2.3 Cantor's Infinite Numbers
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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. -
Mathematics 144 Set Theory Fall 2012 Version
MATHEMATICS 144 SET THEORY FALL 2012 VERSION Table of Contents I. General considerations.……………………………………………………………………………………………………….1 1. Overview of the course…………………………………………………………………………………………………1 2. Historical background and motivation………………………………………………………….………………4 3. Selected problems………………………………………………………………………………………………………13 I I. Basic concepts. ………………………………………………………………………………………………………………….15 1. Topics from logic…………………………………………………………………………………………………………16 2. Notation and first steps………………………………………………………………………………………………26 3. Simple examples…………………………………………………………………………………………………………30 I I I. Constructions in set theory.………………………………………………………………………………..……….34 1. Boolean algebra operations.……………………………………………………………………………………….34 2. Ordered pairs and Cartesian products……………………………………………………………………… ….40 3. Larger constructions………………………………………………………………………………………………..….42 4. A convenient assumption………………………………………………………………………………………… ….45 I V. Relations and functions ……………………………………………………………………………………………….49 1.Binary relations………………………………………………………………………………………………………… ….49 2. Partial and linear orderings……………………………..………………………………………………… ………… 56 3. Functions…………………………………………………………………………………………………………… ….…….. 61 4. Composite and inverse function.…………………………………………………………………………… …….. 70 5. Constructions involving functions ………………………………………………………………………… ……… 77 6. Order types……………………………………………………………………………………………………… …………… 80 i V. Number systems and set theory …………………………………………………………………………………. 84 1. The Natural Numbers and Integers…………………………………………………………………………….83 2. Finite induction -
641 1. P. Erdös and A. H. Stone, Some Remarks on Almost Periodic
1946] DENSITY CHARACTERS 641 BIBLIOGRAPHY 1. P. Erdös and A. H. Stone, Some remarks on almost periodic transformations, Bull. Amer. Math. Soc. vol. 51 (1945) pp. 126-130. 2. W. H. Gottschalk, Powers of homeomorphisms with almost periodic propertiesf Bull. Amer. Math. Soc. vol. 50 (1944) pp. 222-227. UNIVERSITY OF PENNSYLVANIA AND UNIVERSITY OF VIRGINIA A REMARK ON DENSITY CHARACTERS EDWIN HEWITT1 Let X be an arbitrary topological space satisfying the TVseparation axiom [l, Chap. 1, §4, p. 58].2 We recall the following definition [3, p. 329]. DEFINITION 1. The least cardinal number of a dense subset of the space X is said to be the density character of X. It is denoted by the symbol %{X). We denote the cardinal number of a set A by | A |. Pospisil has pointed out [4] that if X is a Hausdorff space, then (1) |X| g 22SW. This inequality is easily established. Let D be a dense subset of the Hausdorff space X such that \D\ =S(-X'). For an arbitrary point pÇ^X and an arbitrary complete neighborhood system Vp at p, let Vp be the family of all sets UC\D, where U^VP. Thus to every point of X, a certain family of subsets of D is assigned. Since X is a Haus dorff space, VpT^Vq whenever p j*£q, and the correspondence assigning each point p to the family <DP is one-to-one. Since X is in one-to-one correspondence with a sub-hierarchy of the hierarchy of all families of subsets of D, the inequality (1) follows. -
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. -
Cardinal Invariants Concerning Functions Whose Sum Is Almost Continuous Krzysztof Ciesielski West Virginia University, [email protected]
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by The Research Repository @ WVU (West Virginia University) Faculty Scholarship 1995 Cardinal Invariants Concerning Functions Whose Sum Is Almost Continuous Krzysztof Ciesielski West Virginia University, [email protected] Follow this and additional works at: https://researchrepository.wvu.edu/faculty_publications Part of the Mathematics Commons Digital Commons Citation Ciesielski, Krzysztof, "Cardinal Invariants Concerning Functions Whose Sum Is Almost Continuous" (1995). Faculty Scholarship. 