Abstract Consequence and Logics
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Notes on Set Theory, Part 2
§11 Regular cardinals In what follows, κ , λ , µ , ν , ρ always denote cardinals. A cardinal κ is said to be regular if κ is infinite, and the union of fewer than κ sets, each of whose cardinality is less than κ , is of cardinality less than κ . In symbols: κ is regular if κ is infinite, and κ 〈 〉 ¡ κ for any set I with I ¡ < and any family Ai i∈I of sets such that Ai < ¤¦¥ £¢ κ for all i ∈ I , we have A ¡ < . i∈I i (Among finite cardinals, only 0 , 1 and 2 satisfy the displayed condition; it is not worth including these among the cardinals that we want to call "regular".) ℵ To see two examples, we know that 0 is regular: this is just to say that the union of finitely ℵ many finite sets is finite. We have also seen that 1 is regular: the meaning of this is that the union of countably many countable sets is countable. For future use, let us note the simple fact that the ordinal-least-upper-bound of any set of cardinals is a cardinal; for any set I , and cardinals λ for i∈I , lub λ is a cardinal. i i∈I i α Indeed, if α=lub λ , and β<α , then for some i∈I , β<λ . If we had β § , then by i∈I i i β λ ≤α β § λ λ < i , and Cantor-Bernstein, we would have i , contradicting the facts that i is β λ β α β ¨ α α a cardinal and < i . -
Equivalents to the Axiom of Choice and Their Uses A
EQUIVALENTS TO THE AXIOM OF CHOICE AND THEIR USES A Thesis Presented to The Faculty of the Department of Mathematics California State University, Los Angeles In Partial Fulfillment of the Requirements for the Degree Master of Science in Mathematics By James Szufu Yang c 2015 James Szufu Yang ALL RIGHTS RESERVED ii The thesis of James Szufu Yang is approved. Mike Krebs, Ph.D. Kristin Webster, Ph.D. Michael Hoffman, Ph.D., Committee Chair Grant Fraser, Ph.D., Department Chair California State University, Los Angeles June 2015 iii ABSTRACT Equivalents to the Axiom of Choice and Their Uses By James Szufu Yang In set theory, the Axiom of Choice (AC) was formulated in 1904 by Ernst Zermelo. It is an addition to the older Zermelo-Fraenkel (ZF) set theory. We call it Zermelo-Fraenkel set theory with the Axiom of Choice and abbreviate it as ZFC. This paper starts with an introduction to the foundations of ZFC set the- ory, which includes the Zermelo-Fraenkel axioms, partially ordered sets (posets), the Cartesian product, the Axiom of Choice, and their related proofs. It then intro- duces several equivalent forms of the Axiom of Choice and proves that they are all equivalent. In the end, equivalents to the Axiom of Choice are used to prove a few fundamental theorems in set theory, linear analysis, and abstract algebra. This paper is concluded by a brief review of the work in it, followed by a few points of interest for further study in mathematics and/or set theory. iv ACKNOWLEDGMENTS Between the two department requirements to complete a master's degree in mathematics − the comprehensive exams and a thesis, I really wanted to experience doing a research and writing a serious academic paper. -
Isabelle's Logics: FOL and ZF
ZF ∀ = Isabelle α λ β → Isabelle's Logics: FOL and ZF Lawrence C. Paulson Computer Laboratory University of Cambridge [email protected] With Contributions by Tobias Nipkow and Markus Wenzel1 31 October 1999 1Markus Wenzel made numerous improvements. Philippe de Groote con- tributed to ZF. Philippe No¨eland Martin Coen made many contributions to ZF. The research has been funded by the EPSRC (grants GR/G53279, GR/H40570, GR/K57381, GR/K77051) and by ESPRIT project 6453: Types. Abstract This manual describes Isabelle's formalizations of many-sorted first-order logic (FOL) and Zermelo-Fraenkel set theory (ZF). See the Reference Manual for general Isabelle commands, and Introduction to Isabelle for an overall tutorial. Contents 1 Syntax definitions 1 2 First-Order Logic 3 2.1 Syntax and rules of inference ................... 3 2.2 Generic packages ......................... 4 2.3 Intuitionistic proof procedures .................. 4 2.4 Classical proof procedures .................... 9 2.5 An intuitionistic example ..................... 10 2.6 An example of intuitionistic negation .............. 12 2.7 A classical example ........................ 13 2.8 Derived rules and the classical tactics .............. 15 2.8.1 Deriving the introduction rule .............. 16 2.8.2 Deriving the elimination rule ............... 17 2.8.3 Using the derived rules .................. 18 2.8.4 Derived rules versus definitions ............. 20 3 Zermelo-Fraenkel Set Theory 23 3.1 Which version of axiomatic set theory? ............. 23 3.2 The syntax of set theory ..................... 24 3.3 Binding operators ......................... 26 3.4 The Zermelo-Fraenkel axioms .................. 29 3.5 From basic lemmas to function spaces .............. 32 3.5.1 Fundamental lemmas .................. -
Vasiliev and the Foundations of Logic
