The Foundation Axiom and Elementary Self-Embeddings of the Universe
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Does the Category of Multisets Require a Larger Universe Than
Pure Mathematical Sciences, Vol. 2, 2013, no. 3, 133 - 146 HIKARI Ltd, www.m-hikari.com Does the Category of Multisets Require a Larger Universe than that of the Category of Sets? Dasharath Singh (Former affiliation: Indian Institute of Technology Bombay) Department of Mathematics, Ahmadu Bello University, Zaria, Nigeria [email protected] Ahmed Ibrahim Isah Department of Mathematics Ahmadu Bello University, Zaria, Nigeria [email protected] Alhaji Jibril Alkali Department of Mathematics Ahmadu Bello University, Zaria, Nigeria [email protected] Copyright © 2013 Dasharath Singh et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In section 1, the concept of a category is briefly described. In section 2, it is elaborated how the concept of category is naturally intertwined with the existence of a universe of discourse much larger than what is otherwise sufficient for a large part of mathematics. It is also remarked that the extended universe for the category of sets is adequate for the category of multisets as well. In section 3, fundamentals required for adequately describing a category are extended to defining a multiset category, and some of its distinctive features are outlined. Mathematics Subject Classification: 18A05, 18A20, 18B99 134 Dasharath Singh et al. Keywords: Category, Universe, Multiset Category, objects. 1. Introduction to categories The history of science and that of mathematics, in particular, records that at times, a by- product may turn out to be of greater significance than the main objective of a research. -
Set Theory, by Thomas Jech, Academic Press, New York, 1978, Xii + 621 Pp., '$53.00
BOOK REVIEWS 775 BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 3, Number 1, July 1980 © 1980 American Mathematical Society 0002-9904/80/0000-0 319/$01.75 Set theory, by Thomas Jech, Academic Press, New York, 1978, xii + 621 pp., '$53.00. "General set theory is pretty trivial stuff really" (Halmos; see [H, p. vi]). At least, with the hindsight afforded by Cantor, Zermelo, and others, it is pretty trivial to do the following. First, write down a list of axioms about sets and membership, enunciating some "obviously true" set-theoretic principles; the most popular Hst today is called ZFC (the Zermelo-Fraenkel axioms with the axiom of Choice). Next, explain how, from ZFC, one may derive all of conventional mathematics, including the general theory of transfinite cardi nals and ordinals. This "trivial" part of set theory is well covered in standard texts, such as [E] or [H]. Jech's book is an introduction to the "nontrivial" part. Now, nontrivial set theory may be roughly divided into two general areas. The first area, classical set theory, is a direct outgrowth of Cantor's work. Cantor set down the basic properties of cardinal numbers. In particular, he showed that if K is a cardinal number, then 2", or exp(/c), is a cardinal strictly larger than K (if A is a set of size K, 2* is the cardinality of the family of all subsets of A). Now starting with a cardinal K, we may form larger cardinals exp(ic), exp2(ic) = exp(exp(fc)), exp3(ic) = exp(exp2(ic)), and in fact this may be continued through the transfinite to form expa(»c) for every ordinal number a. -
Calibrating Determinacy Strength in Levels of the Borel Hierarchy
CALIBRATING DETERMINACY STRENGTH IN LEVELS OF THE BOREL HIERARCHY SHERWOOD J. HACHTMAN Abstract. We analyze the set-theoretic strength of determinacy for levels of the Borel 0 hierarchy of the form Σ1+α+3, for α < !1. Well-known results of H. Friedman and D.A. Martin have shown this determinacy to require α+1 iterations of the Power Set Axiom, but we ask what additional ambient set theory is strictly necessary. To this end, we isolate a family of Π1-reflection principles, Π1-RAPα, whose consistency strength corresponds 0 CK exactly to that of Σ1+α+3-Determinacy, for α < !1 . This yields a characterization of the levels of L by or at which winning strategies in these games must be constructed. When α = 0, we have the following concise result: the least θ so that all winning strategies 0 in Σ4 games belong to Lθ+1 is the least so that Lθ j= \P(!) exists + all wellfounded trees are ranked". x1. Introduction. Given a set A ⊆ !! of sequences of natural numbers, consider a game, G(A), where two players, I and II, take turns picking elements of a sequence hx0; x1; x2;::: i of naturals. Player I wins the game if the sequence obtained belongs to A; otherwise, II wins. For a collection Γ of subsets of !!, Γ determinacy, which we abbreviate Γ-DET, is the statement that for every A 2 Γ, one of the players has a winning strategy in G(A). It is a much-studied phenomenon that Γ -DET has mathematical strength: the bigger the pointclass Γ, the stronger the theory required to prove Γ -DET. -
