V : Number systems and set theory Any reasonable framework for mathematics should include the fundamental number systems which arise in the subject: 1. The natural numbers N (also known as the nonnegative integers). 2. The (signed) integers Z obtained by adjoining negative numbers to N. 3. The rational numbers Q obtained by adjoining reciprocals of nonzero integers to Z. 4. The real numbers R, which should include fundamental constructions like nth roots of positive rational numbers for an arbitrary integer n > 1, and also –1 –2 –k ⋅ ⋅ ⋅ all “infinite decimals” of the form b1 10 + b2 10 + … + bk 10 + … where each bi belongs to {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. Up to this point we have tacitly assumed that such number systems are at our disposal. However, in both the naïve and axiomatic approaches to set theory it is eventually necessary to say more about them. The naïve approach. In naïve set theory it is necessary to do two things. First, one must describe the properties that the set – theoretic versions of these number systems should satisfy. Second, something should be said to justify our describing such systems as THE natural numbers, THE integers, THE rational numbers, and THE real numbers. This usage suggests that we have completely unambiguous descriptions of the number systems in terms of their algebraic and other properties. One way of stating this is that any system satisfying all the conditions for one of the standard systems N, Z, Q or R should be the same as N, Z, Q or R for all mathematical purposes, with some explicit means for mechanical translation from the given system to the appropriate standard model. In less formal terms, it we have any systems X which satisfy all the fundamental properties of one of the systems N, Z, Q or R, then X is essentially a mathematical clone of the appropriate number system. There are good theoretical and philosophical reasons for asking such questions about the essential uniqueness of the number systems, but these question also have some important practical implications for the development of mathematics. If there would be two systems that satisfy the basic properties of N, Z, Q or R but differ from a standard model in some significant fashion, then clearly we might get different versions of mathematics depending upon which example is chosen. To illustrate this, suppose we decided to develop a version of the real numbers in which infinite base 10 “decimal expansions” are replaced by expansions with some other number base, say 16 (to conform with the internal arithmetic of some computer) or 60 (as in Babylonian mathematics). We expect that everything should work the same regardless of the 70 numerical base we choose for expressing quantities, but at some point it is necessary to confirm that our expectation is fulfilled. Later in this unit we shall describe precisely the notion of a mathematical clone. For the time being we note that examples of this concept have already been encountered in Section IV.6 when we talked about whether two partially ordered sets have the same order type. Given two such partially ordered sets, the 1 – 1 order preserving correspondence from one to another can be viewed as a formal mathematical way of saying that either of the partially ordered sets is a clone of the other. Our coverage in this unit will mainly concern the first item described in the naïve approach; namely, the formal properties of the number systems and the mathematical statements of their uniqueness properties. Later in these notes (and largely for reference purposes) we shall explain why the basic properties describe these number systems in a totally unambiguous manner. The axiomatic approach. In axiomatic set theory it is necessary to assume the existence of systems with the given properties and to prove these properties describe them unambiguously (the latter proceeds exactly the same as in naïve set theory). One new issue in the axiomatic approach is the goal of keeping the basic assumptions for set theory as simple as possible. Assuming the existence of four separate but clearly interrelated number systems is a convenient first step, but at some point it is natural to ask if we really need to make such a long list of assumptions in order to set everything up. Aside from possible aesthetic considerations, there is the practical consideration that long lists of assumptions raise questions whether there might be some logical inconsistency; after all, the whole idea of a proof by contradiction is that one makes so many assumptions that the conclusions end up contradicting each other, and it would undermine everything if such contradictions could be derived from the axioms for set theory itself. We shall address some of these issues in the final unit of the notes. Some more specific objectives Much of this unit is devoted to summarizing familiar properties of the four basic number systems, so we shall indicate some points that are less elementary and particularly important. In Section 1 the most significant new item is the statement of the Peano Axioms for the natural numbers, and in Section 2 the discussion of finite induction and recursive definitions in the framework of set theory is one of the main topics in the unit. The formulas for counting the numbers of elements in various finite sets in Section 3 start with familiar ideas, and they give systematic rules that are important both for their own sake and for the remaining units of the course. Finally, the description of the real numbers in Section 4 is fundamentally important. Although this description is fairly concise, it contains everything that is needed to justify the standard facts about real numbers and to develop calculus in a mathematically rigorous fashion. The latter development is covered in subsequent courses. Although the justification of the usual expansions for real numbers is also somewhat peripheral to the present course, for the sake of completeness we shall explain how our formal description of the real numbers yields their familiar properties which are used in everyday work, both inside and outside of mathematics. 71 V . 1 : The natural numbers and integers (Halmos, §§ 11 – 13; Lipschutz, §§ 2.1, 2.7 – 2.9) In many respects the positive integers form the most basic number system in all of the mathematical sciences. Some reasons for this are historical or philosophical, but logical considerations are particularly important for the systematic development of mathematics. Clearly we would like our descriptions of number systems to summarize their basic algebraic properties concise but understandable. In particular, it simplifies things considerably if we can say that addition, subtraction and multiplication are always defined. Since the positive integers are not closed under subtraction, clearly they do not fulfill this condition. Therefore we shall begin by describing the integers, and we shall view the positive integers as a subset of the integers with certain special properties. The important algebraic properties of the integers split naturally into three classes, two of which are fairly general and one of which is more focused. Basic rules for addition and multiplication. Formally, these are the conditions defining an abstract type of mathematical system known as a commutative ring with unit. FIRST AXIOM GROUP FOR THE INTEGERS. The integers are a set Z, and they have binary operations A : Z × Z → Z, normally expressed in the form A(u, v) = u + v, and M : Z × Z → Z, normally expressed in the form M(u, v) = u v or u ⋅ v or u × v, which satisfy the following algebraic conditions: 1. (Associative Laws). For all a, b, c in Z, (a + b) + c = a + (b + c) and (a b) c = a (b c). 2. (Commutative Laws). For all a, b in Z, a + b = b + a and a b = b a. 3. (Distributive Law). For all a, b, c in Z, a (b + c) = a b + a c. 4. (Existence of 0 and 1). There are distinct elements 0, 1 in Z such that for all a we have a + 0 = a, a × 0 = 0 and a × 1 = a. 5. (Existence of negatives or additive inverses). For each a in Z there is an element – a in Z such that a + (– a) = 0. Notational footnote: The notation Z for the integers has become fairly standard in mathematical writings, and it is apparently derived from the German word for numbers (Zahlen) and/or cyclic (zyklisch). We shall need the following basic consequences of the preceding algebraic conditions: Proposition 1. If a belongs to a system satisfying the properties listed above, then we have (– a) (– b) = a b. 72 Proof. The following are special cases of the axioms: 0 = a 0 = a [b + (– b)] = a b + a (– b) 0 = 0 (– b) = [a + (– a)] (– b) = a (– b) + (– a) (– b) The preceding results also show that a b = – [a (– b)] = (– a) (– b). Basic rules for ordering. When combined with the previous conditions, these yield a type of mathematical system known as an ordered integral domain. SECOND AXIOM GROUP FOR THE INTEGERS. There is a linear ordering on Z such that the following hold: 1. If a > 0 and b > 0, then a + b > 0 and a b > 0. 2. For all a, b in Z, we have a > b if and only if a – b > 0. Well – ordering of positive elements. This is the assumption that the set N of nonnegative elements in Z, often called the natural numbers, is well – ordered with respect to the standard linear ordering. WELL - ORDERING AXIOM FOR THE POSITIVE INTEGERS. The set N of all x in Z such that x ≥ 0 is well – ordered.