DOCSLIB.ORG

Chapter 6: Cardinal Numbers and the Axiom of Choice Equinumerosity March 18 & 20, 2014

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Chapter 6: Cardinal Numbers and the Axiom of Choice Equinumerosity March 18 & 20, 2014 Elements of Set Theory Chapter 6: Cardinal Numbers and The Axiom of Choice Equinumerosity March 18 & 20, 2014 Lecturer: Fan Yang 1/28 2/28 What is the size of a set? X = f , , , g Y = f , , , , g ! = f0; 1; 2; 3; 4;::: g Do A and B have the same size? Does A have more elements than B? Are there exactly as many houses as people? Y has more elements than X. Yes, since there are 5 houses and 5 people. The infinite set ! has more elements than the finite sets X and Y . Yes, since there is a one-to-one correspondence between the two Do ! and Z have the same size? What about Q and Z? sets. 0 1 2 3 ! Definition 6.1 -3 -2 -1 0 1 2 3 Z A set A is said to be equinumerous or equipotent to a set B (written 3 1 11 − 2 2 4 A ≈ B) iff there is a bijection from A onto B. -3 -2 -1 0 1 2 3 Q A bijection from A onto B is also called a one-to-one correspondence Given two infinite sets A and B, how to compare their sizes? between sets A and B. 3/28 4/28 Example 6.1: Let A = fa; b; c; dg and B = fa; b; cg. Then A 6≈ B, since there is no bijection from A onto B. In general, for any two finite sets X and Y , if Y ⊂ X, then X 6≈ Y . Consider the following infinite sets. Clearly, Even ⊂ !. ! = f0; 1; 2; 3; 4; 5; 6;::: g, Even = f0; 2; 4; 6; 8; 10; 12;::: g = f2n j n 2 !g. Example 6.2: ! ≈ Even. Are there exactly as many knives as forks? Proof. The function f : ! ! Even defined by taking Yes, as there is a one-to-one correspondence between knives and forks. f (n) = 2n is a bijection. Indeed, f is injective, since n 6= m =) 2n 6= 2m. f is surjective, since for all m 2 Even, m = 2n for some n 2 !, and f (n) = 2n = m. 5/28 6/28 Example 6.3: [Galileo] Similarly, ! ≈ Sq, where ! = f0; 1; 2; 3; 4; 5;::: g, Example 6.5: ! × ! ≈ !. Sq = f0; 1; 4; 9; 16; 25;::: g = fn2 j n 2 !g. Proof. The function f : ! × ! ! ! defined as 1 f (m; n) = [(m + n)2 + 3m + n] since there is a bijection f : ! ! Sq defined as f (n) = n2. 2 is a bijection. Example 6.4: ! ≈ ! n f0g, since the function f : ! ! ! n f0g defined as ! f (n) = n+ is a bijection. 10 0 1 2 3 4 ! 11 6 12 1 2 3 4 ! n f0g 3 7 Remark: For infinite sets A; B, 1 4 8 13 A ⊂ B 6=) A 6≈ B ! 0 2 5 9 14 ! 7/28 8/28 Example 6.6: Q ≈ !. Example 6.7: (0; 1) ≈ R, where (0; 1) = fx 2 R j 0 < x < 1g. Proof. In the following picture, we specify a bijection f : ! ! Q. 3 3 3 3 3 1 2 3 4 5 Proof. By picture. Here (0; 1) has been bent into a semicircle with center P. Each point in (0; 1) is paired with its projection (from P) on [10] 2 2 2 [9] 2 2 the real line. 1 2 3 4 5 [1] 1 1 [2] 1 [8] 1 1 1 2 3 4 5 0 1 [0] 0 0 0 0 0 1 2 3 4 5 :::::: x [4] 1 1 [3] 1 [7] 1 1 − 1 − 2 − 3 − 4 − 5 [5] 2 2 2 [6] 2 2 − 1 − 2 − 3 − 4 − 5 r R 3 3 3 3 3 − 1 − 2 − 3 − 4 − 5 9/28 10/28 Recall : for any sets X and Y , An alternative proof: Claim: the mapping f :(0; 1) ! defined by R X putting Y = ff : X ! Y j f is a functiong: 1 1 f (x) = − x 1 − x Example 6.8: For any set A, we have }A ≈ A2. is a bijection. 