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CARDINAL and ORDINAL NUMBERS Contents 1. the Natural CARDINAL AND ORDINAL NUMBERS JAMES MURPHY Abstract. This paper will present a brief set-theoretic construction of the natural numbers before discussing in detail the ordinal and cardinal numbers. It will then investigate the relationship between the two proper classes, in particular the similar difficulties in discussing the size of the classes. We will end with a short section on the cardinalities of well-known infinite sets with which the reader is likely to be familiar. Contents 1. The Natural Numbers 1 2. Ordinal Numbers 2 3. Ordinal Arithmetic 8 4. Cardinal Numbers 10 5. Cardinal Arithmetic 14 6. Cardinality of Sets 16 Acknowledgments 17 References 18 1. The Natural Numbers Although there are several ways to construct the natural numbers, this paper will use a method that defines each natural number as a set which contains each of its predecessors. Before we can make this approach rigorous, we need a definition. Definition 1.1. For a set x, we define the successor of x, x+, to be the set obtained by adjoining x to the elements of x. In other words, x+ = fx [ fxgg. We can now begin to define the natural numbers. However, we must consider how to start, that is, how to define the first natural number, 0. Since our method is based around defining each natural number with regards to its predecessors, and since 0 has no predecessors in the naturals, we define 0 to be the empty set: 0 = ;. We then define 1, 2 and 3 in the way alluded earlier: 1 = 0+=f0g 2 = 1+=f0; 1g 3 = 2+=f0; 1; 2g Date: DEADLINE AUGUST 21, 2009. 1 2 JAMES MURPHY This method of defining the natural numbers is useful and consistent with our notation for all finite natural numbers, that is, the set N. However, it is not yet clear that this construction of successors can be carried out in one set indefinitely. That is, it is not clear that there exists a non-empty set which contains the successor of each of its elements. We need a set-theoretic axiom for this. Axiom 1. There exists a set containing 0 and containing the successor of each of its elements. This statement of existence is often called The Axiom of Infinity. Such a set A, defined such that 0 2 A and x+ 2 A if x 2 A, is called a successor set. We will next prove that there exists a smallest successor set. Theorem 1.2. There exists a smallest successor set. Proof. Let ! be the intersection of every successor set. Then ! is a successor set itself. For if not, then for some x 2 !, x+ 2= !. But since ! is the intersection of all successor sets, then for some such successor set, x 2 ! but x+ 2= !. This is a contradiction of the definition of successor set. Then ! is a successor set and is, by construction, a subset of all successor sets. It is therefore the smallest successor set. The reader worried that the intersection of all successor sets might not exist should consider the following, more precise definition of !. Take a successor set, α, and consider the set of its subsets, P (α). Then look at the set Aα ⊆ P (α) such that every element of A is a successor set. If we look at the intersection of Aα for all successor sets α, then any trouble with dealing with the intersection of all successor sets is alleviated. This comment is only relevant to those very familiar with set theory, in particular with the theory of proper classes. For all other readers, this comment is not worth fretting over. A natural number is, by definition, an element of !. This construction of ! makes rigorous the intuitive description of the natural numbers as f0; 1; 2; 3; ::g, where the ellipsis represent the so on ad infinitum normally used to describe the natural numbers. 2. Ordinal Numbers Before we can begin this new section, we must present an extremely important definition. We assume that the reader is familiar with the concept of a relation and has seen some examples of a relation, such as <, ≤ and 2. Definition 2.1. A set X is well-ordered by the relation R if the following principles hold: 1.) For every x and y in X, if we have xRy then we cannot have yRx. This means that R is asymmetric on X. 2.) For every x and y in X, exactly one of xRy, yRx and x = y holds. This means that R is connected on X. 3.) For all x, y and z in X, if xRy and yRz, then xRz. This means that R is transitive on X. 4.) Every non-empty subset of X has an R-least element. CARDINAL AND ORDINAL NUMBERS 3 We call a set W together with a relation that well orders it, <, a well-ordering. This is often stated by saying the (W, <) is a well-ordering. Our definition of the naturals is ordered by inclusion, since we defined a nat- ural number n as the set of all natural numbers less that n, that is, we defined n=f0; 1; 2; :::; n − 2; n − 1g. We now want to use this key property of the natural numbers and ! to define numbers larger than !. Since we are defining this new type of number by succession as with the natural numbers, we want the set to be well-ordered by inclusion too. Before we can give a precise definition for this new type of number, which we will call an ordinal number, or more simply an ordinal, we need a couple of definitions. Definition 2.2. A set z is transitive if whenever x and y are sets such that x 2 y and y 2 z, we have x 2 z. Definition 2.3. Let z be a set. We define a relation 2z by 2z = f(x; y) 2 z × z : x 2 yg. We can now define what exactly we mean by an ordinal number and give an example of an ordinal number we have already encountered. Definition 2.4. An ordinal is a set α which is transitive and well-ordered by 2α. Theorem 2.5. ! is an ordinal. Proof. Theorem 1.3 shows that ! is transitive. To see that ! is well-ordered by 2!, let α be a non-empty subset of !. Then we assert that α has a least element, T namely x = β2α β, that is, the intersection of all elements of α. x 6= ;, since 0 2 β for every β 2 α. Now consider γ, the largest element of x. This number must exist, for otherwise every element of α has no largest element, meaning that α cannot be a subset of !, which consists of only natural numbers, each of which have an 2-greatest element. Then by construction, x contains every natural number less than γ. If this were not true, then for some β 2 α, there is some y 2 β and z such that z+ = y but z 62 β, which is absurd based on Definition 1.1. This shows that x is itself a natural number. In fact, it is the natural number γ + 1, again by Definition 1.1. For each β 2 α, the 2-greatest element of β is unique, based on our construction of the naturals. Thus, if x = γ +1, then there must be β 2 α such that the 2-greatest element of this β is γ. This shows that f0; 1; 2; :::; γg = γ+1 = x 2 α. Since x 2 β for all β 2 α, x is the smallest β 2 α, making x the least element of α. This shows that ! is well-ordered by 2!, which completes the proof. In the preceding proof, we used the notation ... to indicate a set of natural numbers which includes every natural number in between 2 and γ. We will now prove a few theorems that characterize ordinal numbers. Theorem 2.6. If α is an ordinal and β 2 α, then β is an ordinal. Proof. To see that β is transitive, we let x and y be sets with x 2 y and y 2 β. Since y 2 β, β 2 α and α is an ordinal and thus transitive, it follows that y 2 α. Since x 2 y and y 2 α, it follows that x 2 α. Now since x; y; β 2 α x 2 y, y 2 β and the relation 2α is transitive on α, we have x 2 β. Thus, β is transitive. Notice that β ⊆ α because β 2 α and α is transitive. Therefore, 2β is the restriction of 2α to the subset β ⊆ α. Since 2α is a well-ordering on α, it follows that 2β is a well-ordering on β. Hence, β is an ordinal. 4 JAMES MURPHY Corollary 2.7. Every n 2 ! is an ordinal. Lemma 2.8. If α is an ordinal, then α2 = α. Proof. Suppose that α is an ordinal and α 2 α. Since α 2 α, 2α is not asymmetric on α. Thus, 2α is not a well-ordering on α, so α is not an ordinal, which is a contradiction. Theorem 2.9. Suppose that α and β are ordinals. Then exactly one of the follow- ing is true: α 2 β, α = β, or β 2 α. Proof. We will first prove that at least one of α 2 β, α = β, or β 2 α holds.
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