SET THEORY Andrea K. Dieterly A Thesis Submitted to the Graduate College of Bowling Green State University in partial fulfillment of the requirements for the degree of MASTER OF ARTS August 2011 Committee: Warren Wm. McGovern, Advisor Juan Bes Rieuwert Blok i Abstract Warren Wm. McGovern, Advisor This manuscript was to show the equivalency of the Axiom of Choice, Zorn's Lemma and Zermelo's Well-Ordering Principle. Starting with a brief history of the development of set history, this work introduced the Axioms of Zermelo-Fraenkel, common applications of the axioms, and set theoretic descriptions of sets of numbers. The book, Introduction to Set Theory, by Karel Hrbacek and Thomas Jech was the primary resource with other sources providing additional background information. ii Acknowledgements I would like to thank Warren Wm. McGovern for his assistance and guidance while working and writing this thesis. I also want to thank Reiuwert Blok and Juan Bes for being on my committee. Thank you to Dan Shifflet and Nate Iverson for help with the typesetting program LATEX. A personal thank you to my husband, Don, for his love and support. iii Contents Contents . iii 1 Introduction 1 1.1 Naive Set Theory . 2 1.2 The Axiom of Choice . 4 1.3 Russell's Paradox . 5 2 Axioms of Zermelo-Fraenkel 7 2.1 First Order Logic . 7 2.2 The Axioms of Zermelo-Fraenkel . 8 2.3 The Recursive Theorem . 13 3 Development of Numbers 16 3.1 Natural Numbers and Integers . 16 3.2 Rational Numbers . 20 3.3 Real Numbers . 22 4 Cardinal Numbers 25 4.1 Cardinal Numbers . 25 4.2 Equivalence of Axioms . 32 Bibliography 34 1 Chapter 1 Introduction A set is an abstraction that is produced in the mind: the set of primary colors, the set of odd integers, or a set of dishes. When we talk about a collection, the mind grasps or pictures the elements together with some relationship between the objects. In a general sense, a set is a collection of objects. However as deduced in Russell's Paradox certain collections cannot be sets. Since we want to be able to define a set by either listing its objects or defining its members by some basic property, we are going to move beyond this basic idea. Axiomatic set theory is the art of explicitly expressing mathematics with rigid rules of logic and the property of membership in a set. We say that an object is a member of a set or an element of a set, denoted 2. This is the very essence of set theory along with logical expressions: \and" ... \or" ... \then" ... \if and only if" and quantifiers: \for all" and \there exists". Our intuition supplies concepts that we want to be included in the theory. Set equality, creation of new sets from existing sets, even the empty set need to be defined by or derived from our set of axioms. Many have worked and laid the foundations of set theory. Georg Cantor's ideas about infinity were radical in his time, but now are widely accepted. The axioms that we will focus on were developed by Ernst Zermelo in 1908 and additional contributions were made by Abraham Fraenkel in 1922. This collection of axioms is known as ZF. 2 1.1 Naive Set Theory We begin by naively recalling important sets which will be defined formally later. Our first set is the set of natural numbers f1,2,3,...g, the numbers that we recited back to parents and teachers, and learned to put in a one-to-one correspondence with objects. Somewhere in the learning process, debt entered the picture; we added negative values and introduced the set of integers f...,-3,-2,-1,0,1,2,3,...g. The question also arose, \what happens when we equally divide five pieces of bread among three people?" The answer leads to another set: an integer divided by an integer - the set of rational numbers. It was assumed that now every number had been defined, until somewhere around the time of Pythagoras when it was shown that the square root of two could not be written as a rational number but it was known to exist, so the set of irrational numbers was defined. With the union of the set of rational numbers with the set of irrational numbers, all numbers were defined on a number line, and the set of real numbers was first described. Hence we will call the real numbers: the continuum. Georg Cantor published a series of papers between 1873 and 1897 with goals of charac- terizing the sizes of infinite sets. He wrote about the distinctions between countable and uncountable sets of real numbers. He defined the power of the set as a size of its measure; he believed in the existence of infinite sets that shared characteristics with finite sets and transfinite numbers that