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Home , Abraham Fraenkel, Absolute Infinite, Axiom of extensionality, Axiom of infinity, Axiom of pairing, Axiom of power 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 of Choice, Zorn’ Lemma and Zermelo’s Well-Ordering Principle. Starting with a brief history of the development of set history, this work introduced the of Zermelo-Fraenkel, common applications of the axioms, and set theoretic descriptions of sets of . The book, Introduction to , by Karel Hrbacek and 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 ...... 2 1.2 The ...... 4 1.3 Russell’s ...... 5

2 Axioms of Zermelo-Fraenkel 7 2.1 First ...... 7 2.2 The Axioms of Zermelo-Fraenkel ...... 8 2.3 The Recursive ...... 13

3 Development of Numbers 16 3.1 Natural Numbers and ...... 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 , 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 of a set, denoted ∈. This is the very essence of set theory along with logical expressions: “and” ... “or” ... “then” ... “” and quantifiers: “for all” and “there exists”. Our intuition supplies that we want to be included in the theory. Set , creation of new sets from existing sets, even the need to be defined by or derived from our set of axioms. Many have worked and laid the foundations of set theory. ’s ideas about infinity were radical in his time, but now are widely accepted. The axioms that we will focus on were developed by in 1908 and additional contributions were made by 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 {1,2,3,...}, 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 {...,-3,-2,-1,0,1,2,3,...}. The question also arose, “what happens when we equally divide five pieces of bread among three people?” The answer leads to another set: an divided by an integer - the set of rational numbers. It was assumed that now every had been defined, until somewhere around the time of Pythagoras when it was shown that the root of two could not be written as a but it was known to exist, so the set of irrational numbers was defined. With the of the set of rational numbers with the set of irrational numbers, all numbers were defined on a number , and the set of real numbers was first described. Hence we will call the real numbers: the . Georg Cantor published a 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 ; he believed in the existence of infinite sets that shared characteristics with finite sets and transfinite numbers that act like ordinary 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 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 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 . 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 of infinities which gives us the that we use today: aleph nought – ℵ0 representing the cardinality of natural numbers and 2ℵ0 representing the cardinality of the continuum [Wapner, 2005, p.12]. What is now known as the , 2ℵ0 =

ℵ1, 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 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 ; 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. 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. listed the Continuum Hypothesis first on his list of twenty-three unsolved problems in 1900 at the congress in Paris. The question, whether there is a set whose cardinality is strictly between that of the integers and that of the real numbers, took over thirty years and the work of two men, Kurt G¨odeland , to solve. G¨odelpublished a paper in 1931 which contained two results that shifted mathematicians’ thoughts about axiomatic systems. His first theorem showed that any axiomatic theory sufficiently defined to 5 discuss the natural numbers is incomplete, that there would always be statements that could be neither proved nor disproved. His second theorem proves that for certain systems of axioms, is not provable within the system itself [Wapner, 2005, 30]. In 1940, G¨odelpublished that the Axiom of Choice and the Continuum Hypothesis were consistent with Zermelo-Fraenkel (ZF) axioms (see chapter two for the formal of the theorems). He tried to prove that they were independent of the ZF axioms (i.. could not be derived from them), but he could not. In 1963, Cohen invented a “technique called , where certain set theoretic models can be extended in such a way so that a given statement becomes true in the extended model.”[Wapner, 2005, 38] Using this technique, he constructed models of set theory where the Axiom of Choice and the Continuum Hypothesis were false. For his work, Paul Cohen was awarded the Fields Medal.

1.3 Russell’s Paradox

Bertrand Russell (1872-1970) was a British mathematician and philosopher who formalized a paradox that led to alarming conclusions to set theory. If set theory does not address his paradox, then we have an inconsistent system and as known from first order logic, any statement would then be a theorem. Let’s explore Russell’s Paradox. Begin by defining the set ρ to be the set of all sets which do not belong to themselves. This seems like a specific property for a collection of sets:

ρ = {x|x is a set and x∈ / x}

Now, either ρ belongs to itself or it does not belong to itself. First, suppose ρ belongs to itself, then it must be one of the sets x such that x∈ / x. Therefore ρ cannot belong to itself: meaning if ρ ∈ ρ, then ρ∈ / ρ. Now, suppose ρ does not belong to itself, then it is an x such that x∈ / x, so it does belong to itself, i.e. if ρ∈ / ρ then ρ ∈ ρ. So in either case, we reach 6 a contradiction. Therefore ρ is not a set even though it could be described by a specific property; Russell’s Paradox puts a limit on what can be called a set. This paradox arises from the self-referring nature of the initial problem. There are some classic examples such as the barber who shaves the men of the town who do not shave themselves and only those men. Who shaves the barber? Another is G¨odel’struth machine that can determine the of any system, but its own. How has set theory resolved this issue? Russell himself proposed a hierarchy of types (or levels) of sets, so that when one collected together some of the sets on one level, that collection was now in the next higher level. Cantor addressed this idea in his absolute infinite. Von Neumann’s ideas are the traditional ones used today where we have distinctions between collections of objects. In particular, collections of sets are known as classes. Certain classes are actually sets, but the others are called proper classes. By Russell’s Paradox the of all sets is a proper . Another example of a proper class is the proper class of all ordinals. All sets will also be classes, but not all classes are sets. Some collections become objects that are just “too big” to be sets. Objects like the collection of all cardinal numbers, or the universe of all sets. 7

Chapter 2

Axioms of Zermelo-Fraenkel

2.1 First Order Logic

Set theory begins with two predicates: equality, =, and set membership, ∈, and our objects are sets. Our atomic statements are a = b and x ∈ y. More complicated statements are built by combining sentences using the boolean connectives: and, or, not, implies; and by using quantifiers: universal, existential. From these we build sentences whose truth is determined by our axioms, logic statements, and which objects are allowed to be sets.. Our axioms, originally written by Zermelo and revised by Fraenkel, are relatively consistent, that is, we cannot derive a contradiction from them. Zermelo’s goal in creating an axiomatic for set theory was not only to underpin the foundations of mathematics, but to create a system that would allow work on more com- plicated structures like the well-ordering of R. For the most part, his axioms are straight forward, non-controversial statements about sets that not only allow much of standard math- ematics to be represented in set , but also provide a more rigorous basis for Cantor’s work on infinite sets [Goldrei, 1996, p. 75]. Most of the axioms in the Zermelo-Fraenkel system (usually abbreviated ZF) state what is intuitively obvious about sets as they are understood in both set theory and in other 8 mathematical disciplines. But the axioms also need to exclude the “sets” that result in Russell’s Paradox.

