Pages: 26

No chapter abstract available.

Pages: 12

In this section we discuss axiomatic systems in mathematics. We explain the notions of “primitive concepts” and “axioms”. We declare as primitive concepts of set theory the words “class”, “set” and “belong to”. These will be the only primitive concepts in our system. We then present and briefly discuss the fundamental Zermelo-Fraenkel axioms of set theory.

Pages: 12

In this section we define the symbols “=” and “ https://www.w3.org/1998/Math/MathML" display="inline"> ⊆ https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math2_1.tif"/> ” and discuss Axioms A1 to A5. We show how Axioms A1 and A2 are used to construct classes. Axioms A3 to A5 are used as tools to construct sets. We distinguish between classes, sets and elements by exhibiting a class which is not an element and a class which is not a set. We also show that all sets are elements. We introduce the concept of “power set of a set” as a set constructing tool.

Pages: 20

No chapter abstract available.

Pages: 9

In this section we define operations on classes that will allow us to better see how classes relate to each other. The concept of unions, intersections and complements of classes and sets are defined. Axiom A6 (Axiom of union) is discussed. This axiom is used to determine when the union of sets is a set. We show how Venn diagrams can serve as a guide when interpreting operations on sets. Some basic laws for the union, intersection and complements of classes and sets are presented in the form of theorems. De Morgan's laws are also stated.

Pages: 9

In this section we define the notion of “ordered pairs” in terms of classes and sets. We then define the Cartesian product of two classes (sets). We also present a few of the basic properties of Cartesian products.

Pages: 32

No chapter abstract available.

Pages: 6

In this section we define a relation R on a class (a set) S. For a relation R on a set S, we define the inverse https://www.w3.org/1998/Math/MathML" display="inline"> R − 1 https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math5_1.tif"/> of the relation R. We also define the domain and the image of R. The composition of two relations R and T is defined, and some of their properties are given.

Pages: 11

In this section we define special types of relations on a class or set: reflexive, symmetric, antisymmetric, asymmetric and transitive relations. Equivalence relations on a class S will be defined as those relations which are simultaneously reflexive, symmetric and transitive. We also define “partial order relations” and “strict order relations” on sets and provide examples of these. Equivalence relations and order relations are the two most important types of relations in the study of Set theory.

Pages: 6

In this section we show that an equivalence relation R on a set S can be used to subdivide S into pairwise disjoint subsets. The subsets when viewed together are called a partition of S. We illustrate how such partitions are obtained.

Pages: 7

In this section we continue our discussion of partitions of a set S by an equivalence relation. When a set S is partitioned by an equivalence relation R, the subsets in this partition are called “equivalence classes induced by R”. These equivalence classes when viewed together are called a “quotient set of S induced by R”. We then show that any partition of S is induced by some equivalence relation R.

Pages: 32

No chapter abstract available.

Pages: 9

In this section we formally define what we mean by a “function”. This is done using only the set-theoretic concepts developed up to now. We introduce notation to simplify the discussion and definition of these concepts. Examples of simple functions such as the identity function, the characteristic function and constant functions are presented. Given a function, f, on a set A, we define the restriction of this function on a subset, D, of A. We state what we mean by “equal functions”. The expressions “one-to-one”, “injective”, “onto”, “surjective” and “bijective functions” are also defined.

Pages: 8

In this section we define the composition, https://www.w3.org/1998/Math/MathML" display="inline"> g ∘ f https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math10_1.tif"/> , of two functions https://www.w3.org/1998/Math/MathML" display="inline"> f : A → B https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math10_2.tif"/> and https://www.w3.org/1998/Math/MathML" display="inline"> g : B → C https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math10_3.tif"/> . We view “composition of functions” as an operation “ https://www.w3.org/1998/Math/MathML" display="inline"> ∘ https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math10_4.tif"/> ” on two functions f and g. From this perspective we then discuss the main properties of composition of functions (such as non-commutativity and associativity). It is in this particular context that we describe the identity function and the inverse of a function. We also define the concept of “invertible function”.