Cantor, God, and Inconsistent Multiplicities*
STUDIES IN LOGIC, GRAMMAR AND RHETORIC 44 (57) 2016 DOI: 10.1515/slgr-2016-0008 Aaron R. Thomas-Bolduc University of Calgary CANTOR, GOD, AND INCONSISTENT MULTIPLICITIES* Abstract. The importance of Georg Cantor’s religious convictions is often ne- glected in discussions of his mathematics and metaphysics. Herein I argue, pace Jan´e(1995), that due to the importance of Christianity to Cantor, he would have never thought of absolutely infinite collections/inconsistent multiplicities, as being merely potential, or as being purely mathematical entities. I begin by considering and rejecting two arguments due to Ignacio Jan´e based on letters to Hilbert and the generating principles for ordinals, respectively, showing that my reading of Cantor is consistent with that evidence. I then argue that evidence from Cantor’s later writings shows that he was still very religious later in his career, and thus would not have given up on the reality of the absolute, as that would imply an imperfection on the part of God. The theological acceptance of his set theory was very important to Can- tor. Despite this, the influence of theology on his conception of absolutely infinite collections, or inconsistent multiplicities, is often ignored in contem- porary literature.1 I will be arguing that due in part to his religious convic- tions, and despite an apparent tension between his earlier and later writings, Cantor would never have considered inconsistent multiplicities (similar to what we now call proper classes) as completed in a mathematical sense, though they are completed in Intellectus Divino. Before delving into the issue of the actuality or otherwise of certain infinite collections, it will be informative to give an explanation of Cantor’s terminology, as well a sketch of Cantor’s relationship with religion and reli- gious figures. -
Weak Measure Extension Axioms * 1 Introduction
Weak Measure Extension Axioms Joan E Hart Union College Schenectady NY USA and Kenneth Kunen University of Wisconsin Madison WI USA hartjunionedu and kunenmathwiscedu February Abstract We consider axioms asserting that Leb esgue measure on the real line may b e extended to measure a few new nonmeasurable sets Strong versions of such axioms such as realvalued measurabilityin volve large cardinals but weak versions do not We discuss weak versions which are sucienttoprovevarious combinatorial results such as the nonexistence of Ramsey ultralters the existence of ccc spaces whose pro duct is not ccc and the existence of S and L spaces We also prove an absoluteness theorem stating that assuming our ax iom every sentence of an appropriate logical form which is forced to b e true in the random real extension of the universe is in fact already true Intro duction In this pap er a measure on a set X is a countably additive measure whose domain the measurable sets is some algebra of subsets of X The authors were supp orted by NSF Grant CCR The rst author is grateful for supp ort from the University of Wisconsin during the preparation of this pap er Both authors are grateful to the referee for a numb er of useful comments INTRODUCTION We are primarily interested in nite measures although most of our results extend to nite measures in the obvious way By the Axiom of Choice whichwealways assume there are subsets of which are not Leb esgue measurable In an attempt to measure them it is reasonable to p ostulate measure extension axioms -
Faculty Document 2436 Madison 7 October 2013
