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1 Elementary Set Theory
1 Elementary Set Theory Notation: fg enclose a set. f1; 2; 3g = f3; 2; 2; 1; 3g because a set is not defined by order or multiplicity. f0; 2; 4;:::g = fxjx is an even natural numberg because two ways of writing a set are equivalent. ; is the empty set. x 2 A denotes x is an element of A. N = f0; 1; 2;:::g are the natural numbers. Z = f:::; −2; −1; 0; 1; 2;:::g are the integers. m Q = f n jm; n 2 Z and n 6= 0g are the rational numbers. R are the real numbers. Axiom 1.1. Axiom of Extensionality Let A; B be sets. If (8x)x 2 A iff x 2 B then A = B. Definition 1.1 (Subset). Let A; B be sets. Then A is a subset of B, written A ⊆ B iff (8x) if x 2 A then x 2 B. Theorem 1.1. If A ⊆ B and B ⊆ A then A = B. Proof. Let x be arbitrary. Because A ⊆ B if x 2 A then x 2 B Because B ⊆ A if x 2 B then x 2 A Hence, x 2 A iff x 2 B, thus A = B. Definition 1.2 (Union). Let A; B be sets. The Union A [ B of A and B is defined by x 2 A [ B if x 2 A or x 2 B. Theorem 1.2. A [ (B [ C) = (A [ B) [ C Proof. Let x be arbitrary. x 2 A [ (B [ C) iff x 2 A or x 2 B [ C iff x 2 A or (x 2 B or x 2 C) iff x 2 A or x 2 B or x 2 C iff (x 2 A or x 2 B) or x 2 C iff x 2 A [ B or x 2 C iff x 2 (A [ B) [ C Definition 1.3 (Intersection). -
UNIVERSITY of CALIFORNIA, IRVINE DISSERTATION Submitted
UNIVERSITY OF CALIFORNIA, IRVINE THREE ESSAYS ON THE FOUNDATIONS OF SCIENCE DISSERTATION submitted in partial satisfaction of the requirements for the degree of DOCTOR OF PHILOSOPHY in Social Science – Mathematical Behavioral Sciences by Rolf Henry Johansson Dissertation Committee: Professor Louis Narens, Chair Professor Donald Saari Professor Kent Johnson 2014 © 2014 Rolf Henry Johansson DEDICATION For my mother Lalla and my father Kurt for their continuing support. ii TABLE OF CONTENTS Page ACKNOWLEDGEMENTS iv CURRICULUM VITAE v ABTRACT OF THE DISSERTATION viii ESSAY 1: THE UNIQUENESS PROBLEM FOR FINITE SEMIORDERS Abstract 1 1. Introduction 2 2. Preliminaries 8 3. Representation Problems for Semiorders 17 4. Theorems 24 5. Discussion 37 Appendix 40 References 45 ESSAY 2: TROUBLES WITH CONVENTION T Abstract 48 1. Introduction: Convention T and Natural Languages 50 2. Hintikka‟s Counterexample 55 3. Hintikka-Type Counterexamples 59 4. Proposed Solutions 63 4.1 On ambiguity versus context-sensitivity 64 4.2 Formalizing the metalanguage 70 4.3 Paraprasing the conditional 73 4.4 Dispensing with disquotation 76 5. Discussion 78 References 82 ESSAY 3: ELEMENTARY FORMULAS FOR THE nTH PRIME AND FOR THE NUMBER OF PRIMES UP TO A GIVEN LIMIT Abstract 84 1. Introduction 86 2. A Brief History of Prime Representing Functions 89 3. Formulas Based on Wilson‟s Theorem 93 4. Formulas Based on the Sieve of Eratosthenes 96 5. Elementary Formulas for Primes 100 References 107 iii ACKNOWLEDGEMENTS I would like to express my deepest gratitude to my committee chair, Louis Narens, for many stimulating conversations over the years and for guidance throughout my graduate education. -
Intransitivity, Utility, and the Aggregation of Preference Patterns
ECONOMETRICA VOLUME 22 January, 1954 NUMBER 1 INTRANSITIVITY, UTILITY, AND THE AGGREGATION OF PREFERENCE PATTERNS BY KENNETH 0. MAY1 During the first half of the twentieth century, utility theory has been subjected to considerable criticism and refinement. The present paper is concerned with cer- tain experimental evidence and theoretical arguments suggesting that utility theory, whether cardinal or ordinal, cannot reflect, even to an approximation, the existing preferences of individuals in many economic situations. The argument is based on the necessity of transitivity for utility, observed circularities of prefer- ences, and a theoretical framework that predicts circularities in the presence of preferences based on numerous criteria. THEORIES of choice may be built to describe behavior as it is or as it "ought to be." In the belief that the former should precede the latter, this paper is con- cerned solely with descriptive theory and, in particular, with the intransitivity of preferences. The first section indicates that transitivity is not necessary to either the usual axiomatic characterization of the preference relation or to plau- sible empirical definitions. The second section shows the necessity of transitivity for utility theory. The third section points to various examples of the violation of transitivity and suggests how experiments may be designed to bring out the conditions under which transitivity does not hold. The final section shows how intransitivity is a natural result of the necessity of choosing among alternatives according to conflicting criteria. It appears from the discussion that neither cardinal nor ordinal utility is adequate to deal with choices under all conditions, and that we need a more general theory based on empirical knowledge of prefer- ence patterns. -
Natural Numerosities of Sets of Tuples