Pages: 6

Suppose we are given a function https://www.w3.org/1998/Math/MathML" display="inline"> f : A → B https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math11_1.tif"/> . In this section we “elevate” this function so that it acts on https://www.w3.org/1998/Math/MathML" display="inline"> P ( A ) https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math11_2.tif"/> , mapping its elements to elements of https://www.w3.org/1998/Math/MathML" display="inline"> P ( B ) https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math11_3.tif"/> according to the rule determined by f. This provides a mechanism by which we can study the image of sets under the function f. We also define the set-valued inverse of a function, https://www.w3.org/1998/Math/MathML" display="inline"> f ← https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math11_4.tif"/> , and provide some examples. We also show how such functions act on unions and intersections of sets.

Pages: 7

In this section we show how a function https://www.w3.org/1998/Math/MathML" display="inline"> f : S → T https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math12_1.tif"/> partitions its domain S. This partition induces an equivalence relation R f on S which in turn leads to the quotient set https://www.w3.org/1998/Math/MathML" display="inline"> S / R f https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math12_2.tif"/> . We then present a theorem which shows how any function can be expressed as the composition of two functions, neither of which is f itself or the identity function.

Pages: 62

No chapter abstract available.

Pages: 17

The main objective in this section is to discuss how the natural numbers, https://www.w3.org/1998/Math/MathML" display="inline"> N = { 0 , 1 , 2 , 3 , … } https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math13_1.tif"/> , are constructed within the Zermelo-Fraenkel axiomatic system. We begin by stating the definitions of “successor set” and “inductive set”. The natural numbers, https://www.w3.org/1998/Math/MathML" display="inline"> N https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math13_2.tif"/> , is then defined as the “smallest” inductive set. A ZF-axiom will guarantee the existence of this “smallest inductive set” called the “natural numbers”. We then show how the Principle of mathematical induction is an immediate consequence of this definition of https://www.w3.org/1998/Math/MathML" display="inline"> N https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math13_3.tif"/> . We define “transitive sets” as sets, A, whose elements are subsets of A. The elements of https://www.w3.org/1998/Math/MathML" display="inline"> N https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math13_4.tif"/> are then shown to be “transitive sets”. We then prove a few properties possessed by all natural numbers.

Pages: 10

In this section we introduce the notion of “well-ordered set”. We show that the set of all natural numbers, when equipped with the membership ordering relation “ https://www.w3.org/1998/Math/MathML" display="inline"> ∈ https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math14_1.tif"/> ”, is a well-ordered set. We then define “bounded set” and “the maximal element of a set”. Finally we show that bounded subsets of https://www.w3.org/1998/Math/MathML" display="inline"> N https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math14_2.tif"/> must contain a maximal element. We then use https://www.w3.org/1998/Math/MathML" display="inline"> N https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math14_3.tif"/> to construct various other sets, some of which are also well-ordered.

Pages: 11

In this section we define addition, subtraction and multiplication of the natural numbers in a set-theoretic context. We then show that with these definitions, we obtain the expected results. Addition and multiplication on https://www.w3.org/1998/Math/MathML" display="inline"> N https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math15_1.tif"/> are defined recursively.

Pages: 11

In this section we define both the integers, https://www.w3.org/1998/Math/MathML" display="inline"> Z https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math16_5.tif"/> , and the rational numbers, https://www.w3.org/1998/Math/MathML" display="inline"> Q https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math16_6.tif"/> . The integers are presented as a quotient set of https://www.w3.org/1998/Math/MathML" display="inline"> N × N https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math16_7.tif"/> , while the rationals are presented as a quotient set of https://www.w3.org/1998/Math/MathML" display="inline"> Z × ( Z − { 0 } ) https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math16_8.tif"/> . Addition, subtraction and multiplication on each of these are defined within the set-theoretic context. Order relations are defined on each of https://www.w3.org/1998/Math/MathML" display="inline"> Z https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math16_9.tif"/> and https://www.w3.org/1998/Math/MathML" display="inline"> Q https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math16_10.tif"/> so that they are linearly ordered in the way we are accustomed to.

