebo, for helpful supervision of my project, but also for introducing me to the philosophical ideas about the infinite in a phi- losophy of language seminar, at the University of Oslo, autumn, 2012. His ideas and teaching has been of great inspiration. Also, attending the PPP-seminars (Plurals, Predicates and Paradox), led by Linnebo, in 2013, was of great interest, and helped me single out the topic I wanted to write about.
    [Show full text]