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 367, Number 1, January 2015, Pages 275–292 S 0002-9947(2014)06136-9 Article electronically published on July 2, 2014 NATURAL NUMEROSITIES OF SETS OF TUPLES MARCO FORTI AND GIUSEPPE MORANA ROCCASALVO Abstract. We consider a notion of “numerosity” for sets of tuples of natural numbers that satisfies the five common notions of Euclid’s Elements, so it can agree with cardinality only for finite sets. By suitably axiomatizing such a no- tion, we show that, contrasting to cardinal arithmetic, the natural “Cantorian” definitions of order relation and arithmetical operations provide a very good algebraic structure. In fact, numerosities can be taken as the non-negative part of a discretely ordered ring, namely the quotient of a formal power series ring modulo a suitable (“gauge”) ideal. In particular, special numerosities, called “natural”, can be identified with the semiring of hypernatural numbers of appropriate ultrapowers of N. Introduction In this paper we consider a notion of “equinumerosity” on sets of tuples of natural numbers, i.e. an equivalence relation, finer than equicardinality, that satisfies the celebrated five Euclidean common notions about magnitudines ([8]), including the principle that “the whole is greater than the part”. This notion preserves the basic properties of equipotency for finite sets. In particular, the natural Cantorian definitions endow the corresponding numerosities with a structure of a discretely ordered semiring, where 0 is the size of the emptyset, 1 is the size of every singleton, and greater numerosity corresponds to supersets. This idea of “Euclidean” numerosity has been recently investigated by V. -
Kenneth Arrow's Contributions to Social
Kenneth Arrow’s Contributions to Social Choice Theory Eric Maskin Harvard University and Higher School of Economics Kenneth Arrow created the modern field of social choice theory, the study of how society should make collection decisions on the basis of individuals’ preferences. There had been scattered contributions to this field before Arrow, going back (at least) to Jean-Charles Borda (1781) and the Marquis de Condorcet (1785). But earlier writers all focused on elections and voting, more specifically on the properties of particular voting rules (I am ignoring here the large literature on utilitarianism – following Jeremy Bentham 1789 – which I touch on below). Arrow’s approach, by contrast, encompassed not only all possible voting rules (with some qualifications, discussed below) but also the issue of aggregating individuals’ preferences or welfares, more generally. Arrow’s first social choice paper was “A Difficulty in the Concept of Social Welfare” (Arrow 1950), which he then expanded into the celebrated monograph Social Choice and Individual Values (Arrow 1951). In his formulation, there is a society consisting of n individuals, indexed in1,..., , and a set of social alternatives A (the different possible options from which society must choose). The interpretation of this set-up depends on the context. For example, imagine a town that is considering whether or not to build a bridge across the local river. Here, “society” comprises the citizens of the town, and A consists of two options: “build the bridge” or “don’t build it.” In the case of pure distribution, where there is, say, a jug of milk and a plate of cookies to be divided among a group of children, the children are the society and A 1 includes the different ways the milk and cookies could be allocated to them. -
On Hybric Order Dimensions 1 Introduction
On hybric order dimensions Athanasios Andrikopoulos Abstract The notion of interval order was introduced by Norbert Wiener [28] in order to clarify the relation between the notion of an instant of time and that of a period of time. This was a problem on which Bertrand Russell [18] worked at the time. Interval orders play an impor- tant role in many areas of pure and applied mathematics, graph theory, computer science and engineering. Special cases of interval order are the semiorder and linear order. All of these notions are especially important in the study of linear-interval and linear-semiorder dimension of a binary relation. This kind of dimension, which we call hybric order dimension, gives a common generalization of linear order and interval order (semiorder) dimension and is ar- guably the most important measure of ordered set complexity. In this paper, we give three main results of the theory of hybric order dimension. More precicely, we obtain necessary and sufficient conditions for a binary relation to have an interval order (resp. linear-interval order, linear-simiorder) extension as well as a realizer of interval orders (resp. linear-interval orders, linear-simiorders). We also obtain a characterization of the interval order (resp. linear-interval order, linear-simiorder) dimension. Since the hybric order dimension of a binary relation is less than its (linear) order dimension, these results will be able to improve known results in graph theory and computer science by finding more efficient algorithms. Keywords: Graph, Extension theorems, Linear-interval order, Linear-semiorder, Realizer, Dimension. 1 Introduction Zeno’s paradox posed for first time the question of whether time should be represented by a discrete or a continuous variable. -