University of Wisconsin Faculty Document 2436 Madison 7 October 2013 MEMORIAL RESOLUTION OF THE FACULTY OF THE UNIVERSITY OF WISCONSIN-MADISON ON THE DEATH OF PROFESSOR EMERITA MARY ELLEN RUDIN Mary Ellen Rudin, Hilldale professor emerita of mathematics, died peacefully at home in Madison on March 18, 2013. Mary Ellen was born in Hillsboro, Texas, on December 7, 1924. She spent most of her pre-college years in Leakey, another small Texas town. In 1941, she went off to college at the University of Texas in Austin, and she met the noted topologist R.L. Moore on her first day on campus, since he was assisting in advising incoming students. He recognized her talent immediately and steered her into the math program, which she successfully completed in 1944, and she then went directly into graduate school at Austin, receiving her PhD under Moore’s supervision in 1949. After teaching at Duke University and the University of Rochester, she joined the faculty of the University of Wisconsin-Madison as a lecturer in 1959 when her husband Walter came here. Walter Rudin died on May 20, 2010. Mary Ellen became a full professor in 1971 and professor emerita in 1991. She also held two named chairs: she was appointed Grace Chisholm Young Professor in 1981 and Hilldale Professor in 1988. She received numerous honors throughout her career. She was a fellow of the American Academy of Arts and Sciences and was a member of the Hungarian Academy of Sciences, and she received honorary doctor of science degrees from the University of North Carolina, the University of the South, Kenyon College, and Cedar Crest College. -
Sets - June 24 Chapter 1 - Sets and Functions
Math 300 - Introduction to Mathematical reasoning Summer 2013 Lecture 1: Sets - June 24 Chapter 1 - Sets and Functions Definition and Examples Sets are a \well-defined” collection of objects. We shall not say what we mean by \well-defined” in this course. At an introductory level, it suffices to think of a Set as a collection of objects. Some examples of sets are - • Collection of students taking Math 300, • Collection of all possible speeds that your car can attain, • Collection of all matrices with real entries (Math 308). Examples We mention some mathematical examples which we shall use throughout the course. Keep these examples in mind. • N - the set of natural numbers. Here, N = f1; 2; 3;:::; g • Z - the set of integers. Here, Z = f;:::; −3; −2; −1; 0; 1; 2; 3;:::; g • Q - the set of rational numbers. The set of rational numbers can described as fractions, where the numerator and denominator of the fraction are both integers and the denominator is not equal to zero. • R - the set of real numbers. • R+ - the set of positive real numbers. The objects of the set are called elements, members or points of the set. Usually a set is denoted by a capital letter. The elements of the set are denoted by lowercase letters. The symbol 2 denotes the phrase \belongs to". Let A be a set and let x be an element of the set A. The statement x 2 A should be read as x belongs to the set A. For example, the statement \1 2 N" should be read as \1 belongs to the set of natural numbers". -
Review of Set-Theoretic Notations and Terminology A.J
Math 347 Review of Set-theoretic Notations and Terminology A.J. Hildebrand Review of Set-theoretic Notations and Terminology • Sets: A set is an unordered collection of objects, called the elements of the set. The standard notation for a set is the brace notation f::: g, with the elements of the set listed inside the pair of braces. • Set builder notation: Notation of the form f::: : ::: g, where the colon (:) has the meaning of \such that", the dots to the left of the colon indicate the generic form of the element in the set and the dots to 2 the right of the colon denote any constraints on the element. E.g., fx 2 R : x < 4g stands for \the set of 2 all x 2 R such that x < 4"; this is the same as the interval (−2; 2). Other examples of this notation are f2k : k 2 Zg (set of all even integers), fx : x 2 A and x 2 Bg (intersection of A and B, A \ B). 2 Note: Instead of a colon (:), one can also use a vertical bar (j) as a separator; e.g., fx 2 R j x < 4g. Both notations are equally acceptable. • Equality of sets: Two sets are equal if they contain the same elements. E.g., the sets f3; 4; 7g, f4; 3; 7g, f3; 3; 4; 7g are all equal since they contain the same three elements, namely 3, 4, 7. • Some special sets: • Empty set: ; (or fg) • \Double bar" sets: 1 ∗ Natural numbers: N = f1; 2;::: g (note that 0 is not an element of N) ∗ Integers: Z = f:::; −2; −1; 0; 1; 2;::: g ∗ Rational numbers: Q = fp=q : p; q 2 Z; q 6= 0g ∗ Real numbers: R • Intervals: ∗ Open interval: (a; b) = fx 2 R : a < x < bg ∗ Closed interval: [a; b] = fx 2 R : a ≤ x ≤ bg ∗ Half-open interval: [a; b) = fx 2 R : a ≤ x < bg • Universe: U (\universe", a \superset" of which all sets under consideration are assumed to be a subset). -
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. -
Set-Theoretic Foundations