2.2 The Axioms of Zermelo-Fraenkel

A.1 The Axiom of Existence. There exists a set which has no elements. The Axiom of Existence says that we assume an empty set exists; this parallels Euclid’s first postulate that a exists. Formally stated: ∃x∀y(y∈ / x). Some scholars state this axiom as: a set exists, which means there exists an x such that x = x. Either way, we have a universe which has in it. Many of the other axioms assert if a set exists, a new set can be defined from the original set. But this axiom guarantees that at least one set exists. We will use the familiar notation, ∅, for the empty set.

A.2 The Axiom of . If every element of X is an element of Y and every element of Y is an element of X, then X = Y . Formally stated: X = Y ⇐⇒ ∀x ∈ X, x ∈ Y ∧ ∀y ∈ Y, y ∈ X. This axiom defines set equality and gives us a way to interpret the predicate =; two sets are equal if they contain exactly the same members. This is what we expect and now we have a procedure to determine if the statement X = Y is true or false. As a consequence of the , there is only one set which contains no elements. So it is proper to refer to it as the empty set.

Lemma 2.2.1 There is only one empty set.

Proof: Assume A and B both have no elements. Then every element of A is an element of B. The statement a ∈ A implies a ∈ B is true. Likewise, every element of B is also an element of A. Therefore, A = B by the Axiom of Extensionality.  9 Definition: Let x and y be sets. Whenever z ∈ x implies z ∈ y, then we say x is a subset of y and write x ⊆ y. When x ⊆ y and x 6= y, then we say x is a proper subset of y, x ⊂ y. Also when x ⊆ y and y ⊆ x, then x = y by the Axiom of Extensionality.

Theorem 2.2.2 Set equality is an equivalence , i.e. = is reflexive, symmetric and transitive.

Proof: Suppose X 6= X, but then there exists x ∈ X such that x∈ / X, contradicting the Axiom of Extensionality. Therefore, X = X and = is reflexive. Suppose X = Y . Then by the Axiom of Extensionality, every element of X is an element of Y , so X ⊆ Y ; also every element of Y is an element of X, so Y ⊆ X. Therefore, Y = X and = is symmetric. Suppose X = Y and Y = Z. For all w ∈ X, w ∈ Y and w ∈ Z so X ⊆ Y and X ⊆ Z. For all w ∈ Z, w ∈ Y and w ∈ X, so Z ⊆ Y and Z ⊆ X. Because X ⊆ Z and Z ⊆ X, then X = Z

and = is transitive. 

Definition:A property P for all elements of a set x is usually defined as a P : x that is true whenever the property holds: the set {x|P(x) = true} where P is its .

A.3 The of Comprehension. Let P(?) be a property of x. For any set A, there is a set B such that x ∈ B if and only if x ∈ A and P(?) holds. This axiom, also known as the Axiom of Separation, is necessarily limited so as not to create self-referring sets. First there must be a set A. After that is established, we can talk about elements of A that share some property. If there is a set x, then of x can be defined or “separated out” based on a property P. By stating first that a set exists we can avoid Russell’s paradox which first states a property and then constructs a “set” whose members satisfy that property. 10 A.4 The . For any sets A and B, there is a set C such that x ∈ C if and only if x = A or x = B. The axiom of pairing provides a way to construct a new set from previously existing sets, a potentially “larger” set than either A or B. Again, note that A and B are assumed to exist already. This introduces the unordered set {A, B}. The set {x, x} = {x} because they contain the same elements so by the Axiom of Ex- tensionality they are equal. By convention we call the set {x} a set. In standard set theory notation, if we want to refer to an (A, B), first we would create the singleton set {A} and the unordered set {A, B} by the axiom of the pair. Then by apply- ing the axiom again, we would pair these two sets together {{A}, {A, B}} and we use the notation (A, B).

Lemma 2.2.3 Given sets x and y there is a unique set z whose elements are x and y.

Proof: By the Axiom of Pairing there is at least one set z whose elements are precisely x and y. Let z0 be another such set. Then any element of z0, x or y, is an element of z. So by

0 the axiom of extensionality z = z , so z is unique. 

Definition: We will define the ordered set (a,b) to be the set {{a}, {a, b}} and say a is the first coordinate and b is the second coordinate.

Lemma 2.2.4 (a, b) = (a0, b0) if and only if a = a0 and b = b0.

Proof: ⇐ If a = a0 and b = b0, then

(a, b) = {{a}, {a, b}} = {{a0}, {a0, b0}} = (a0, b0)

⇒ Assume that(a, b) = (a0, b0), so then {{a}, {a, b}} = {{a0}, {a0, b0}}. If a 6= b, {a} = {a0} and {a, b} = {a0, b0}. So by the first, a = a0 and then {a, b} = {a, b0} implies b = b0 as desired. If a = b, {{a}, {a, a}} = {{a}}. So {a} = {a0}, {a} = {a0, b0}, and we get a = a0 = b0, so

0 0 a = a and b = b is true in this case also.  11 A.5 The . For any set S, there exists a set U such that x ∈ U if and only if x ∈ A for some A ∈ S. This set is called the union of the members of S and denoted S S. If the set S contains members a, b, c, ... then just the members of a, b, c, ... are contained in S S; the order of terms is not important. This axiom allows for sets with more than just two elements which is the limit with the Axiom of Pairing. As a further example of the comprehension principle, now that we know that for sets a and b, a ∪ b is a set. Consider the following:

a ∩ b = {z|(z ∈ a ∧ z ∈ b)}.