Pages: 11

In this section we show how https://www.w3.org/1998/Math/MathML" display="inline"> R https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math17_1.tif"/> is defined within the confines of the ZFC-set-theoretic axiomatic system. With this objective in mind, we begin by defining “initial segments” of rational numbers. These special elements of https://www.w3.org/1998/Math/MathML" display="inline"> P ( Q ) https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math17_2.tif"/> are referred to as “Dedekind cuts”. A linear order relation is defined on these as well as operations of addition and multiplication. We then show that there is a function, f, which maps the real numbers https://www.w3.org/1998/Math/MathML" display="inline"> R https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math17_3.tif"/> to the Dedekind cuts one-to-one and onto, while respecting order, addition and multiplication. Finally we show that the set of all Dedekind cuts satisfies the essential Completeness property of the real numbers.

Pages: 48

No chapter abstract available.

Pages: 13

In this section we give a definition of “infinite set” as put forward by Richard Dedekind. We then establish a few of the most basic properties of infinite sets and compare them to those of finite sets. These properties allow us to characterize finite sets as those sets which are in one-to-one correspondence with some natural number, n. We also show that for any finite set S, https://www.w3.org/1998/Math/MathML" display="inline"> P ( S ) https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math18_1.tif"/> is finite. Finally, we prove a version of the “Recursively defined function theorem”.

Pages: 10

In this section we define the words “equipotent sets”, “countable sets” and “uncountable sets”. We show that https://www.w3.org/1998/Math/MathML" display="inline"> Q https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math19_1.tif"/> and https://www.w3.org/1998/Math/MathML" display="inline"> Z https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math19_2.tif"/> are countable since we can produce a one-to-one correspondence between each of these and https://www.w3.org/1998/Math/MathML" display="inline"> N https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math19_3.tif"/> . We also show that https://www.w3.org/1998/Math/MathML" display="inline"> N × Z https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math19_4.tif"/> and https://www.w3.org/1998/Math/MathML" display="inline"> N × N https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math19_5.tif"/> are countable. Countable unions of countable sets are shown to be countable. We also show that no such one-to-one correspondence can exist between https://www.w3.org/1998/Math/MathML" display="inline"> R https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math19_6.tif"/> and https://www.w3.org/1998/Math/MathML" display="inline"> N https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math19_7.tif"/> , and so https://www.w3.org/1998/Math/MathML" display="inline"> R https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math19_8.tif"/> is uncountable.

Pages: 15

In this section we first show that the equipotence relation “ https://www.w3.org/1998/Math/MathML" display="inline"> ∼ e https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math20_1.tif"/> ” is an equivalence relation on the class https://www.w3.org/1998/Math/MathML" display="inline"> S https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math20_2.tif"/> of all sets. We use this equivalence relation to partition https://www.w3.org/1998/Math/MathML" display="inline"> S https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math20_3.tif"/> into equivalence classes. We show that for any set S, S cannot be equipotent to its power set https://www.w3.org/1998/Math/MathML" display="inline"> P ( S ) https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math20_4.tif"/> . This fact allows us to construct infinitely many distinct classes of mutually equipotent sets. We also show that for any non-empty set S, the two sets https://www.w3.org/1998/Math/MathML" display="inline"> P ( S ) https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math20_5.tif"/> and https://www.w3.org/1998/Math/MathML" display="inline"> 2 S https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math20_6.tif"/> are equipotent. Finally, we show that https://www.w3.org/1998/Math/MathML" display="inline"> P ( N ) https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math20_7.tif"/> is embedded in https://www.w3.org/1998/Math/MathML" display="inline"> R https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math20_8.tif"/> , and https://www.w3.org/1998/Math/MathML" display="inline"> R https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math20_9.tif"/> is embedded in https://www.w3.org/1998/Math/MathML" display="inline"> P ( N ) https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math20_10.tif"/> .