Contemporary Mathematics Volume 690, 2017 http://dx.doi.org/10.1090/conm/690/13872 Set-theoretic foundations Penelope Maddy Contents 1. Foundational uses of set theory 2. Foundational uses of category theory 3. The multiverse 4. Inconclusive conclusion References It’s more or less standard orthodoxy these days that set theory – ZFC, extended by large cardinals – provides a foundation for classical mathematics. Oddly enough, it’s less clear what ‘providing a foundation’ comes to. Still, there are those who argue strenuously that category theory would do this job better than set theory does, or even that set theory can’t do it at all, and that category theory can. There are also those who insist that set theory should be understood, not as the study of a single universe, V, purportedly described by ZFC + LCs, but as the study of a so-called ‘multiverse’ of set-theoretic universes – while retaining its foundational role. I won’t pretend to sort out all these complex and contentious matters, but I do hope to compile a few relevant observations that might help bring illumination somewhat closer to hand. 1. Foundational uses of set theory The most common characterization of set theory’s foundational role, the char- acterization found in textbooks, is illustrated in the opening sentences of Kunen’s classic book on forcing: Set theory is the foundation of mathematics. All mathematical concepts are defined in terms of the primitive notions of set and membership. In axiomatic set theory we formulate . axioms 2010 Mathematics Subject Classification. Primary 03A05; Secondary 00A30, 03Exx, 03B30, 18A15. -
MAGIC Set Theory Lecture 6
MAGIC Set theory lecture 6 David Aspero´ University of East Anglia 15 November 2018 Recall: We defined (V : ↵ Ord) by recursion on Ord: ↵ 2 V = • 0 ; V = (V ) • ↵+1 P ↵ V = V : β<δ if δ is a limit ordinal. • δ { β } S (V : ↵ Ord) is called the cumulative hierarchy. ↵ 2 We saw: Proposition V↵ is transitive for every ordinal ↵. Proposition For all ↵<β, V V . ↵ ✓ β Definition For every x V , let rank(x) be the first ↵ such that 2 ↵ Ord ↵ x V . 2 2 ↵+1 S Definition For every set x, the transitive closure of x, denoted by TC(x), is X : n <! where { n } X = x S • 0 X = X • n+1 n So TC(x)=Sx x x x ... [ [ [ [ S SS SSS [Exercise: TC(x) is the –least transitive set y such that x y. ✓ ✓ In other words, TC(x)= y : y transitive, x y .] { ✓ } T Definition V denotes the class of all sets; that is, V = x : x = x . { } Definition WF = V : ↵ Ord : The class of all x such that x V for { ↵ 2 } 2 ↵ some ordinal ↵. S Note: WF is a transitive class: y x V implies y V since 2 2 ↵ 2 ↵ V↵ is transitive. Proposition If x WF, then there is some ↵ Ord such that x V . ✓ 2 ✓ ↵ Proof. Define the function F(y)=min γ y V . By the assumption { | 2 γ} on x, F is a well-defined function there. By Replacement γ y x : γ = F(y) is a set, and by Union it has a { |9 2 } supremum ↵. -
A Multiverse Perspective in Mathematics and Set Theory: Does Every Mathematical Statement Have a Definite Truth Value?
The universe view The multiverse view Dream Solution of CH is unattainable Further topics Multiverse Mathematics A multiverse perspective in mathematics and set theory: does every mathematical statement have a definite truth value? Joel David Hamkins The City University of New York Mathematics, Philosophy, Computer Science College of Staten Island of CUNY The CUNY Graduate Center Meta-Meta Workshop: Metamathematics and Metaphysics Fudan University, Shanghai June 15, 2013 Meta-Meta Workshop, Shanghai June 15, 2013 Joel David Hamkins, New York The universe view The multiverse view Dream Solution of CH is unattainable Further topics Multiverse Mathematics The theme The theme of this talk is the question: Does every mathematical problem have a definite answer? I shall be particularly interested in this question as it arises in the case of set theory: Does every set-theoretic assertion have a definite truth value? Meta-Meta Workshop, Shanghai June 15, 2013 Joel David Hamkins, New York The universe view The multiverse view Dream Solution of CH is unattainable Further topics Multiverse Mathematics Set theory as Ontological Foundation A traditional view in set theory is that it serves as an ontological foundation for the rest of mathematics, in the sense that other abstract mathematical objects can be construed fundamentally as sets. On this view, mathematical objects—functions, real numbers, spaces—are sets. Being precise in mathematics amounts to specifying an object in set theory. In this way, the set-theoretic universe becomes the realm of all mathematics. Having a common foundation was important for the unity of mathematics. A weaker position remains compatible with structuralism.