The property stated in formal for the z is acceptable by the Axiom of Comprehen- sion, but to use it, z must first be an element of a suitable set. But that is not a problem since a ∪ b is a known set, and it may be used now to define the of two sets. Definition: The intersection of two sets a and b, written as a ∩ b, is the set

{z ∈ a ∪ b|(z ∈ a ∧ z ∈ b)}.

A.6 The Axiom of . For any set S, there exists a set P such that X ∈ P if and only if X ⊆ S. The power set axiom says the collection of all subsets of a given set is again a set. This is a standard tool used in mathematics, but we need the axiom to ensure that the power set is in fact a set. We notate the power set of S as P(S). A.7 The Axiom of Infinity. An inductive set exists. Definition: Given a set x. the successor of x, written as x+, is the set x+ = x ∪ {x}. Definition: The set I is said to be an inductive set if ∅ ∈ I and x ∪ {x} ∈ I whenever x ∈ I. 12 This axiom guarantees an inductive set, so we can define N, our first infinite set. We will use this axiom when we develop natural numbers in the next chapter.

A.8 The Axiom Schema of Replacement. Let P(x, y) be a property such that for every x there is a unique y for which P(x, y) holds. For every A there is a B such that for every x ∈ A there is y ∈ B for which P(x, y) holds. This axiom can be used even if P is a class function. As long as A, the domain, is a set then the of A under P , B will be a set.

A.9 The Axiom of Foundation. For all x, there exists y such that y ∈ x then there exists y such that y ∈ x and there is no z such that z ∈ x and x ∈ y. Another version of the Axiom of Foundation is that every set is well-founded, meaning that every set contains an ∈-minimal element where if y ∈ x, y is ∈-minimal in x if there is no z ∈ x such that z ∈ y. Therefore, no set is an element of itself and no infinite descending exists.

Theorem 2.2.5 Let z be any set with x = z, then z∈ / z.

Proof: As z is a set, so is {z}. By the Axiom of Foundation there is an element y of {z} such that {z} ∩ y = ∅. As the only element of {z} is z, this y can only be z, so that {z} ∩ z = ∅. This means that the element z of the {z} is not in the other set involved in this intersection; therefore, z∈ / z as required.  From the axioms many theorems and lemmas are developed. The natural numbers, rational numbers, integers, and the real numbers are constructed as sets, as well as the mathematical operations that we use in the systems. Two of the tools that are used again and again are the induction theorem and the recursive theorem. 13 2.3 The Recursive Theorem

Consider the following functions with the same {1, 4, 9, 16, 25, 36, ...}.

s : N → N

s0 = 1

2 sn+1 = n

This is an explicit function. Given any n, s may be quickly evaluated. But this next one involves something else: f : N → N

f0 = 1

fn+1 = fn + 2n + 1

Notice the initial condition is followed by a recursive formula. The value of n + 1 may only be computed after the value of n is known. In order to prove the Recursive Theorem we need to set the groundwork.

Definition: An m-step computation t is a finite sequence that is of length (m + 1) where t0 = k and tk+1 = g(tk, k) for all k < m, k ≥ 0. Definition: If f(x) = g(x) for all x ∈ dom f T dom g, then f and g are compatible functions. The set of functions with this property is a compatible system of pairwise functions.

Lemma 2.3.1 If H is a compatible system of functions, then S H is a function that extends all h ∈ H.

Proof: Let (a, b1) ∈ H and (a, b2) ∈ H. Then there are functions h1, h2 ∈ H such that 14

(a, b1) ∈ h1 and (a, b2) ∈ h2. But h1 and h2 are compatible and a ∈ dom h1∩ dom h2, so

b1 = h1(a) = h2(a) = b2.

Note that x ∈ dom (S H) if and only if x ∈dom h for some h ∈ H strictly from the definition S of compatibility. Therefore H extends all h ∈ H. 

Theorem 2.3.2 The Recursive Theorem: For any set A, any a ∈ A and any function g : A × N −→ A, there exists a unique infinite sequence f : N −→ A such that

(a) f0 = a

(b) fn+1 = g(fn, n) for all n ∈ N.

There are five statements that need to be proved.

1. f exists.

2. f is a function.

3. dom f = N and ran f ⊆ A.

4. f satisfies (a) and (b).

5. f is unique.

Proof of the existence of f: Let t be an m-step computation based on a and g if t0 = a for

all k such that 0 ≤ k < m, tk+1 = g(tk, k). Note that t ⊆ N × A. Define F = {t ∈ P(N × A|t is an m-step computation for some m}. Here we used both the Axiom of the Power Set and the Axiom Schema of Comprehension. Define f = S F . Proof f is a function: It is enough to show that the system of functions F is compatible because then F can be pieced together to form a single function f, which extends all F . Let t, u ∈ F , dom t = n ∈ N, dom u = m ∈ N. Assume n ≤ m, then n ∈ m. By using induction we can show that tk = uk. Clearly t0 = a = u0. Next let k be such that k + 1 < n 15

and assume tk = uk. Then tk+1 = g(tk, k) = g(uk, k) = uk+1. Therefore tk = uk for all k < n. So F is a system of compatible functions and therefore, f = S F is the therefore f is a function. Proof dom f = N and ran f ⊆ A: We claim that dom f = N and ran f ⊆ A. By the manner in which f is defined it is obvious that dom f ⊆ N and ran f ⊆ A. To show dom f = N it is enough to prove that for each n ∈ N there is an n-step computation t. By induction: clearly t = {(0, a)} is a 0-step computation. Assume t is an n-step computation. Then the function

+ + t on (n + 1) + 1 is an (n + 1)-step computation {t = t ∪ {(n + 1, g(tn, n))}} because

+ tk = tk if k ≤ n

+ tn+1 = g(tn, n).

