The Continuum Hypothesis by Kyle Rodriguez
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Georg Cantor English Version
GEORG CANTOR (March 3, 1845 – January 6, 1918) by HEINZ KLAUS STRICK, Germany There is hardly another mathematician whose reputation among his contemporary colleagues reflected such a wide disparity of opinion: for some, GEORG FERDINAND LUDWIG PHILIPP CANTOR was a corruptor of youth (KRONECKER), while for others, he was an exceptionally gifted mathematical researcher (DAVID HILBERT 1925: Let no one be allowed to drive us from the paradise that CANTOR created for us.) GEORG CANTOR’s father was a successful merchant and stockbroker in St. Petersburg, where he lived with his family, which included six children, in the large German colony until he was forced by ill health to move to the milder climate of Germany. In Russia, GEORG was instructed by private tutors. He then attended secondary schools in Wiesbaden and Darmstadt. After he had completed his schooling with excellent grades, particularly in mathematics, his father acceded to his son’s request to pursue mathematical studies in Zurich. GEORG CANTOR could equally well have chosen a career as a violinist, in which case he would have continued the tradition of his two grandmothers, both of whom were active as respected professional musicians in St. Petersburg. When in 1863 his father died, CANTOR transferred to Berlin, where he attended lectures by KARL WEIERSTRASS, ERNST EDUARD KUMMER, and LEOPOLD KRONECKER. On completing his doctorate in 1867 with a dissertation on a topic in number theory, CANTOR did not obtain a permanent academic position. He taught for a while at a girls’ school and at an institution for training teachers, all the while working on his habilitation thesis, which led to a teaching position at the university in Halle. -
The Continuum Hypothesis
Mathematics As A Liberal Art Math 105 Fall 2015 Fowler 302 MWF 10:40am- 11:35am BY: 2015 Ron Buckmire http://sites.oxy.edu/ron/math/105/15/ Class 22: Monday October 26 Aleph One (@1) and All That The Continuum Hypothesis: Introducing @1 (Aleph One) We can show that the power set of the natural numbers has a cardinality greater than the the cardinality of the natural numbers. This result was proven in an 1891 paper by German mathematician Georg Cantor (1845-1918) who used something called a diagonalization argument in his proof. Using our formula for the cardinality of the power set, @0 jP(N)j = 2 = @1 The cardinality of the power set of the natural numbers is called @1. THEOREM Cantor's Power Set Theorem For any set S (finite or infinite), cardinality of the power set of S, i.e. P (S) is always strictly greater than the cardinality of S. Mathematically, this says that jP (S)j > jS THEOREM c > @0 The cardinality of the real numbers is greater than the cardinality of the natural numbers. DEFINITION: Uncountable An infinite set is said to be uncountable (or nondenumerable) if it has a cardinality that is greater than @0, i.e. it has more elements than the set of natural numbers. PROOF Let's use Cantor's diagonal process to show that the set of real numbers between 0 and 1 is uncountable. Math As A Liberal Art Class 22 Math 105 Spring 2015 THEOREM The Continuum Hypothesis, i.e. @1 =c The continuum hypothesis is that the cardinality of the continuum (i.e. -
The Continuum Hypothesis and Its Relation to the Lusin Set
THE CONTINUUM HYPOTHESIS AND ITS RELATION TO THE LUSIN SET CLIVE CHANG Abstract. In this paper, we prove that the Continuum Hypothesis is equiv- alent to the existence of a subset of R called a Lusin set and the property that every subset of R with cardinality < c is of first category. Additionally, we note an interesting consequence of the measure of a Lusin set, specifically that it has measure zero. We introduce the concepts of ordinals and cardinals, as well as discuss some basic point-set topology. Contents 1. Introduction 1 2. Ordinals 2 3. Cardinals and Countability 2 4. The Continuum Hypothesis and Aleph Numbers 2 5. The Topology on R 3 6. Meagre (First Category) and Fσ Sets 3 7. The existence of a Lusin set, assuming CH 4 8. The Lebesque Measure on R 4 9. Additional Property of the Lusin set 4 10. Lemma: CH is equivalent to R being representable as an increasing chain of countable sets 5 11. The Converse: Proof of CH from the existence of a Lusin set and a property of R 6 12. Closing Comments 6 Acknowledgments 6 References 6 1. Introduction Throughout much of the early and middle twentieth century, the Continuum Hypothesis (CH) served as one of the premier problems in the foundations of math- ematical set theory, attracting the attention of countless famous mathematicians, most notably, G¨odel,Cantor, Cohen, and Hilbert. The conjecture first advanced by Cantor in 1877, makes a claim about the relationship between the cardinality of the continuum (R) and the cardinality of the natural numbers (N), in relation to infinite set hierarchy. -