822. https://researchrepository.wvu.edu/faculty_publications/822 This Article is brought to you for free and open access by The Research Repository @ WVU. It has been accepted for inclusion in Faculty Scholarship by an authorized administrator of The Research Repository @ WVU. For more information, please contact [email protected]. Cardinal invariants concerning functions whose sum is almost continuous. Krzysztof Ciesielski1, Department of Mathematics, West Virginia University, Mor- gantown, WV 26506-6310 ([email protected]) Arnold W. Miller1, York University, Department of Mathematics, North York, Ontario M3J 1P3, Canada, Permanent address: University of Wisconsin-Madison, Department of Mathematics, Van Vleck Hall, 480 Lincoln Drive, Madison, Wis- consin 53706-1388, USA ([email protected]) Abstract Let A stand for the class of all almost continuous functions from R to R and let A(A) be the smallest cardinality of a family F ⊆ RR for which there is no g: R → R with the property that f + g ∈ A for all f ∈ F . We define cardinal number A(D) for the class D of all real functions with the Darboux property similarly. -
Elements of Set Theory
Elements of set theory April 1, 2014 ii Contents 1 Zermelo{Fraenkel axiomatization 1 1.1 Historical context . 1 1.2 The language of the theory . 3 1.3 The most basic axioms . 4 1.4 Axiom of Infinity . 4 1.5 Axiom schema of Comprehension . 5 1.6 Functions . 6 1.7 Axiom of Choice . 7 1.8 Axiom schema of Replacement . 9 1.9 Axiom of Regularity . 9 2 Basic notions 11 2.1 Transitive sets . 11 2.2 Von Neumann's natural numbers . 11 2.3 Finite and infinite sets . 15 2.4 Cardinality . 17 2.5 Countable and uncountable sets . 19 3 Ordinals 21 3.1 Basic definitions . 21 3.2 Transfinite induction and recursion . 25 3.3 Applications with choice . 26 3.4 Applications without choice . 29 3.5 Cardinal numbers . 31 4 Descriptive set theory 35 4.1 Rational and real numbers . 35 4.2 Topological spaces . 37 4.3 Polish spaces . 39 4.4 Borel sets . 43 4.5 Analytic sets . 46 4.6 Lebesgue's mistake . 48 iii iv CONTENTS 5 Formal logic 51 5.1 Propositional logic . 51 5.1.1 Propositional logic: syntax . 51 5.1.2 Propositional logic: semantics . 52 5.1.3 Propositional logic: completeness . 53 5.2 First order logic . 56 5.2.1 First order logic: syntax . 56 5.2.2 First order logic: semantics . 59 5.2.3 Completeness theorem . 60 6 Model theory 67 6.1 Basic notions . 67 6.2 Ultraproducts and nonstandard analysis . 68 6.3 Quantifier elimination and the real closed fields . -
21 Cardinal and Ordinal Numbers. by W. Sierpióski. Monografie Mate
BOOK REVIEWS 21 Cardinal and ordinal numbers. By W. Sierpióski. Monografie Mate- matyczne, vol. 34. Warszawa, Panstwowe Wydawnictwo Naukowe, 1958. 487 pp. Not since the publication in 1928 of his Leçons sur les nombres transfinis has Sierpióski written a book on transfinite numbers. The present book, embodying the fruits of a lifetime of research and ex perience in teaching the subject, is therefore most welcome. Although generally similar in outline to the earlier work, it is an entirely new book, and more than twice as long. The exposition is leisurely and thickly interspersed with illuminating discussion and examples. The result is a book which is highly instructive and eminently readable. Whether one takes the chapters in order or dips in at random he is almost sure to find something interesting. Many examples and ap plications are included in the form of exercises, nearly all accom panied by solutions. The exposition is from the standpoint of naive set theory. No axioms, other than the axiom of choice, are ever stated explicitly, although Zermelo's system is occasionally referred to. But the role of the axiom of choice is a central theme throughout the book. For a student who wishes to learn just when and how this axiom is needed this is the best book yet written. There is an excellent chapter de voted to theorems equivalent to the axiom of choice. These include not only well-ordering, trichotomy, and Zorn's principle, but also several less familiar propositions: Lindenbaum's theorem that of any two nonempty sets one is equivalent to a partition of the other; Vaught's theorem that every family of nonempty sets contains a maximal disjoint family; Tarski's theorem that every cardinal has a successor, and other propositions of cardinal arithmetic; Kurepa's theorem that the proposition that every partially ordered set con tains a maximal family of incomparable elements is an equivalent when joined with the ordering principle, i.e., the proposition that every set can be ordered. -