Therefore, each n ∈ N is the domain of some computation t ∈ F so N ⊆ S t ∈ F . So N = dom t = dom f.

Proof f satisfies (a) and (b): Clearly, f0 = a for all t ∈ F . To show fn+1 = g(fn, n) for any n ∈ N, let t be a (n + 1)-step computation, then tk = fk for all k ∈ dom t, so fn+1 = tn+1 = g(fn, n). Proof f is unique: Let h be another function such that h : N −→ A so

• h0 = a

• hn+1 = g(hn, n) for all n ∈ N.

We need to show that fn = hn for all n. Using induction f0 = a = h0. Assume fn = hn and by induction then fn+1 = g(fn, n) = g(hn, n) = hn+1. Therefore f = h as claimed.

Since we have proved the five conditions, we have proved the recursive theorem.  16

Chapter 3

Development of Numbers

Sets of numbers are not difficult to describe with language. The sets of natural numbers, integers, rational numbers, and the real numbers can be described by examples. It is desirable to describe these sets in set theory terms. We will need a defined order on the sets with operations that fit our intuition and the mathematics that we do on them.

3.1 Natural Numbers and Integers

Our goal is to order the natural numbers by ∈-relation which is one of the two basic operators in set theory. Remember that given a set X, the successor of x, written as x+, is the set x+ = x ∪ {x} and by the Axiom of Foundation x 6= x+. In the construction of the natural numbers, we are primarily bringing two axioms to- gether: the empty set exists and an inductive set exists. Since the empty set exists, we can talk about the successor of the empty set, which is the set containing the empty set or 17 ∅+ = ∅ ∪ {∅} = {∅}.

Then the successor of this set:

∅++ = (∅+)+ = {∅} ∪ {{∅}} = {∅}+ = {∅, {∅}}.

And again:

∅+++ = (∅++)+ = {∅, {∅}} ∪ {{∅, {∅}}} = {∅, {∅}}+ = {∅, {∅}, {∅, {∅, {∅}}}}.

Immediately, we see that x ⊂ x+ for all x and also that x ∈ x+. Since we have an inductive set, we can continue this process infinitely.

Lemma 3.1.1 x+ has one more element than x.

Proof: By definition x+ = x ∪ {x} so x ⊂ x+, but it appears that x ∈ x+ (the single element {x} in the union). For this to be an extra element, we need to show that x∈ / x. If x ∈ x, then {x} ⊂ x implies x ∪ {x} = x. But then x = x+ which contradicts the Axiom of Foundation.

+ + Because x ∈ x and x∈ / x, there is one more element in x than in x. 

Definition: The set X is transitive if for all y ∈ X, y ⊆ X. Notice that this means if z ∈ y and y ∈ X then z ∈ X.

Lemma 3.1.2 A set X is transitive if and only if X ⊆ P(X).

Proof: ⇒ Let X be transitive. Then for all y ∈ X, y ⊆ X, so y ∈ P(X) and since X ⊆ X, we can conclude X ⊆ P(X). 

⇐ Suppose X ⊆ P(X). X ∈ P(X) and for all y ∈ P(X), y ⊆ X. For any y0 ∈ X, y0 ∈

P(X), so y0 ⊆ X as desired and X is transitive. 

Lemma 3.1.3 If {yi}i ∈ I is a family of inductive sets, then their intersection is an induc- tive set. 18

Proof: Notice that since ∩{yi} ∈ yi the intersection is a set. Next, since for all i ∈ I, ∅ ∈

+ + yi, ∅ ∈ ∩{yi}i ∈ I. Whenever x ∈ y ∩ {yi}i ∈ I then x ∈ {yi} for all i ∈ I, so x ∈ ∩{yi}, so we fulfill the definition of an inductive set.  Definition: The set of Natural Numbers is the intersection of all inductive sets, that is:

\ N = {I|I is an inductive set}.

Each natural number is the set of all its predecessors. Combining successors and the familiar expressions of natural numbers, we see the relation between them.

1 =0+ = 0 ∪ {0} = {0}

2 =1+ = 1 ∪ {1} = {0, 1}

3 =2+ = 2 ∪ {2} = {0, 1, 2}

...

n =(n − 1)+ = n − 1 ∪ {n − 1} = {0, 1, 2, . . . , n − 1}

n+ =n ∪ {n} = {0, 1, 2, . . . , n − 1, n}

There is one element in 1, two elements in 2,three elements in 3,. . . , n elements in n, and n + 1 elements in n+.

Theorem 3.1.4 N is ∈ -transitive. For all m, n, p ∈ N if m ∈ n and n ∈ p, then m ∈ p.

Proof: Throughout this solution, we will denote ∅ as 0, the set which contains no elements. We will prove by induction on p for fixed m and n. Define A ⊆ N such that A = {p ∈ N : if m ∈ n ∈ p, then m ∈ p}. We will show A is an inductive set so A = N.  Since m ∈ n ∈ 0 is false by our assumption, then m ∈ 0 is true. So 0 ∈ A. Let p ∈ A; we need to show that p+ ∈ A. So suppose m ∈ n ∈ p+. Since p+ = p ∪ {p} and n ∈ p+, we have two cases: either n ∈ p or n ∈ {p} i.e. n = p. In the first case n ∈ p, we have m ∈ n ∈ p, and p ∈ A, so m ∈ p. But p ⊆ p+ so that m ∈ p+ as needed. In the second case, n = p and 19 we are given m ∈ p, so m ∈ p. Just as before we can conclude that m ∈ p+. Together we have shown m ∈ p+ and so p+ ∈ A; therefore, A is inductive, so A = N which shows that ∈ is transitive on N.

Lemma 3.1.5 ∈ is irreflexive on N, i.e. for all n ∈ N, n∈ / n.