Aaboe, Asger Episodes from the Early History of Mathematics QA22 .A13 Abbott, Edwin Abbott Flatland: a Romance of Many Dimensions QA699 .A13 1953 Abbott, J
James J. Gehrig Memorial Library _________Table of Contents_______________________________________________ Section I. Cover Page..............................................i Table of Contents......................................ii Biography of James Gehrig.............................iii Section II. - Library Author’s Last Name beginning with ‘A’...................1 Author’s Last Name beginning with ‘B’...................3 Author’s Last Name beginning with ‘C’...................7 Author’s Last Name beginning with ‘D’..................10 Author’s Last Name beginning with ‘E’..................13 Author’s Last Name beginning with ‘F’..................14 Author’s Last Name beginning with ‘G’..................16 Author’s Last Name beginning with ‘H’..................18 Author’s Last Name beginning with ‘I’..................22 Author’s Last Name beginning with ‘J’..................23 Author’s Last Name beginning with ‘K’..................24 Author’s Last Name beginning with ‘L’..................27 Author’s Last Name beginning with ‘M’..................29 Author’s Last Name beginning with ‘N’..................33 Author’s Last Name beginning with ‘O’..................34 Author’s Last Name beginning with ‘P’..................35 Author’s Last Name beginning with ‘Q’..................38 Author’s Last Name beginning with ‘R’..................39 Author’s Last Name beginning with ‘S’..................41 Author’s Last Name beginning with ‘T’..................45 Author’s Last Name beginning with ‘U’..................47 Author’s Last Name beginning -
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. -
Set.1 the Power of the Continuum
set.1 The Power of the Continuum sol:set:pow: In second-order logic we can quantify over subsets of the domain, but not over explanation sec sets of subsets of the domain. To do this directly, we would need third-order logic. For instance, if we wanted to state Cantor's Theorem that there is no injective function from the power set of a set to the set itself, we might try to formulate it as \for every set X, and every set P , if P is the power set of X, then not P ⪯ X". And to say that P is the power set of X would require formalizing that the elements of P are all and only the subsets of X, so something like 8Y (P (Y ) $ Y ⊆ X). The problem lies in P (Y ): that is not a formula of second-order logic, since only terms can be arguments to one-place relation variables like P . We can, however, simulate quantification over sets of sets, if the domain is large enough. The idea is to make use of the fact that two-place relations R relates elements of the domain to elements of the domain. Given such an R, we can collect all the elements to which some x is R-related: fy 2 jMj : R(x; y)g is the set \coded by" x. Conversely, if Z ⊆ }(jMj) is some collection of subsets of jMj, and there are at least as many elements of jMj as there are sets in Z, then there is also a relation R ⊆ jMj2 such that every Y 2 Z is coded by some x using R. -
Unsolvable Problems, the Continuum Hypothesis, and the Nature of Infinity
Unsolvable problems, the Continuum Hypothesis, and the nature of infinity W. Hugh Woodin Harvard University January 9, 2017 V : The Universe of Sets The power set Suppose X is a set. The powerset of X is the set P(X ) = fY Y is a subset of X g: Cumulative Hierarchy of Sets The universe V of sets is generated by defining Vα by induction on the ordinal α: 1. V0 = ;, 2. Vα+1 = P(Vα), S 3. if α is a limit ordinal then Vα = β<α Vβ. I If X is a set then X 2 Vα for some ordinal α. I V0 = ;, V1 = f;g, V2 = f;; f;gg. I These are just the ordinals: 0, 1, and 2. I V3 has 4 elements. I This is not the ordinal 3 (in fact, it is not an ordinal). I V4 has 16 elements. I V5 has 65; 536 elements. I V1000 has a lot of elements. V! is infinite, it is the set of all (hereditarily) finite sets. The conception of V! is mathematically identical to the conception of the structure (N; +; ·). Beyond the basic axioms: large cardinal axioms The axioms I The ZFC axioms of Set Theory specify the basic axioms for V . I These axioms are naturally augmented by additional principles which assert the existence of \very large" infinite sets. I These additional principles are called large cardinal axioms. I There is a proper class of measurable cardinals. I There is a proper class of strong cardinals. I There is a proper class of Woodin cardinals. -