Chapter 4 Vasiliev and the Foundations of Logic Otávio Bueno Abstract Nikolai Vasiliev offered a systematic approach to the development of a class of non-classical logics, which he called “Imaginary Logics”. In this paper, IexaminecriticallysomeofthecentralfeaturesofVasiliev’sapproachtological theory, suggesting its relevance to contemporary debates in the philosophy of logic. IarguethatthereismuchofsignificantvalueinVasiliev’swork,whichdeserves close philosophical engagement. Keywords Vasiliev • logical pluralism • Revisability • a priori • negation 4.1 Introduction: Six Central Features of Vasiliev’s Approach to Logical Theory Nikolai Vasiliev’s approach to logical theory has a number of features. Six of them, in particular, are worth highlighting: (a) logical pluralism (there is a plurality of logics, depending on the subject matter under consideration); (b) logical revisability (certain logical laws can be revised depending on the subject matter); (c) logical non-a priorism (certain logical laws are empirically based); (d) logical contingency (given the empirical nature of some logical laws, they are ultimately contingent; in this context, issues regarding the scope of logic are also examined, with the accompanying distinction between laws of objects and laws of thought); (e) the nature of negation (negation is characterized via incompatibility; it is not just difference, nor is it grounded on absence, and it is inferred rather than perceived); and (f) logical commitment (why Vasiliev is not a dialetheist, after all). In this paper, I will examine each of these features, and suggest the relevance of many of Vasiliev’s proposals to contemporary philosophical reflection about the foundations of logic. Although the terms I use to describe some of the views O. Bueno (!) Department of Philosophy, University of Miami, Coral Gables, FL 33124, USA e-mail: [email protected] ©SpringerInternationalPublishingAG2017 43 V. -
Axioms of Set Theory and Equivalents of Axiom of Choice Farighon Abdul Rahim Boise State University, [email protected]
Boise State University ScholarWorks Mathematics Undergraduate Theses Department of Mathematics 5-2014 Axioms of Set Theory and Equivalents of Axiom of Choice Farighon Abdul Rahim Boise State University, [email protected] Follow this and additional works at: http://scholarworks.boisestate.edu/ math_undergraduate_theses Part of the Set Theory Commons Recommended Citation Rahim, Farighon Abdul, "Axioms of Set Theory and Equivalents of Axiom of Choice" (2014). Mathematics Undergraduate Theses. Paper 1. Axioms of Set Theory and Equivalents of Axiom of Choice Farighon Abdul Rahim Advisor: Samuel Coskey Boise State University May 2014 1 Introduction Sets are all around us. A bag of potato chips, for instance, is a set containing certain number of individual chip’s that are its elements. University is another example of a set with students as its elements. By elements, we mean members. But sets should not be confused as to what they really are. A daughter of a blacksmith is an element of a set that contains her mother, father, and her siblings. Then this set is an element of a set that contains all the other families that live in the nearby town. So a set itself can be an element of a bigger set. In mathematics, axiom is defined to be a rule or a statement that is accepted to be true regardless of having to prove it. In a sense, axioms are self evident. In set theory, we deal with sets. Each time we state an axiom, we will do so by considering sets. Example of the set containing the blacksmith family might make it seem as if sets are finite. -
Paradox and Foundation Zach Weber Submitted in Total Fulfilment of The
Paradox and Foundation Zach Weber Submitted in total fulfilment of the requirements of the degree of Doctor of Philosophy May 2009 School of Philosophy, Anthropology and Social Inquiry The University of Melbourne This is to certify that - the thesis comprises only my original work towards the PhD, - due acknowledgement has been made in the text to all other material used, - the thesis is less than 100,000 words in length. Preface Dialethic paraconsistency is an approach to formal and philosophical theories in which some but not all contradictions are true. Advancing that program, this thesis is about paradoxes and the foundations of mathematics, and is divided accordingly into two main parts. The first part concerns the history and philosophy of set theory from Cantor through the independence proofs, focusing on the set concept. A set is any col- lection of objects that is itself an object, with identity completely determined by membership. The set concept is called naive because it is inconsistent. I argue that the set concept is inherently and rightly paradoxical, because sets are both intensional and extensional objects: Sets are predicates in extension. All consistent characterizations of sets are either not explanatory or not coherent. To understand sets, we need to reason about them with an appropriate logic; paraconsistent naive set theory is situated as a continuation of the original foundational project. The second part produces a set theory deduced from an unrestricted compre- hension principle using the weak relevant logic DLQ, dialethic logic with quantifiers. I discuss some of the problems involved with embedding in DLQ, especially related to identity and substitution. -