1 1 1 1 For any a; b 2 (0; 1), if f (a) = f (b), namely, if a − 1−a = b − 1−b , then Proof. Define a function H : }A ! A2 by taking (b − a) 1 + 1 = 0 thereby a = b. Hence f is injective. ab (1−a)(1−b) H(B) = fB; For any r 2 , there is x = p 2 2 (0; 1) such that where f : A ! f0; 1g is the characteristic function of B defined as R r 2 B r+ 4+r +2 ( 1 if x 2 B; 1 1 1 1 f (x) = f (x ) = − = − 2 B r x 1 − x x 1 − x 0 if x 2 A n B: r p r r p r r + 4 + r 2 − 2 r + 4 + r 2 + 2 Claim: H is a bijection. = · p 2 r + 4 + r 2 Indeed, H is injective, since for any B; C ⊆ A, p 2 2 (r + 4 + r 2) − 2 H(B) = H(C) =) fB = fC =) 8x 2 A(fB(x) = fC(x)) = p = r: 2(r + 4 + r 2) =) 8x 2 A(x 2 B $ x 2 C) =) B = C; A Hence f is surjective. H is surjective, since for any function g 2 2, there is B = fx 2 A j g(x) = 1g ⊆ A such that H(B) = fB = g. 11/28 12/28 Theorem 6B (Cantor 1873) (a) The set ! is not equinumerous to the set R of real numbers. (b) No set is equinumerous to its power set. Theorem 6A Proof. (a) [Diagonal argument] Given any map f : ! ! R. We show For any sets A; B and C: that there exists z 2 R such that z 2= ran f . (a) A ≈ A. f (0) = 236: 0012 0 1 2 ::: (b) If A ≈ B, then B ≈ A. f (1) = −7: 73 374 7 4 ::: (c) If A ≈ B and B ≈ C, then A ≈ C. f (2) = 3: 1 41 15 5 ::: Proof. Exercise. f (3) = 0: 5 2 46 6 ::: . That is, ≈ is an “equivalence relation” on the class of all sets. The integer part of z is 0, and the (n + 1)st decimal place of z is 3 unless the (n + 1)st decimal place of f (n) is 3, in which case the (n + 1)st decimal place of z is 4. For example, in the case shown, z = 0:3433 ::: Clearly, z 6= f (n) for all n, as it differs from f (n) in the (n + 1)st decimal place. Hence z 2= ran f . 13/28 14/28 (b) Suppose g : A ! }A is surjective. Let B = fx 2 A j x 2= g(x)g: Since B 2 }A and g is surjective, there exists x0 2 A such that g(x0) = B: But then, by the definition of B, Finite Sets x0 2 B () x0 2= g(x0) = B; which is a contradiction. Hence there is no surjection from A onto }A, thus A 6≈ }A. Remark: We will soon be able to prove R ≈ }!. R has smaller size than }R, which has smaller size than }}R, etc. 15/28 16/28 Definition 6.2 3 4 5 2 A set is finite iff it is equinumerous to some natural number. Otherwise it is infinite. 1 That is, for any set A: 0 A is finite iff 9n 2 !(A ≈ n) A is infinite iff 8n 2 !(A 6≈ n) Example 6.9: The set A = fa0; a1; a2; a3g is finite, since A ≈ 4 via the bijection f : 4 ! A defined as: f (n) = an. 4 = f0; 1; 2; 3g A = fa0; a1; a2; a3g Example 6.10: Any finite set A is not equinumerous to an infinite set B. There are finitely many forks in the picture. Why? Proof. Suppose A ≈ B. As A is finite, A ≈ n for some n 2 !. From the Because there are exactly6 forks. This, in turn, is because there is a transitivity of ≈, it follows that B ≈ n, contradicting B being infinite. one-to-one correspondence between the natural number Next, we want to check that each finite set A is equinumerous to a 6 = f0; 1; 2; 3; 4; 5g and the set F of forks, or in other words, 6 ≈ F. unique natural number n. This requires the Pigeonhole Principle. 17/28 18/28 Proof. (of Pigeonhole Principle) We shall show that for any n 2 !, if f : n ! n Pigeonhole Principle: is a one-to-one function, then ran f = n (not a proper subset of n), namely, that the set T is inductive: If n items are put into m pigeonholes with n > m, T = fn 2 ! j ran f = n for any one-to-one function f : n ! ng: then at least one pigeon- Base case: 0 2 T , since the only function f : 0 ! 0 is the empty function hole must contain more f = ;, and ran ; = ;. than one item. Inductive step: Assume k 2 T and f : k + ! k + is a one-to-one function. We + + Here n = 10 and m = 9. show that ran f = k . Note that f k is a one-to-one function from k into k . Case 1: ran (f k) ⊆ k. Then f k : k ! k is a one-to-one function, as + + f : k ! k is 1-1. Thus, the assumption k 2 T implies ran f k = k. Pigeonhole Princinple Moreover, since f is 1-1, we must have f (k) = k.