Reverse Mathematics, Trichotomy, and Dichotomy
Reverse mathematics, trichotomy, and dichotomy Fran¸coisG. Dorais, Jeffry Hirst, and Paul Shafer Department of Mathematical Sciences Appalachian State University, Boone, NC 28608 February 20, 2012 Abstract Using the techniques of reverse mathematics, we analyze the log- ical strength of statements similar to trichotomy and dichotomy for sequences of reals. Capitalizing on the connection between sequential statements and constructivity, we find computable restrictions of the statements for sequences and constructive restrictions of the original principles. 1 Axiom systems and encoding reals We will examine several statements about real numbers and sequences of real numbers in the framework of reverse mathematics and in some formalizations of weak constructive analysis. From reverse mathematics, we will concentrate on the axiom systems RCA0, WKL0, and ACA0, which are described in detail by Simpson [8]. Very roughly, RCA0 is a subsystem of second order arith- metic incorporating ordered semi-ring axioms, a restricted form of induction, 0 and comprehension for ∆1 definable sets. The axiom system WKL0 appends K¨onig'stree lemma restricted to 0{1 trees, and the system ACA0 appends a comprehension scheme for arithmetically definable sets. ACA0 is strictly stronger than WKL0, and WKL0 is strictly stronger than RCA0. We will also make use of several formalizations of subsystems of con- structive analysis, all variations of E-HA!, which is intuitionistic arithmetic 1 (Heyting arithmetic) in all finite types with an extensionality scheme. Unlike the reverse mathematics systems which use classical logic, these constructive systems omit the law of the excluded middle. For example, we will use ex- tensions of E\{HA!, a form of Heyting arithmetic with primitive recursion restricted to type 0 objects, and induction restricted to quantifier free for- mulas. -
The Dimension of Semiorders a Partially Ordered Set Or Poset Is An
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF COMBINATORIAL THEORY, Series A25, 5&61 (1978) The Dimension of Semiorders I. RABINOVITCH * Department of Statistics, Uniuersity of Waterloo, Waterfoo, Ontario Communicated by the Managing Editors Received December 26. 1974 The structure of semiorders and interval orders is investigated and various characterizations are given. These ideas are then applied to prove that the dimen- sion of a semiorder is at most 3, to characterize semiorders of dimension 3 and height 2, and to prove Hiraguchi’s Theorem. 1. PRELIMINARIES A partially ordered set or poset is an ordered pair (X, P) where P is an irreflexive, asymmetric, and transitive relation on X. In this paper X will always be a finite set. If Y C X then (Y, P Iy) is called the restriction of (X, P) to Y where P Iy = P n (Y x Y). If P and Q are both partial orders on the same set X with P C Q then Q is called an extension of P and P is called a subrelation of Q. Two elements x and y of X are incomparable if neither (x, I?) E P nor ( y, X) E P. This is written x izr y. An antichain of (X, P) is a subset of pairwise incomparable elements. A chain is a subset with no two elements incomparable. The height of (X, P), written h(P), is one less than the number of elements in a chain of maximum size. -
Some Set Theory We Should Know Cardinality and Cardinal Numbers
SOME SET THEORY WE SHOULD KNOW CARDINALITY AND CARDINAL NUMBERS De¯nition. Two sets A and B are said to have the same cardinality, and we write jAj = jBj, if there exists a one-to-one onto function f : A ! B. We also say jAj · jBj if there exists a one-to-one (but not necessarily onto) function f : A ! B. Then the SchrÄoder-BernsteinTheorem says: jAj · jBj and jBj · jAj implies jAj = jBj: SchrÄoder-BernsteinTheorem. If there are one-to-one maps f : A ! B and g : B ! A, then jAj = jBj. A set is called countable if it is either ¯nite or has the same cardinality as the set N of positive integers. Theorem ST1. (a) A countable union of countable sets is countable; (b) If A1;A2; :::; An are countable, so is ¦i·nAi; (c) If A is countable, so is the set of all ¯nite subsets of A, as well as the set of all ¯nite sequences of elements of A; (d) The set Q of all rational numbers is countable. Theorem ST2. The following sets have the same cardinality as the set R of real numbers: (a) The set P(N) of all subsets of the natural numbers N; (b) The set of all functions f : N ! f0; 1g; (c) The set of all in¯nite sequences of 0's and 1's; (d) The set of all in¯nite sequences of real numbers. The cardinality of N (and any countable in¯nite set) is denoted by @0. @1 denotes the next in¯nite cardinal, @2 the next, etc. -
Logic and Proof Release 3.18.4