Pages: 8

In this section we state and prove the Schröder-Bernstein theorem. We then illustrate some of its consequences. In particular, we use it as a tool to prove that https://www.w3.org/1998/Math/MathML" display="inline"> R https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math21_1.tif"/> and https://www.w3.org/1998/Math/MathML" display="inline"> P ( N ) https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math21_2.tif"/> are equipotent. We finally show that https://www.w3.org/1998/Math/MathML" display="inline"> N N https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math21_3.tif"/> and https://www.w3.org/1998/Math/MathML" display="inline"> R https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math21_4.tif"/> are equipotent.

Pages: 40

No chapter abstract available.

Pages: 10

In this section we state the Continuum hypothesis and the Generalized continuum hypothesis; we discuss their meaning and consequences. We “informally” define the class of cardinal numbers, https://www.w3.org/1998/Math/MathML" display="inline"> C https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math22_1.tif"/> , introduce the “aleph” notation for cardinal numbers and define addition, multiplication and exponentiation of cardinal numbers. Finally, we show that the class of all cardinal numbers is a proper class.

Pages: 7

We have defined addition and multiplication of cardinal numbers in such a way that when adding or multiplying finite cardinal numbers, we obtain precisely the same answers as the ones obtained when performing these operations with natural numbers in the conventional way. In this section we verify that these two operations are well-defined even when adding and multiplying infinite cardinals. We then verify that addition and multiplication of cardinals satisfy, in many cases, the same properties as addition and multiplication of natural numbers. But not all of their properties generalize from the natural numbers to infinite cardinal numbers.

Pages: 9

In this section we show that cardinal exponentiation is well-defined. We then prove three of the most basic properties of cardinal exponentiation as well as inequalities involving cardinal exponentiation. We also show that https://www.w3.org/1998/Math/MathML" display="inline"> | R R | = c c = 2 c https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math24_1.tif"/> .

Pages: 12

In this section we examine a few sets whose cardinality equals the cardinality of https://www.w3.org/1998/Math/MathML" display="inline"> R https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math25_1.tif"/> . In particular, we determine the cardinality of the set of all one-to-one functions mapping https://www.w3.org/1998/Math/MathML" display="inline"> N https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math25_2.tif"/> to https://www.w3.org/1998/Math/MathML" display="inline"> R https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math25_3.tif"/> and the cardinality of the set of all continuous real-valued functions. We also study the well-known Cantor set, discuss its construction and prove that it has cardinality c.

Pages: 74

No chapter abstract available.

Pages: 16

In this section we review a few basic notions about well-ordered sets. We define special subsets of well-ordered sets called “initial segments”. “Order isomorphisms” are defined as strictly increasing one-to-one functions between well-ordered sets. Initial segments of well-ordered sets are themselves well-ordered sets. We prove basic properties of order isomorphisms between initial segments and well-ordered sets. We then define the relation, https://www.w3.org/1998/Math/MathML" display="inline"> ≤  WO https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math26_1.tif"/> , on well-ordered sets as follows: “S https://www.w3.org/1998/Math/MathML" display="inline"> ≤  WO https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math26_2.tif"/> T if S and T are order isomorphic or one is order isomorphic to an initial segment of the other”. This section provides the fundamental background for the study of the important set-theoretic topic of “ordinal numbers”.

Pages: 18

In this section we provide some motivation for the construction of what we will call the “ordinal numbers”. We will see that https://www.w3.org/1998/Math/MathML" display="inline"> N https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math27_1.tif"/> , as well as all of its elements, are ordinal numbers. When https://www.w3.org/1998/Math/MathML" display="inline"> N https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math27_2.tif"/> is viewed as an ordinal number, it is represented by the symbol, ω. We review the notions of “transitive sets” and “ https://www.w3.org/1998/Math/MathML" display="inline"> ∈ https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math27_3.tif"/> - well-ordering”. We then define “the immediate successor” of an element of a linearly ordered set, and show that the immediate successor of an ordinal number is an ordinal number. We also exhibit an ordinal number successor formula, https://www.w3.org/1998/Math/MathML" display="inline"> α + = α ∪ { α } https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math27_4.tif"/> , and show how it is used to recursively construct the most ordinals. We then prove a few basic properties of ordinal numbers. From this we deduce that all pairs of distinct ordinals are https://www.w3.org/1998/Math/MathML" display="inline"> ∈ https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math27_5.tif"/> -comparable. We define “limit ordinals”, show how these are constructed and provide methods to recognize them.