act like ordinary counting numbers [Drake and D.Singh, 1996, p.1]. Cantor talked about the Absolute Infinite which he identified with God who is beyond mathematics. Today, the collection of all ordinals which is an example of von Neumann's proper classes, has been compared to the absolute infinite [Drake and D.Singh, 1996, p.2]. Cantor began his investigations by defining the cardinality of finite sets. If two finite sets have the same cardinality, then their elements can be put in a one-to-one correspondence with each other. He extended this idea to infinite sets as sets that can be put into a one- to-one correspondence to a proper subset of itself [Wapner, 2005, p.8]. For example, the set of even integers is a subset of the set of integers, but since a one-to-correspondence exists between them, they have the same cardinality. 3 Cantor is known for showing that the rational numbers have the same cardinality as the natural numbers with the method that has come to be called the Cantor diagonalization argument. In his argument he showed that the real numbers do not have the same cardinality as the natural numbers and even though his argument relies heavily on the Axiom of Choice, more recent proofs have been \choice-free". So Cantor established the existence of at least two infinities, the second which is the cardinality of the continuum. He hypothesized a hierarchy of infinities which gives us the notation that we use today: aleph nought { @0 representing the cardinality of natural numbers and 2@0 representing the cardinality of the continuum [Wapner, 2005, p.12]. What is now known as the Continuum Hypothesis, 2@0 = @1; was one of his obsessions that may have led to his mental breakdown at the end of his life [Dauben, 1993, Internet version 2004, p.1]. Cantor's early biographers portrayed him as suffering longer and longer periods of mental breakdowns due to the conflicts about his theories and the resistance they aroused from the mathematical community. Poincare declared the theory of transfinite numbers as a \disease" which hopefully math could recover. Kronecker attacked Cantor personally, calling him a \scientific charlatain" and a \corrupter of youth." It has come to light that Cantor was indeed suffering from cyclic manic-depression. It has been suggested that Cantor used the seasons of his illness to produce a great deal during the manic cycle and then during a following depression he was able to overcome his own doubts to the solidness of his theories [Dauben, 1993, Internet version 2004, p.1-2]. By the end of his life, Cantor's work was becoming accepted. Hilbert described his work as ... \the finest product of mathematical genius and and one of the supreme achievements of purely intellectual human activity" [O'Connor and Robertson, October 1998, p.1]. 4 1.2 The Axiom of Choice The Axiom of Choice is the workhorse of so many theorems; though unaware, Cantor em- ployed it in his diagonalization proof. But if mathematicians allow this axiom, then they must accept the startling conclusions. If S is a collection of non-empty sets, then a new set of elements, called a choice set, can be formed by choosing one element from each set in the collection. This does not seem like an unreasonable idea. Surely, if S is a finite collection, the existence of a choice set follows from a finite number of steps. We can actually construct the set. But if S is infinite, does a choice set exist? Sometimes yes and sometimes no. Bertrand Russell clarified this with his shoes and socks example: \to choose one sock from each of infinitely many pairs of socks requires the Axiom of Choice, but for shoes the axiom is not needed." We can create an explicit choice set for shoes with an algorithm such as always choose the right shoe and so the set is clearly defined. But an algorithm that would yield a clearly defined set is not available for choosing one sock from infinitely many pairs of socks. We need the Axiom of Choice to create a unique set, and we must assume its existence as an axiom [Wapner, 2005, p.16ff]. The Axiom of Choice has led to some very surprising conclusions. For example, the ability to reassemble a single sphere into two new spheres identical to the first (The Banach-Tarski Paradox) or the existence of a set of real numbers which is not Lebesgue measurable. These conclusions are why some mathematicians avoid this axiom. Others embrace it because it yields results that we could not get otherwise. David Hilbert listed the Continuum Hypothesis first on his list of twenty-three unsolved problems in 1900 at the congress in Paris.