Proof: Define A, a subset of N, as A = {n ∈ N : n∈ / n}. As 0 = ∅, which contains no elements, 0 ∈/ 0, so 0 ∈ A. Now suppose n ∈ A; we need to show n+ ∈ A. By means of contradiction, suppose n+ ∈ n+. Since n+ = n ∪ {n}, then either n+ ∈ n or n+ = n. First if n+ ∈ n, and we know that n ∈ n ∪ {n} ∈ n+, we can conclude n ∈ n+ ∈ n. But ∈ is transitive implying n ∈ n, but this contradicts n ∈ A. Second if n+ = n, then we have n ∈ n ∪ {n} = n+ = n showing n ∈ n contradicting n ∈ A. Since we have a contradiction in

+ + + both cases, n ∈/ n . So n ∈ A, A is inductive and A = N. So ∈ is irreflexive on N. 

Lemma 3.1.6 That for all m, n ∈ N, exactly one of the following holds: m ∈ n, m = n, or n ∈ m.

Proof: By induction on n for a fixed m, we will show that m ∈ n, m = n, or n ∈ m for all n ∈ N. For n = 0, we cannot have m ∈ 0 = ∅. If m happens to be 0, we are done: m = 0 = n. If m 6= 0, 0 ∈ m as a natural number is the set of its predecessors. So the result holds for n = 0. Now suppose our induction hypothesis holds for n. We need to show that m ∈ n+, m = n+, or n+ ∈ m. In the case m ∈ n, then as n ⊂ n+, we have m ∈ n+. In the case m = n, then as n ∈ n+, we have m ∈ n+. Lastly, in the case n ∈ m means that m 6= ∅ and that m = k+ for some k ∈ N. So n ∈ m gives n ∈ k ∪ {k}, so either n ∈ k or n = k. If n ∈ k then n+ ∈ k+, i.e. n+ ∈ m. If n = k, then n+ = k+, i.e. n+ ∈ m. In all the cases, we

+ + + have m ∈ n , m = n , or n ∈ m as required. 

The common definition of the integers is that they are the set of natural numbers in union with their additive inverses. We can use the union of sets and our definition of natural numbers to define the integers in set theory. Once we define an order on natural numbers then we can define operations. 20 Consider the elements of 2 x N ∈ N x N where 2 = {0, 1}. We will designate the elements of S as negative elements using S = {(0, a)|a ∈ N} and designate the elements of T as positive elements and zero using T = {(1, b)|b ∈ N}. Let I = S\{(0, 0)} ∪ T . Now we can define an order on the elements of I, n = (a, b), m = (c, d) such that

  a < c;   n < m if a = c = 0 and d < b;    a = c = 1 and b < d.

and

n = m if {a = c and b = d.

In the set of integers Z evert element has both a successor and a precessor. Let f : I −→ ZS\(0, 0) 3 s = (0, a) 7−→ −a, T 3 t = (1, b) 7−→ b; f is a that does that.

Z = {..., −3, −2, −1, 0, 1, 2, 3,...}

3.2 Rational Numbers

The set of rational numbers is our first dense linearly ordered set. Now that the integers are defined, we will consider elements of Z × Z − {0}. One of the issues that must be considered is the when are equal, so we will define nonsymmetric equivalence classes. Let a, b, c, d, e, f, g, p, q ∈ Z. Let (p, q) ∈ Z × Z such that q 6= 0 and let f : Z × Z −→ Q such that (p, q) 7−→ p/q. We will consider (a, b) and (c, d) to be representatives of the same if ad = bc. 1. This relation is reflexive because (a, b) = (a, b) since ab = ba. 2. This relation is transitive because (a, b) = (c, d) and (c, d) = (e, f) implies that (a, b) = (e, f). 21

ad = bc and cf = de so c = de/f by substitution then acf/e = bc =⇒ af/e = b

=⇒ af = be so our conclusion holds (a, b) = (e, f).

Let Q0 = Z × (Z − {0}) and let Q be the set of equivalence classes on Q0, which we will define as the set of rational numbers. We need to define an order on the rationals. For r = (a, b) and s = (c, d), we will say

  r < 0/1 < s;    r, s > 0/1 then choose elements of their equivalence class r < s if  such that r = (a, b) = (e, g) and s = (c, d) = (f, g) if | e |<| f |;    r, s < 0/1 choose as above and if | f |<| e | .

Theorem 3.2.1 The set, Z, of integers and the set, Q, of rational are countable.

Proof: Z is infinite because N ⊆ Z. Let i : Z −→ Q0 −→ i(a) = (a, 1) which is a one-to-one mapping. So it is enough to prove that Q is countable. The set of fractions Q0 is countable,

0 and since Q ⊆ Q then Q is countable. 

Definition: An ordered set (X, <) is dense if it has at least two elements and if for all a, b ∈ X, a < b implies that there exists x ∈ X such that a < x < b.

Theorem 3.2.2 Q is dense.

Proof: Let r, s be rational such that r < s; assume that r = a/b and s = c/d and that b > 0 and d > 0. Let ad + bc x = 2bd that is x = (r + s)/2 then r < x < s as desired.  22 3.3 Real Numbers

The real numbers have many complexities that need to be addressed. We know of their existence; there is no rational number such that x2 = 2, so our set of rational numbers has gaps or missing elements. The real numbers need to be complete i.e. having no gaps. To help us understand more about the real numbers, we will find the next definitions useful. Definition:A cut is a pair (L, U) of sets such that

1. L and U are disjoint nonempty subsets of P and L ∪ U = P .

2. If l ∈ L and u ∈ U, then l < u.

Definition: A cut (L, U) is a gap if L does not have a greatest element and U does not have a least element. Definition: A cut (L, U) is a if L does not have a greatest element. There are two types of Dedekind cuts (L, U):

1. Those where R = {x ∈ Q|x ≥ q} for some q ∈ Q; we denote (L, U) = [q].

2. Gaps.

Richard Dedekind’s (1831-1916) definition of the real numbers is based on the picture of how an , r, sits among the rational numbers on the . There is a rational number to the left and right of r; r cuts Q into two subsets L, the rational numbers less than r, and U, the rational numbers greater than r. So without presupposing that an irrational number exists and using today’s language, the set of real numbers R is the set of all Dedekind cuts. The completion of the rational numbers is the main step in developing the real numbers. This is a special case of the general construction of a completion of an arbitrary dense linearly ordered set without endpoints.