Load more

Recommended publications
  • 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).
    [Show full text]
  • 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.
    [Show full text]
  • Chapter 1. Combinatorial Theory 1.3: No More Counting Please 1.3.1
    Chapter 1. Combinatorial Theory 1.3: No More Counting Please Slides (Google Drive) Alex Tsun Video (YouTube) In this section, we don't really have a nice successive ordering where one topic leads to the next as we did earlier. This section serves as a place to put all the final miscellaneous but useful concepts in counting. 1.3.1 Binomial Theorem n We talked last time about binomial coefficients of the form k . Today, we'll see how they are used to prove the binomial theorem, which we'll use more later on. For now, we'll see how they can be used to expand (possibly large) exponents below. You may have learned this technique of FOIL (first, outer, inner, last) for expanding (x + y)2. We then combine like-terms (xy and yx). (x + y)2 = (x + y)(x + y) = xx + xy + yx + yy [FOIL] = x2 + 2xy + y2 But, let's say that we wanted to do this for a binomial raised to some higher power, say (x + y)4. There would be a lot more terms, but we could use a similar approach. 1 2 Probability & Statistics with Applications to Computing 1.3 (x + y)4 = (x + y)(x + y)(x + y)(x + y) = xxxx + yyyy + xyxy + yxyy + ::: But what are the terms exactly that are included in this expression? And how could we combine the like-terms though? Notice that each term will be a mixture of x's and y's. In fact, each term will be in the form xkyn−k (in this case n = 4).
    [Show full text]
  • 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.
    [Show full text]
  • Sets and Classes As Many
    SETS AND CLASSES AS MANY by John L. Bell INTRODUCTION Set theory is sometimes formulated by starting with two sorts of entities called individuals and classes, and then defining a set to be a class as one, that is, a class which is at the same time an individual, as indicated in the diagram: Individuals Sets Classes If on the other hand we insist—as we shall here—that classes are to be taken in the sense of multitudes, pluralities, or classes as many, then no class can be an individual and so, in particular, the concept of set will need to be redefined. Here by “class as many” we have in mind what Erik Stenius refers to in [5] as set of, which he defines as follows: If we start from a Universe of Discourse given in advance, then we may define a set-of things as being many things in this UoD or just one thing - or even no things, if we want to introduce this way of speaking. Stenius draws a sharp distinction between this concept and that of 2 set as a thing. As he says, The distinction between a set-as-a-thing and a set of corresponds to the Russellian distinction between a “class as one” and a “class as many” (Principles of Mathematics, p. 76). Only I use the expressions ‘set-as-a- thing” and “set-of” instead, in order to stress the (grammatical and) ontological character of the distinction; and also because of the difficulty that a set-of need not consist of many things; it can comprise just one thing or no things.
    [Show full text]
  • 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.
    [Show full text]
  • Lecture 6: Entropy
    Matthew Schwartz Statistical Mechanics, Spring 2019 Lecture 6: Entropy 1 Introduction In this lecture, we discuss many ways to think about entropy. The most important and most famous property of entropy is that it never decreases Stot > 0 (1) Here, Stot means the change in entropy of a system plus the change in entropy of the surroundings. This is the second law of thermodynamics that we met in the previous lecture. There's a great quote from Sir Arthur Eddington from 1927 summarizing the importance of the second law: If someone points out to you that your pet theory of the universe is in disagreement with Maxwell's equationsthen so much the worse for Maxwell's equations. If it is found to be contradicted by observationwell these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of ther- modynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation. Another possibly relevant quote, from the introduction to the statistical mechanics book by David Goodstein: Ludwig Boltzmann who spent much of his life studying statistical mechanics, died in 1906, by his own hand. Paul Ehrenfest, carrying on the work, died similarly in 1933. Now it is our turn to study statistical mechanics. There are many ways to dene entropy. All of them are equivalent, although it can be hard to see. In this lecture we will compare and contrast dierent denitions, building up intuition for how to think about entropy in dierent contexts. The original denition of entropy, due to Clausius, was thermodynamic.
    [Show full text]
  • 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.
    [Show full text]
  • 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 ............................................