Logic and Proof Release 3.18.4 Jeremy Avigad, Robert Y. Lewis, and Floris van Doorn Sep 10, 2021 CONTENTS 1 Introduction 1 1.1 Mathematical Proof ............................................ 1 1.2 Symbolic Logic .............................................. 2 1.3 Interactive Theorem Proving ....................................... 4 1.4 The Semantic Point of View ....................................... 5 1.5 Goals Summarized ............................................ 6 1.6 About this Textbook ........................................... 6 2 Propositional Logic 7 2.1 A Puzzle ................................................. 7 2.2 A Solution ................................................ 7 2.3 Rules of Inference ............................................ 8 2.4 The Language of Propositional Logic ................................... 15 2.5 Exercises ................................................. 16 3 Natural Deduction for Propositional Logic 17 3.1 Derivations in Natural Deduction ..................................... 17 3.2 Examples ................................................. 19 3.3 Forward and Backward Reasoning .................................... 20 3.4 Reasoning by Cases ............................................ 22 3.5 Some Logical Identities .......................................... 23 3.6 Exercises ................................................. 24 4 Propositional Logic in Lean 25 4.1 Expressions for Propositions and Proofs ................................. 25 4.2 More commands ............................................ -
Intransitivity in Theory and in the Real World
Entropy 2015, 17, 4364-4412; doi:10.3390/e17064364 OPEN ACCESS entropy ISSN 1099-4300 www.mdpi.com/journal/entropy Article Intransitivity in Theory and in the Real World Alexander Y. Klimenko School of Mechanical and Mining Engineering, The University of Queensland, Brisbane QLD 4072, Australia; E-Mail: [email protected] Academic Editor: Ali E. Abbas Received: 30 March 2015 / Accepted: 15 June 2015 / Published: 19 June 2015 Abstract: This work considers reasons for and implications of discarding the assumption of transitivity—the fundamental postulate in the utility theory of von Neumann and Morgenstern, the adiabatic accessibility principle of Caratheodory and most other theories related to preferences or competition. The examples of intransitivity are drawn from different fields, such as law, biology and economics. This work is intended as a common platform that allows us to discuss intransitivity in the context of different disciplines. The basic concepts and terms that are needed for consistent treatment of intransitivity in various applications are presented and analysed in a unified manner. The analysis points out conditions that necessitate appearance of intransitivity, such as multiplicity of preference criteria and imperfect (i.e., approximate) discrimination of different cases. The present work observes that with increasing presence and strength of intransitivity, thermodynamics gradually fades away leaving space for more general kinetic considerations. Intransitivity in competitive systems is linked to complex phenomena that would be difficult or impossible to explain on the basis of transitive assumptions. Human preferences that seem irrational from the perspective of the conventional utility theory, become perfectly logical in the intransitive and relativistic framework suggested here. -
W. E. Armstrong on Intransitivity of Indifference and His Influence In
W. E. Armstrong on Intransitivity of Indifference and his Influence in Choice Theory Gustavo O. Aggio∗ Trabalho submetido a` “Area´ 1: Historia´ do Pensamento Economicoˆ e Metodologia” Abstract In this paper we reconstruct the trajectory of a branch of choice theory in that the relation of indif- ference is intransitive. This trajectory begins with the work of W.E. Armstrong, a British economist that worked in this theme from 1939 to 1958. In 1956 the mathematician R.D. Luce formalized the Armstrong’s idea and this formalization influenced others authors in several areas like economics, psychology, operational research, and philosophy, among others. In this reconstruction we also ana- lyzed and synthesized the data of published papers with references to Armstrong’s papers and Luce paper. Keywords: Intransitivity; W.E. Armstrong; R.D. Luce; Semiorder. JEL codes: B21, B16, B31. Resumo: Neste artigo nos´ reconstru´ımos a trajetoria´ de um ramo da teoria da escolha na qual a relac¸ao˜ de indiferenc¸a e´ intransitiva. Esta trajetoria´ comec¸a com o trabalho de W.E. Armstrong, um economista britanicoˆ que trabalhou neste tema entre 1939 e 1958. Em 1956 o matematico´ R.D. Luce formalizou a ideia de Armstrong e esta formalizac¸ao˜ influenciou outros autores em varias´ areas como econo- mia, psicologia, pesquisa operacional e filosofia, entre outras. Nesta reconstruc¸ao˜ nos´ tambem´ analisamos e sintetizamos os dados sobre artigos publicados com referencias aos artigos de Arm- strong e de Luce. Palavras chave: Intransitividade; W.E. Armstrong; R.D.Luce; Semiorder. ∗Department of Economic Theory - Institute of Economics - University of Campinas.