Pages: 22

In this section we discuss the class, https://www.w3.org/1998/Math/MathML" display="inline"> O https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math28_1.tif"/> , of all ordinal numbers. It will be seen to be an https://www.w3.org/1998/Math/MathML" display="inline"> ∈ https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math28_2.tif"/> -well-ordered transitive proper class of sets. Initial segments of https://www.w3.org/1998/Math/MathML" display="inline"> O https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math28_3.tif"/> are shown to be ordinals. Once we prove the “Principle of transfinite induction” we show that every well-ordered set is order isomorphic to some ordinal. We then show that for any set, S, there exists an ordinal which cannot be embedded in S. We introduce what is known as the “Hartogs number of a set” and use this concept to construct a strictly increasing sequence of uncountable ordinals indexed with the ordinals.

Pages: 16

In this section we formally define “initial ordinals”. We then prove that the class of all initial ordinals is precisely the class https://www.w3.org/1998/Math/MathML" display="inline"> I = ω 0 ∪ { ω α : α ∈ O } https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math29_1.tif"/> defined at the end of the previous section. We state and prove (by invoking the Axiom of choice) the Well-ordering theorem. Finally, we define the “cardinal numbers” as being the class of all initial ordinals.

Pages: 50

No chapter abstract available.

Pages: 15

In this section we prove that the Axiom of choice is equivalent to the Well-ordering principle. We provide a few mathematical statements whose proof requires the Axiom of choice. Finally we present Zorn's lemma and show that it is equivalent to the Axiom of choice. A proof of the fact that every vector space has a basis is given by invoking Zorn's lemma.

Pages: 6

In this section we state the Axiom of regularity and present some of its equivalent forms. We prove that the Axiom of regularity is equivalent to the statement “Every set has an https://www.w3.org/1998/Math/MathML" display="inline"> ∈ https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math31_1.tif"/> -minimal element”. We also show that in the presence of the Axiom of regularity, no set can be an element of itself. We define “well-founded sets” and show that in the presence of the Axiom of choice, the Axiom of regularity is equivalent to the statement “Every set is well-founded”. The transitive closure of a set is defined.

Pages: 16

We show how to construct, incrementally, a class of sets whose union, V, contains all sets. The class, V, is referred to as the “Von Neumann's universe of sets”. The Axiom of regularity is used to show that this class, https://www.w3.org/1998/Math/MathML" display="inline"> V https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math32_1.tif"/> , indeed contains all sets. We then define the “cumulative hierarchy” and the “rank of a set”.

Pages: 11

In this section we define the “countable chain condition” on a partially ordered set https://www.w3.org/1998/Math/MathML" display="inline"> ( P , ≤ ) https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math33_1.tif"/> . We then define those subsets of https://www.w3.org/1998/Math/MathML" display="inline"> ( P , ≤ ) https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math33_2.tif"/> called “filters”. We introduce an axiom which is indepe of ZFC called Martin's axiom, of particular interest when ¬CH is assumed. We then list a few consequences of this axiom. NOTE: This chapter is presented as a matter of interest and is mostly destined to readers well versed on topological spaces and more advanced topics in real analysis.

Pages: 20

No chapter abstract available.

Pages: 8

In this section we define the operation of addition on the ordinal numbers. We then show its most basic properties and provide a few examples.

Pages: 10

In this section we define the “lexicographic ordering” of the Cartesian product of two well-ordered sets. We then define the multiplication of two ordinals α and β as https://www.w3.org/1998/Math/MathML" display="inline"> α × β = ord B × A https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003586395/4b760900-5c01-4f24-9115-a0800b001426/content/math35_1.tif"/> where A and B are their respective set representatives. We then list a few basic properties of ordinal multiplication. This is followed by a definition of exponentiation and a presentation of a few exponentiation properties.