    [Show full text]
  • Preliminary Knowledge on Set Theory
    Huan Long Shanghai Jiao Tong University Basic set theory Relation Function Spring 2018 Georg Cantor(1845-1918) oGerman mathematician o Founder of set theory Bertrand Russell(1872-1970) oBritish philosopher, logician, mathematician, historian, and social critic. Ernst Zermelo(1871-1953) oGerman mathematician, foundations of mathematics and hence on philosophy David Hilbert (1862-1943) o German mathematicia, one of the most influential and universal mathematicians of the 19th and early 20th centuries. Kurt Gödel(1906-1978) oAustrian American logician, mathematician, and philosopher. ZFC not ¬CH . Paul Cohen(1934-2007)⊢ oAmerican mathematician, 1963: ZFC not CH,AC . Spring 2018 ⊢ Spring 2018 By Georg Cantor in 1870s: A set is an unordered collection of objects. ◦ The objects are called the elements, or members, of the set. A set is said to contain its elements. Notation: ◦ Meaning that: is an element of the set A, or, Set A contains ∈ . Important: ◦ Duplicates do not matter. ◦ Order does not matter. Spring 2018 a∈A a is an element of the set A. a A a is NOT an element of the set A. Set of sets {{a,b},{1, 5.2}, k} ∉ the empty set, or the null set, is set that has no elements. A B subset relation. Each element of A is also an element of B. ∅ A=B equal relation. A B and B A. ⊆ A B ⊆ ⊆ A B strict subset relation. If A B and A B ≠ |A| cardinality of a set, or the number of distinct elements. ⊂ ⊆ ≠ Venn Diagram V U A B U Spring 2018 { , , , } a {{a}} ∈ ∉ {{ }} ∅ ∉∅ {3,4,5}={5,4,3,4} ∅ ∈ ∅ ∈ ∅ S { } ∅⊆ S S ∅ ⊂ ∅ |{3, 3, 4, {2, 3},{1,2,{f}} }|=4 ⊆ Spring 2018 Union Intersection Difference Complement Symmetric difference Power set Spring 2018 Definition Let A and B be sets.
    [Show full text]
  • Elements of Set Theory
    Elements of Set Theory P´eterMekis Department of Logic, ELTE Budapest December 11, 2017 Preface This is an as yet incomplete draft of what is going to be the lecture notes for a course under the same title for students of the Logic and the Theory of Science MA program of ELTE Budapest. The text is written for students that are logically and philosophically minded, but do not necessarily have a mathematical background. However, it requires familiarity with the basics of first-order logic. A reader who is skilled in reading mathematical texts will find most of the proofs too detailed. Note that any proof can be skipped and taken as an exercise to avoid boredom. Instead of the more well-known system of Zermelo{Fraenkel set theory, these notes use the framework of G¨odel{Bernays set theory (GB). Beyond the personal taste of the author, the reason for this choice is that there is a follow-up of this course under the title Alternative set theories, the basic task of which is to investigate the philosophical motivations and implications of various versions of the set/class distinction, which is at the heart of any axiomatic set theory, and which is more convenient to discuss in a language where one can quantify over classes. A mathematically minded reader may be surprised by the length of discussing certain basic topics, even without any apparent practical profit, and even at the cost of giving a rather sketchy presentation of more advanced results. An example is the detailed com- parison of three different ways to represent natural numbers in GB.
    [Show full text]
  • 2.5. INFINITE SETS Now That We Have Covered the Basics of Elementary
    2.5. INFINITE SETS Now that we have covered the basics of elementary set theory in the previous sections, we are ready to turn to infinite sets and some more advanced concepts in this area. Shortly after Georg Cantor laid out the core principles of his new theory of sets in the late 19th century, his work led him to a trove of controversial and groundbreaking results related to the cardinalities of infinite sets. We will explore some of these extraordinary findings, including Cantor’s eponymous theorem on power sets and his famous diagonal argument, both of which imply that infinite sets come in different “sizes.” We also present one of the grandest problems in all of mathematics – the Continuum Hypothesis, which posits that the cardinality of the continuum (i.e. the set of all points on a line) is equal to that of the power set of the set of natural numbers. Lastly, we conclude this section with a foray into transfinite arithmetic, an extension of the usual arithmetic with finite numbers that includes operations with so-called aleph numbers – the cardinal numbers of infinite sets. If all of this sounds rather outlandish at the moment, don’t be surprised. The properties of infinite sets can be highly counter-intuitive and you may likely be in total disbelief after encountering some of Cantor’s theorems for the first time. Cantor himself said it best: after deducing that there are just as many points on the unit interval (0,1) as there are in n-dimensional space1, he wrote to his friend and colleague Richard Dedekind: “I see it, but I don’t believe it!” The Tricky Nature of Infinity Throughout the ages, human beings have always wondered about infinity and the notion of uncountability.
    [Show full text]