Theorem 3.3.1 Let (P, <) be a dense linearly ordered set without endpoints. Then there exists an order preserving injection of P into C as such P ⊆ C and: 23 (a) P is dense in C.

(b) C does not have endpoints.

Moreover, this complete linearly ordered set (C, ≺) is unique up to over P . If (C∗, ≺ ∗) is a complete linearly ordered set which satisfies (a)-(b), then there is an isomor- phism h between (C, ≺) and (C∗, ≺ ∗) such that h(x) = x for each x ∈ P. The linearly ordered set (C, ≺) is called the completion of (P, <).

Proof of Uniqueness of Completion: Let (C, ≺) and (C∗, ≺ ∗) be two complete linearly ordered sets satisfying (a)-(b). We need to show that there is an isomorphism h of C on

C∗ such that h(x) = x for each x ∈ P. If c ∈ C, let SC = {p ∈ P |p ≤ c} and let

SC∗ = {p ∈ P |p ≤ ∗c∗} for c∗ ∈ C ∗ . If S is a nonempty subset of P bounded from above, let supS be the supremum of S in (C, ≺) and sup*S be the supremum of S in (C∗, ≺ ∗).

Note that supSC = c, sup*SC∗ = c∗. We define the mapping h as h(c) =sup*SC . Clearly h is a mapping of C into C∗; to show that h is onto, let c∗ ∈ C∗ be arbitrary. Then c∗ =sup*SC∗ and if we let c = supC∗, then SC = SC∗ and c∗ = h(c). If c ≺ d, then because P is dense in C, there is p ∈ P such that c ≺ p ≺ d implying that sup*Sc ≺ ∗p ≺ ∗ sup*d and h(c) ≺ ∗h(d).

Finally, if x ∈ P , then x = supSx = sup*Sx and so h(x) = x. Proof of the existence of completion: Consider the set C of all Dedekind cuts (A, B) in (P, <) and order C as follows:

(A, B)  (A0,B0) if and only if A ⊆ A0.

(a’) P 0 is dense in (C, ≺). Let c, d ∈ C be such that c ≺ d with c = (A, B) and d = (A0,B0) and A ⊂ A0. Let p ∈ P be such that p ∈ A0 and p∈ / A. Moreover, we can assume that p is not the least element of B. Then (A, B) ≺ [p] ≺ (A0,B0) and so P 0 is dense in C.

(b’) C does not have endpoints. If (A, B) ∈ C, then there is p ∈ B that is not the least element of B, and we have (A, B) ≺ [p]. Hence C does not have a greatest element 24 and in a similar manner it may be shown that C does not have a least element.

To show that C is complete, let S be a nonempty subset of C, bounded from above. There- fore, there is (A0,B0) ∈ C such that A ⊆ A0 whenever (A, B) ∈ S. To find the supremum of S, let [ \ AS = {A|(A, B) ∈ S} and BS = P − AS = {B|(A, B) ∈ S}.

(AS,BS) is a Dedekind cut because BS is nonempty since B0 ⊆ BS and AS does not have a

0 greatest element since none of the A s does. Since AS ⊇ A for each (A, B) ∈ S, (AS,BS) is ˆ ˆ an upper bound of S; let us show that (AS,BS) is the least upper bound of S. If (A, B) is ˆ S any upper bound of S, then A ⊆ A for all (A, B) ∈ S; and consequently, AS = {A|(A, B) ∈ ˆ ˆ ˆ S} ⊆ A, so (AS,BS)  (A, B). Thus (AS,BS) is the supremum of S. 25

Chapter 4

Cardinal Numbers

Georg Cantor is credited with formalizing the concepts of ordinal and cardinal numbers. Ordinal numbers are associated with ordering such as: he finished the race in third place while cardinal numbers are used to represent the cardinality (size) of a set as mentioned before. In finite cases there is little distinction between cardinal numbers and ordinal numbers. A counting number could be used to represent the size of a set or the position of an element in a sequence. For every finite position described in a sequence, we can construct a set of equal size. This is not true for infinite where the distinctions between ordinals and cardinals are necessary.

4.1 Cardinal Numbers

The cardinality of a finite set is a natural number, that is, the number of elements in the set, and the transfinite cardinal numbers describe the size of infinite sets. We use the notation

ℵ0 which is read as aleph nought to represent the size on the set of natural numbers, and c to represent the size of the continuum or set of real numbers. There are other transfinite numbers because Cantor’s Theorem which will be proved shortly says that the cardinality of any set A is less than the cardinality of the P(A). Two sets, X and Y , have the same cardinality or are equipotent if there is a one-to-one mapping f of X onto Y and is notated 26 |X| = |Y |. If |X| ≤ |Y |, then there exists a one-to-one mapping f of X into Y . Definition: A set X is said to be finite if there is a bijection f : n −→ X for some natural number n. X is said to be infinite if there is no such bijection, for any n.

Theorem 4.1.1 A is equipotent to A : |A| = |A|.

Proof: IdA is a one-to-one mapping of A onto A. 

Theorem 4.1.2 If A is equipotent to B, then B is equipotent to A: if |A| = |B|, then |B| = |A|.

Proof: If f is a one-to-one mapping of A onto B, f −1 is a one-to-one mapping of B onto A.



Theorem 4.1.3 If A is equipotent to B and B is equipotent to C, then A is equipotent to C: if |A| = |B| and |B| = |C|, then |A| = |C|.

Proof: If f is a one-to-one mapping of A onto B and g is a one-to-one mapping of B onto

C, then f ◦ g is a one-to-one mapping of A onto C.  The set of cardinal numbers is a linearly ordered set; any two elements are comparable by ≤. With the above properties plus |A| ≤ |B| we will establish the necessary basis, but to show asymmetry, we will need the Cantor-Bernstein Theorem.