Aristotle on Non-Contradiction: Philosophers Vs
Journal of Ancient Philosophy J. anc. philos. (Engl. ed.), São Paulo, v.7, n.2. p. 51-74, 2013. ISSN 1981-9471 - FFLCH/USP DOI: http://dx.doi.org/10.11606/issn.1981-9471.v7i2p51-74 www.revistas.usp.br/filosofiaantiga Aristotle on Non-Contradiction: Philosophers vs. Non-Philosophers Jean-Louis Hudry USP, Brazil Abstract: Aristotle’s principle of non-contradiction (PNC) has been interpreted by Łukasiewicz through three distinct formulations, namely ontological, logical, and psychological. Many have criticized Łukasiewicz’s position, but they still maintain that Aristotle defends distinct formulations. In contrast, this paper shows that Aristotle suggests only one formulation of the PNC. This unique formulation belongs to philosophy as the first science, so that the philosophers think of the PNC as a necessarily true principle, owing to their meta-physical cognition of the nature of things. Yet, there is another way to understand this formulation. Indeed, the non-philosophers believe in the PNC, without being able to understand its necessary truth, due to their ignorance of philosophy. Thus, Aristotle has to convince them that the PNC is the most certain opinion of all, and his dialectical justifications are purposely weak, as they are only concerned with the defense of a common opinion. In Chapter 3 of Metaphysics Gamma, Aristotle introduces the so-called principle of non-contradiction (hereafter PNC).1 It is important to put this principle into context in order to understand why and how Aristotle introduces it: It is proper for the one who best cognizes (gnôrizonta) each genus to be able to state the most certain principles (archas) of an actual thing, so that the one who cognizes beings, qua beings (tôn ontôn hêi onta), will also be able to state the most certain principles of all things. -
Local Theory of Sets As a Foundation for Category Theory and Its Connection with the Zermelo–Fraenkel Set Theory
Journal of Mathematical Sciences, Vol. 138, No. 4, 2006 LOCAL THEORY OF SETS AS A FOUNDATION FOR CATEGORY THEORY AND ITS CONNECTION WITH THE ZERMELO–FRAENKEL SET THEORY V. K. Zakharov, E. I. Bunina, A. V. Mikhalev, and P. V. Andreev UDC 510.223, 512.581 CONTENTS Introduction..............................................5764 1. First-Order Theories and LTS ....................................5767 1.1. Language of first-order theories ................................5767 1.2. Deducibility in a first-order theory . ............................5768 1.3.Aninterpretationofafirst-ordertheoryinasettheory...................5770 2.LocalTheoryofSets.........................................5772 2.1.Properaxiomsandaxiomschemesofthelocaltheoryofsets................5772 2.2.Someconstructionsinthelocaltheoryofsets........................5775 3. Formalization of the Naive Category Theory within the Framework oftheLocalTheoryofSets.....................................5776 3.1.Definitionofalocalcategoryinthelocaltheoryofsets...................5776 3.2. Functors and natural transformations and the “category of categories” and the “category of functors” in the local theory of sets generated by them . ................5777 4. Zermelo–Fraenkel Set Theory and Mirimanov–von Neumann Sets ................5779 4.1.TheproperaxiomsandaxiomschemesoftheZFsettheory................5779 4.2.OrdinalsandcardinalsintheZFsettheory.........................5782 4.3.Mirimanov–vonNeumannsetsintheZFsettheory.....................5786 5. Universes, Ordinals, Cardinals, and Mirimanov–von -
The Logical Analysis of Key Arguments in Leibniz and Kant
University of Mississippi eGrove Electronic Theses and Dissertations Graduate School 2016 The Logical Analysis Of Key Arguments In Leibniz And Kant Richard Simpson Martin University of Mississippi Follow this and additional works at: https://egrove.olemiss.edu/etd Part of the Philosophy Commons Recommended Citation Martin, Richard Simpson, "The Logical Analysis Of Key Arguments In Leibniz And Kant" (2016). Electronic Theses and Dissertations. 755. https://egrove.olemiss.edu/etd/755 This Thesis is brought to you for free and open access by the Graduate School at eGrove. It has been accepted for inclusion in Electronic Theses and Dissertations by an authorized administrator of eGrove. For more information, please contact [email protected]. THE LOGICAL ANALYSIS OF KEY ARGUMENTS IN LEIBNIZ AND KANT A Thesis presented in partial fulfillment of requirements for the degree of Master of Arts in the Department of Philosophy and Religion The University of Mississippi By RICHARD S. MARTIN August 2016 Copyright Richard S. Martin 2016 ALL RIGHTS RESERVED ABSTRACT This paper addresses two related issues of logic in the philosophy of Gottfried Leibniz. The first problem revolves around Leibniz’s struggle, throughout the period of his mature philosophy, to reconcile his metaphysics and epistemology with his antecedent theological commitments. Leibniz believes that for everything that happens there is a reason, and that the reason God does things is because they are the best that can be done. But if God must, by nature, do what is best, and if what is best is predetermined, then it seems that there may be no room for divine freedom, much less the human freedom Leibniz wished to prove. -