Forcing? Thomas Jech
WHAT IS... ? Forcing? Thomas Jech What is forcing? Forcing is a remarkably powerful case that there exists no proof of the conjecture technique for the construction of models of set and no proof of its negation. theory. It was invented in 1963 by Paul Cohen1, To make this vague discussion more precise we who used it to prove the independence of the will first elaborate on the concepts of theorem and Continuum Hypothesis. He constructed a model proof. of set theory in which the Continuum Hypothesis What are theorems and proofs? It is a use- (CH) fails, thus showing that CH is not provable ful fact that every mathematical statement can from the axioms of set theory. be expressed in the language of set theory. All What is the Continuum Hypothesis? In 1873 mathematical objects can be regarded as sets, and Georg Cantor proved that the continuum is un- relations between them can be reduced to expres- countable: that there exists no mapping of the set sions that use only the relation ∈. It is not essential N of all integers onto the set R of all real numbers. how it is done, but it can be done: For instance, Since R contains N, we have 2ℵ0 > ℵ , where 2ℵ0 0 integers are certain finite sets, rational numbers and ℵ are the cardinalities of R and N, respec- 0 are pairs of integers, real numbers are identified tively. A question arises whether 2ℵ0 is equal to with Dedekind cuts in the rationals, functions the cardinal ℵ1, the immediate successor of ℵ0. -
The Continuum Hypothesis
Gauge Institute Journal, Volume 4, No 1, February 2008, H. Vic Dannon The Continuum Hypothesis H. Vic Dannon [email protected] September 2007 1 Gauge Institute Journal, Volume 4, No 1, February 2008, H. Vic Dannon Abstract We prove that the Continuum Hypothesis is equivalent to the Axiom of Choice. Thus, the Negation of the Continuum Hypothesis, is equivalent to the Negation of the Axiom of Choice. The Non-Cantorian Axioms impose a Non-Cantorian definition of cardinality, that is different from Cantor’s cardinality imposed by the Cantorian Axioms. The Non-Cantorian Theory is the Zermelo-Fraenkel Theory with the Negation of the Axiom of Choice, and with the Negation of the Continuum Hypothesis. This Theory has distinct infinities. Keywords: Continuum Hypothesis, Axiom of Choice, Cardinal, Ordinal, Non-Cantorian, Countability, Infinity. 2000 Mathematics Subject Classification 03E04; 03E10; 03E17; 03E50; 03E25; 03E35; 03E55. 2 Gauge Institute Journal, Volume 4, No 1, February 2008, H. Vic Dannon Contents Preface……………………………………………………...... 4 1 Hilbert’s 1st problem: The Continuum Hypothesis...6 Preface to 2…………………………………………………...14 2 Non-Cantorian Cardinal Numbers………………….......16 Preface to 3……………………………………………………38 3 Rationals Countability and Cantor’s Proof....................39 Preface to 4……………………………………………………50 4 Cantor’s Set and the Cardinality of the Reals............ 52 Preface to 5……………………………………………………75 5 Non-Cantorian Set Theory…………………………….....76 Preface to 6…………………………………………………....86 6 Cardinality, Measure, Category………………………....87 Preface to 7…………………………………………………...100 7 Continuum Hypothesis, Axiom of Choice, and Non- Cantorian Theory………….……………………………....101 3 Gauge Institute Journal, Volume 4, No 1, February 2008, H. Vic Dannon Preface The Continuum Hypothesis says that there is no infinity between the infinity of the natural numbers, and the infinity of the real numbers. -
The Continuum Hypothesis and Forcing