Theorem 4.1.4 The Cantor-Bernstein Theorem: If |X| ≤ |Y | and |Y | ≤ |X|, then |X| = |Y |.

Proof: In this proof we will us R[A] to denote the image of set A under relation R, which is the set of all y from the range of R related in R to some element of A. Let f be a one-to-one function that maps X into Y and g be a one-to-one functions that maps Y into X. To show |X| = |Y | we need a one to function that X onto Y . Lets first apply f and then g; g ◦ f maps X into X and is one-to-one with g[f[X]] ⊆ g[Y ] ⊆ X. Moreover since f and g are one-to-one, we have |X| = |g[f[X]]|. This follows from the following lemma. 27

Figure 4.1: Diagram to assist in visualizing proof.

Lemma 4.1.5 If A1 ⊆ B ⊆ A and |A1| = |A|, then |B| = |A|.

Proof: Let f be a one-to-one mapping of A onto A1. By we define two sequence of sets:

A0,A1, ..., An, ... and B0,B1, ..., Bn, ...

In our diagram the An’s are squares and the Bn’s are disks. Let A0 = A and B0 = B and for each n,

(?))An+1 = f[An] and Bn+1 = f[Bn].

Since A0 ⊇ B0 ⊇ A1, it follows from ? by induction that for each n, An ⊇ BnAn+1. For each 28 S∞ n, we let Cn = An − Bn and define C = n=0 Cn,D = A − C.C is the shaded part of the S∞ diagram. By ?, we have f[Cn] = Cn+1 and so f[Cn] = n=0 Cn. Now we are ready to define a one-to-one mappingg ˆ of A onto B:

  (x) if x ∈ C, gˆ(x) =  g(x) if x ∈ D.

Bothg ˆ  C andg ˆ  D are one-to-one functions and their ranges are disjoint. Thusg ˆ is a one-to-one function and maps A onto f[C] ∪ D = B. 

Theorem 4.1.6 Cantor’s Theorem. For any set A, we have |A| ≤ |P(A)|.

Proof: We consider the function g : A → P(A) which sends each element of a ∈ A into the set consisting of the singleton {a} ∈ P(A), so g(a) = {a} is one-to-one; therefore, |A| ≤ |P(A). We need to show that A cannot be mapped onto P(A) for any set A. So let f be a mapping of some set A into P(A). We claim that the following set B is not in the range of f:

B = {a ∈ A|a∈ / f(a)}.

If B = f(z) for some z ∈ A, then either z ∈ B or z∈ / B. If z ∈ B, then z∈ / f(z) = B, and if z∈ / B then z ∈ f(z) = B, a contradiction in both cases. Thus there is no mapping of A onto P(A) and P(A) has a greater cardinality than A.

Is there a β between ℵ0 and c? The Continuum Hypothesis states that there is no cardinal number β such that ℵ0 < β < c. In 1963, Paul Cohen showed that the Continuum Hypothesis is independent of our ZF axioms in a similar sense that Euclid’s Fifth Postulate on parallel lines in independent of the other axioms in geometry. The set P = {1, 2, 3, ...} is the set of counting numbers or positive integers. Using the set P we will apply the following definitions. Definition: A set D is said to be denumerable or countably infinite if D has the same cardinality as P. 29 Definition: A set is countable if it is finite or denumerable and a set is nondenumerable or uncountably infinite if it is not countable. From these definitions there are easily seen conclusions. (a) Any infinite sequence of distinct elements of the form: a1, a2, a3, ... is countable infinite because is basically a func-

tion f(n) = an whose domain is P. If an are distinct the function is one-to-one and onto. Examples: {1, 1/2, 1/3,..., 1/n, . . .}

{1, 4, 9, . . . , n2,...}

{(1, 1), (2, 1/2), (3, 1/3),..., (n, 1/n),...}

(b) The product set P × P can be written as:

{(1, 1), (2, 1), (1, 2), (1, 3), (2, 2),...}

The sequence can be determined by following the arrows in the diagram and therefor P × P is countably infinite.

Figure 4.2: Cantor’s diagonalizational argument.

(c) Recall that the natural numbers or nonnegative integers are N = {0, 1, 2, 3,...} = 30 P ∪ {0}. Now each a ∈ P can be written uniquely in the form

a = 2r(2s + 1) where r, s ∈ N. The function f : P → N × N defined by f(a) = (r, s) is one-to-one and onto. Thus N×N has the same cardinality as P and is denumerable and note that P×P ⊂ N×N. Other works have included these very useful theorems and corollary.

Theorem 4.1.7 Every infinite set contains a subset which is denumerable.

Theorem 4.1.8 A subset of a denumerable set is finite or denumerable.

Corollary 4.1.9 A subset of a is countable.

Theorem 4.1.10 A countable union of countable sets is countable.

This next theorem tells us that there are just as many rational numbers as there are positive integers and that Q has the same cardinality as P.

Theorem 4.1.11 The set Q of rational numbers is denumerable.

Proof: Let Q+ be the set of positive rational numbers and Q− be the set of negative rational numbers. Then Q = Q+ ∪ {0} ∪ Q−. Let f : Q → P × P be defined by

f(p/q) = (p, q) where p/q is any element of Q+ expressed as the ratio of two relatively positive integers. Then f is one-to-one and so Q+ has the same cardinality of a subset of P × P. By the preceeding examples

Theorem 4.1.12 The cardinality of the continuum is equal to the powerset of the natural numbers: c = 2ℵ0 . 31 Proof: Let R be the set of real numbers and let P(Q) be the power set of the set of rational numbers. Let the function f : R → P(Q) be defined by

f(a) = {x|x ∈ Q, x < a}.

That is, f maps each a into the set of rational numbers less than a. We shall show that f is one-to-one. Let a, b ∈ R, a 6= b and say a < b. Since Q is dense in R, then there exists a rational number r such that

a < r < b.