Leibniz and the Principle of Sufficient Reason
Leibniz and the Principle of Sufficient Reason by Owen Pikkert A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Department of Philosophy University of Toronto © Copyright by Owen Pikkert 2018 Abstract Leibniz and the Principle of Sufficient Reason Owen Pikkert Doctor of Philosophy Department of Philosophy University of Toronto 2018 Leibniz’s principle of sufficient reason (PSR) is the claim that everything has an explanation. It rules out brute facts, inexplicable primitives, and purely random events. On the usual view, Leibniz grounds the PSR in purely descriptive truths. On my view, Leibniz grounds the PSR in this being the best of all possible worlds. God only creates the best, and a world in which the PSR is true is better than a world in which it is false. For the PSR ensures that the world has an explanatory structure, the investigation of which facilitates human happiness. This way of grounding the PSR faces at least two problems. The first problem is that it presupposes that the PSR is a contingent principle, even though most commentators take it to be necessary. But I argue that Leibniz is indeed committed to the contingency of the PSR. I demonstrate this by showing how, for Leibniz, PSR-violating entities such as vacua, atoms, and indiscernible bodies are possible but not actual. I also argue that the contingency of the ii PSR does not conflict with Leibniz’s other modal commitments. In particular, it does not conflict with the modal status of his principle of the identity of indiscernibles, nor with the modal status of his theory of truth. -
Early Russell on Types and Plurals of Analytical Philosophy Kevin C
Journal FOR THE History Early Russell ON TYPES AND PlurALS OF Analytical Philosophy KeVIN C. Klement VOLUME 2, Number 6 In 1903, in The Principles of Mathematics (PoM), Russell endorsed an account of classes whereupon a class fundamentally is to be Editor IN Chief considered many things, and not one, and used this thesis to SandrA Lapointe, McMaster University explicate his first version of a theory of types, adding that it formed the logical justification for the grammatical distinction Editorial BoarD between singular and plural. The view, however, was short- Juliet Floyd, Boston University lived—rejected before PoM even appeared in print. However, GrEG Frost-Arnold, Hobart AND William Smith Colleges aside from mentions of a few misgivings, there is little evidence Henry Jackman, YORK University about why he abandoned this view. In this paper, I attempt to Chris Pincock, Ohio State University clarify Russell’s early views about plurality, arguing that they Mark TExtor, King’S College London did not involve countenancing special kinds of plural things RicharD Zach, University OF Calgary distinct from individuals. I also clarify what his misgivings about these views were, making it clear that while the plural understanding of classes helped solve certain forms of Russell’s PrODUCTION Editor paradox, certain other Cantorian paradoxes remained. Finally, Ryan Hickerson, University OF WESTERN OrEGON I aim to show that Russell’s abandonment of something like plural logic is understandable given his own conception of Editorial Assistant logic and philosophical aims when compared to the views and Daniel Harris, CUNY GrADUATE Center approaches taken by contemporary advocates of plural logic. -
First-Order Logic in a Nutshell Syntax
First-Order Logic in a Nutshell 27 numbers is empty, and hence cannot be a member of itself (otherwise, it would not be empty). Now, call a set x good if x is not a member of itself and let C be the col- lection of all sets which are good. Is C, as a set, good or not? If C is good, then C is not a member of itself, but since C contains all sets which are good, C is a member of C, a contradiction. Otherwise, if C is a member of itself, then C must be good, again a contradiction. In order to avoid this paradox we have to exclude the collec- tion C from being a set, but then, we have to give reasons why certain collections are sets and others are not. The axiomatic way to do this is described by Zermelo as follows: Starting with the historically grown Set Theory, one has to search for the principles required for the foundations of this mathematical discipline. In solving the problem we must, on the one hand, restrict these principles sufficiently to ex- clude all contradictions and, on the other hand, take them sufficiently wide to retain all the features of this theory. The principles, which are called axioms, will tell us how to get new sets from already existing ones. In fact, most of the axioms of Set Theory are constructive to some extent, i.e., they tell us how new sets are constructed from already existing ones and what elements they contain. However, before we state the axioms of Set Theory we would like to introduce informally the formal language in which these axioms will be formulated.