Cantor's Discovery of Transfinite Numbers
HISTORIA MATHEMATICA 22 (1995), 33-42 "What Fermented in Me for Years": Cantor's Discovery of Transfinite Numbers Jos~ FERREIROS History of Science and Technology, 543 Stephens Hall, Unioersity of California at Berkeley, Berkeley, California 94720 Transfinite (ordinal) numbers were a crucial step in the development of Cantor's set theory. The new concept depended in various ways on previous problems and results, and especially on two questions that were at the center of Cantor's attention in September 1882, when he was about to make his discovery. First, the proof of the Cantor-Bendixson theorem motivated the introduction of transfinite numbers, and at the same time suggested the "principle of limitation," which is the key to the connection between transfinite numbers and infinite powers. Second, Dedekind's ideas, which Cantor discussed in September 1882, seem to have played an important heuristic role in his decision to consider the "symbols of infinity" that he had been using as true numbers, i.e., as autonomous objects; to this end Cantor introduced in his work, for the first time, ideas on (well) ordered sets. © 1995 Academic Press, Inc. Los ntimeros (ordinales) transfinitos constituyeron un paso clave en el desarrollo de la teorfa de conjuntos de Cantor. La nueva idea dependi6 en varias formas de resultados y problemas previos, pero especialmente de dos cuestiones que ocuparon a Cantor en septiem- bre de 1882, estando a punto de realizar su descubrimiento. En primer lugar, el teorema de Cantor-Bendixson, cuya demostraci6n motiv6 la introducci6n de los ntimeros transfinitos, y a la vez sugiri6 el "principio de limitaci6n" que constituye la clave de la conexi6n entre ntimeros transfinitos y potencias infinitas. -
Applications of Ultrafilters in Ergodic Theory and Combinatorial Number
Vrije Universiteit Amsterdam Jagiellonian University Master Thesis Applications of Ultrafilters in Ergodic Theory and Combinatorial Number Theory Supervisors: Author: Dr. Tanja Eisner Jakub Konieczny Prof. Dr. Ale Jan Homburg Dr Piotr Niemiec A thesis submitted in fulfilment of the requirements for the degree of Master of Mathematics in the Department of Mathematics, Vrije Universiteit Amsterdam Instytut Matematyki, Jagiellonian University arXiv:1310.1056v2 [math.DS] 16 Oct 2013 October 2013 Declaration of Authorship I, Jakub Konieczny, declare that this thesis titled “Applications of Ultrafilters in Ergodic Theory and Combinatorial Number Theory” is my own. I confirm that: This work was done wholly or mainly while in candidature for a research degree at these Universities. Where I have consulted the published work of others, this is always clearly at- tributed. Where I have quoted from the work of others, the source is always given. With the exception of such quotations, this thesis is entirely my own work. I have acknowledged all main sources of help. Signed: Date: i “Thanks to my solid academic training, today I can write hundreds of words on virtually any topic without possessing a shred of information, which is how I got a good job in journalism." Dave Barry VRIJE UNIVERSITEIT AMSTERDAM JAGIELLONIAN UNIVERSITY Abstract Faculty of Sciences, Department of Mathematics Wydział Matematyki i Informatyki, Instytut Matematyki Master of Mathematics Applications of Ultrafilters in Ergodic Theory and Combinatorial Number Theory by Jakub Konieczny Ultrafilters are very useful and versatile objects with applications throughout mathemat- ics: in topology, analysis, combinarotics, model theory, and even theory of social choice. Proofs based on ultrafilters tend to be shorter and more elegant than their classical coun- terparts. -
Set Theory: Taming the Infinite