Then r ∈ f(b) and r∈ / f(a); so f(b) 6= f(a). Therefore, f is one-to-one and |R ≤ |P(Q)|.

ℵ0 Since |R| = c and |Q| = ℵ0, we have c ≤ 2 . Now let C(P) be the family of characteristic functions on P, and f : P → {0, 1} which is equivalent to P(P). Let I = [0, 1], the closed unit and let the function F : C(P) → I be defined

F (f) = 0.f(1)f(2)f(3) ... an infinite decimal consisting of zeros and/or ones. Suppose f, g ∈ C(P) and f 6= g. Then the decimals would be different and so F (f) 6= F (g). According, F is one-to-one. Therefore,

PQ ≤ |C(P)| ≤ |I|.

Since Q = ℵ0 and I| = c, we have

2ℵ0 ≤ c.

Both inequalities give us

c = 2ℵ0 .

 32 4.2 Equivalence of Axioms

The Axiom of Choice, the Zermelo’s Well-Ordering Principle and Zorn’s Lemma are equiv- alent axioms. AXIOM of CHOICE: For every family F of nonempty sets, there is a function f such that f(S) ∈ S for each set S in the family F. WELL-ORDERING PRINCIPLE: Zermelo’s Theorem: Every set can be well-ordered. ZORN’S LEMMA Maximal Principle: Let (P, <) be a nonempty partially ordered set and let every chain in P have an upper bound. Then P has a maximal element. Definition An ordering of< of a set S is a well-ordering if every non-empty X ⊆ S has a least element in the ordering <. Definition A subset C of a partially ordered set (P, <) is a chain inP if C is linearly ordered by <;u is an upper bound of C if c ≤ u for each c ∈ C. For some a ∈ P, a is a maximal element if a < x for no x ∈ P .

Theorem 4.2.1 The Axiom of Choice, the Well-Ordering Principle, and Zorn’s Lemma are equivalent.

Proof: i. Axiom of Choice implies the Well-Ordering Principle. Assume the Axiom of Choice and let A be a nonempty set. Let F be the family of all nonempty subsets of

A. Let f be a for F. Let a0 = f(a). Define recursively for each ordinal

αa1 = f(A \{a0})a2 = f(A \{a0, a1})

0 aα = f(A \{aβ : β < α}) whenever the set is nonempty. Let A = {a ∈ A : ∃α ∈ ORD, a = aα}. By the Axiom of Replacement f 0(A0) is a set of ordinals and so by construction f 0(A0) is an ordinal, say τ. Now if A \ A0 6= ∅ then our recursive definition would allow us to choose

0 an aτ = f(A \{aσ : σ < τ}) - a contradiction. Therefore A = A so we have enumerated

0 A = {aα : α < τ} which is well-ordered A by the map f ii. Well-Ordering Principle implies Zorn’s Lemma. Assume every set maybe well-ordered. 33 Let (P, <) be a partially ordered set in which every chain has an upper bound. Let P =

{aβ : β < α} be an α-sequence on P so there is a bijection between P and an ordinal. Let

p0 = a0

p1 = a1

define recursively

pβ = aβγ

where βγ is the least ordinal so that a βγ is an upper bound of the chain {pβ : α < β} = B.

This construction ends when a pβ is selected which happens to be maximal or ends. The construction must end or else B \{pβ : α < β}= 6 ∅ which means B is equal to the class of ordinals contradicting B ⊆ P which has an α-sequence. iii. Zorn’s Lemma implies the Axiom of Choice. It will suffice to show that every system of nonempty sets S has some choice function. Let F be the system of all functions f for which the dom(f) ⊆ S and f(X) ∈ X holds for all X ∈ dom(f). (Examples of such functions are provided by ∅ or {(X, x)} for any x ∈ X ∈ S.

The set F is ordered by inclusion ⊆. Also F0 is a linearly ordered subset of (F, ⊆) since S either f ⊆ g or g ⊆ f holds for any f, g, ∈ F0. Let f0 = F0; f0 is a function since it extends all f ∈ F0, f0 ∈ F and f0 is an upper bound on F0 in (F, ⊆). The assumptions of Zorn’s Lemma are satisfied, so (F, ⊆) has a maximal element f. The proof is complete if the dom(f) = S. If not, select X ∈ S − dom(f) and x ∈ X. Then,

fb = f ∪ {(X, x)} ∈ F and fb ⊃ f, contradicting the maximality of f.  34

Bibliography

Joseph W Dauben. Georg Cantor and the Battle for Transfinite Set Theory. Proceedings of the 9th ACMS Conference. North-Holland Publishing Company, Westmont College, Santa Barbara, CA, internet version, journal of acms 2004 edition, 1993, Internet version 2004.

URL http://www.acmsonline.org/Dauben-Cantor.pdf.

F.R. Drake and D.Singh. Intermediate Set Theory. John Wiley & Sons, Baffins Lane, Chichester, West Sussex PO19 IUD, England, 1996. ISBN 0 471 96494 8.

Derek Goldrei. Classic Set Theory: a guided independent study. Chapman & Hall, New York, NY, 1996. ISBN 0 412 60610 0.

Karel Hrbacek and Thomas Jech. Introduction to Set Theory. Number 220 in Monographs and textbooks in pure and applied mathematics. Marcel Dekker, Inc., New York, NY, third edition, 1999. ISBN 0-8247-7915-0.

Warren Wm McGovern. Cantor’s dream. Set Theory, August 2003.

J.J. O’Connor and E.F. Robertson. Georg Ferdinand Ludwig Philipp Cantor. School of Mathematics and Statistics, Univeristy of St. Andrews, Scottland, October 1998. URL

http://www-history.mcs.st-andrews.ac.uk/Biographies/Cantor.html.

Leonard M. Wapner. The Pea and the Sun: a Mathematical Paradox. A K Peters, Ltd., 888 Worcester Street, Sute 230, Wellesley MA 02482, 2005. ISBN 1-56881-213-2.