The Continuum Hypothesis and Forcing Connor Lockhart December 2018 Abstract In this paper we introduce the problem of the continuum hypothesis and its solution via Cohen forcing. First, we introduce the basics of first order logic and standard ZFC set theory before elaborating on ordinals, cardinals and the forcing concept. The motivation of this paper is exposi- tory and should be approachable for anyone familiar with first order logic and set theory. Contents 1 Introduction and the Continuum Hypothesis 2 2 First Order Logic 2 2.1 Formal Languages . 2 2.2 Model Theory . 3 3 Set Theory and the ZFC axioms 6 3.1 List and Motivation of the Zermelo-Fraenkel Axioms . 6 4 Ordinals and Cardinals 9 4.1 Orderings and Ordinal numbers . 9 4.2 Ordinal Arithmetic . 10 4.3 Cardinal Numbers . 11 5 Forcing 11 5.1 Tools of Forcing . 12 5.2 Useful Forcing Lemmas . 15 6 Independence of the Continuum Hypothesis 16 7 Acknowledgments 17 1 1 Introduction and the Continuum Hypothesis The continuum hypothesis (also referred to as CH) was first formulated in 1878 by Georg Cantor following his work on the foundations of set theory. Its for- mulation is often stated as There is no set whose cardinality is strictly between that of the integers and the real numbers. This can also be reformulated to state that the successor cardinal to @0 is the cardinality of the reals. Such was suspected, but not proven, by Cantor and his contemporaries. The first major advance on the problem was presented by G¨odelin 1940 showing its consistency with ZFC axioms, and independence was finally shown in 1963 by Cohen. -
The Continuum Hypothesis Is True... for All Practical Purposes
The Continuum Hypothesis is True... for all Practical Purposes Sarah E. Horton,∗ Alicia R. Richardson,† Natalie M. Scala,‡ and Andrea C. Watkins§ July 18, 2003 Abstract Let G be an uncountable subset of the real numbers. It is shown that if G is a closed set, open set, or Fσ set, then the cardinality of G is the same as the cardinality of the real numbers. In particular, the Continuum Hypothesis is true for the basic sets in the construction of the Borel sets. 1 Introduction Set theory is the language of mathematics. Everything mathematicians know and use today can be traced back to set theory and the eight axioms of Ernst Zer- melo and Abraham Fraenkel, denoted ZF. These basic suppositions are used to develop the natural, rational, and real numbers and, thus, mathematics. With- out a foundation, it would be impossible to define any concepts or formulate any theorems or lemmas. More importantly, it is necessary to establish undefined concepts in order to prevent a never-ending cycle of philosophic speculation and doubt about our processes. Due to the intuitive and powerful nature of ZF, set theorists agree upon these axioms without formal justification. Fortunately, there are models, or structures, of set theory in which ZF is true. Seeing ZF work in a model helps to give credibility to the system of axioms. Furthermore, there are models that are consistent with some of the axioms and not with others. These models illustrate the relative independence of the axioms. Similarly, there are statements that are independent of ZF. That is, there exist models of ZF where the statement is true and there exist models of ∗Department of Mathematics and Statistics, Stephen F. -
The Axiom of Determinacy
Virginia Commonwealth University VCU Scholars Compass Theses and Dissertations Graduate School 2010 The Axiom of Determinacy Samantha Stanton Virginia Commonwealth University Follow this and additional works at: https://scholarscompass.vcu.edu/etd Part of the Physical Sciences and Mathematics Commons © The Author Downloaded from https://scholarscompass.vcu.edu/etd/2189 This Thesis is brought to you for free and open access by the Graduate School at VCU Scholars Compass. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of VCU Scholars Compass. For more information, please contact [email protected]. College of Humanities and Sciences Virginia Commonwealth University This is to certify that the thesis prepared by Samantha Stanton titled “The Axiom of Determinacy” has been approved by his or her committee as satisfactory completion of the thesis requirement for the degree of Master of Science. Dr. Andrew Lewis, College of Humanities and Sciences Dr. Lon Mitchell, College of Humanities and Sciences Dr. Robert Gowdy, College of Humanities and Sciences Dr. John Berglund, Graduate Chair, Mathematics and Applied Mathematics Dr. Robert Holsworth, Dean, College of Humanities and Sciences Dr. F. Douglas Boudinot, Graduate Dean Date © Samantha Stanton 2010 All Rights Reserved The Axiom of Determinacy A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science at Virginia Commonwealth University. by Samantha Stanton Master of Science Director: Dr. Andrew Lewis, Associate Professor, Department Chair Department of Mathematics and Applied Mathematics Virginia Commonwealth University Richmond, Virginia May 2010 ii Acknowledgment I am most appreciative of Dr. Andrew Lewis. I would like to thank him for his support, patience, and understanding through this entire process.