This is page 54 Printer: Opaque this CHAPTER 2 Set Theory: Taming the Infinite 2.1 Introduction “I see it, but I don’t believe it!” This disbelief of Georg Cantor in his own creations exemplifies the great skepticism that his work on infinite sets in- spired in the mathematical community of the late nineteenth century. With his discoveries he single-handedly set in motion a tremendous mathematical earthquake that shook the whole discipline to its core, enriched it immea- surably, and transformed it forever. Besides disbelief, Cantor encountered fierce opposition among a considerable number of his peers, who rejected his discoveries about infinite sets on philosophical as well as mathematical grounds. Beginning with Aristotle (384–322 b.c.e.), two thousand years of West- ern doctrine had decreed that actually existing collections of infinitely many objects of any kind were not to be part of our reasoning in philosophy and mathematics, since they would lead directly into a quagmire of logical con- tradictions and absurd conclusions. Aristotle’s thinking on the infinite was in part inspired by the paradoxes of Zeno of Elea during the fifth century b.c.e. The most famous of these asserts that Achilles, the fastest runner in ancient Greece, would be unable to surpass a much slower runner, provided that the slower runner got a bit of a head start. Namely, Achilles would then first have to cover the distance between the starting positions, dur- ing which time the slower runner could advance a certain distance. Then Achilles would have to cover that distance, while the slower runner would again advance, and so on. -
Cardinals and the Size of Infinite Sets 1 Review of Bijections
Cardinals and the size of infinite sets These notes revise some lectured material and have some questions for you to complete yourself. (You might find that some of the boxes are too small|just staple extra sheets at the back.) Note that these are not complete lecture notes|they cover most but not all things mentioned in the lectures|and they do not replace the notes you take in lectures. 1 Review of bijections Question 1 Recall the definition of a bijection between two sets. Question 2 (a) Find a bijection between the natural numbers N and the set of even natural numbers f2n j n 2 Ng. (b) Do the same for N and the set of integers Z. Question 3 Find a bijection (0; 1) ! R between the open interval (0; 1) (which is a subset of R) and R itself. [Hint: the function tan(x) can help by giving a bijection from some interval to R|that's not quite the right interval, but you'll be able to fix that problem somehow.] 1 2 The size of finite sets A set A is finite with n elements if A = fa1; : : : ; ang where the elements ai are all distinct. We often denote the number of elements of such sets A by jAj = n. It is clear in this case that there is a bijection f1; 2; : : : ; ng ! A given by i 7! ai: For finite sets, this is a simple observation. But for infinite sets, it will be used as a definition that will provide the starting point of our study. -
The Arithmetic of Cardinal Numbers
The arithmetic of cardinal numbers We wish to extend the familiar operations of addition, multiplication and expo- nentiation of the natural numbers so as to apply to all cardinal numbers. We start with addition and multiplication, then go on consider the unary exponen- tiation function assigning 2 n to each number n and finally the general binary exponentiation operation assigning mn to numbers n, m . Addition and Multiplication Lemma 2 For finite sets A, B , (i) If A ∩ B = ∅ then A ∪ B is finite and |A ∪ B| = |A| + |B|, (ii) A × B is finite and |A × B| = |A| · | B|. Proof see Example 1.9. For sets A, B let A + B = ( {0} × A) ∪ ({1} × B). We call this the disjoint union of A and B. Lemma 3 If A ∼ C and B ∼ D then (i) A + B ∼ C + D, (ii) A × B ∼ C × D. Proof Let f : A ∼ C, g : B ∼ D. Then h : A + B → C + D and e : A × B → C × D defined by h(0 , a ) = (0 , f (a)) for a ∈ A, h (1 , b ) = (1 , g (b)) for b ∈ B e(a, b ) = ( f(a), g (b)) for a ∈ A, b ∈ B are bijections so the result follows. Definition 10 For cardinal numbers n, m let (i) n + m = |A + B| for some/any sets A, B such that n = |A| and m = |B|, (ii) n · m = |A × B| for some/any sets A, B such that n = |A| and m = |B|, Note that the previous lemma ensures that these definitions make sense. Proposition 21 For cardinal numbers n, m and any sets A, B such that A ∩ B = ∅ and n = |A| and m = |B| we have n + m = |A ∪ B|.