CHAPTER I THE FOUNDATIONS OF SET THEORY It is assumed that the reader has seen a development of mathematics based on some principles roughly resembling the axioms listed in Q 7 of the Introduction. In this chapter we review such a development, stressing some foundational points which will be important for later work. 91. Why axioms? Most mathematicians have little need for a precise codification of the set theory they use. It is generally understood which principles are correct beyond any doubt, and which are subject to question. For example, it is generally agreed that the Continuum Hypothesis (CH) is not a basic prin- ciple, but rather an open conjecture, and we are all able, without the benefit of any formal axiomatization, to tell which of our theorems we have proved absolutely and which depend upon the (as yet undecided) truth or falsity of CH. However, in this book we are concerned with establishing results like: “CH is neither provable nor refutable from ordinary set-theoretic prin- ciples”. In order to make that statement precise, we must say exactly what those principles are; in this book, we have defined them to be the axioms of ZFC listed in the Introduction. The assertion: “CH is neither provable nor refutable from ZFC is now a well-defined statement which we shall establish in Chapters VI and VII. The question remains as to whether the axioms of ZFC do embody all the “ordinary set-theoretic principles”. In this chapter we shall develop them far enough to be able to see how one can derive from them all of current conventional mathematics. Of course, future generations of mathematicians may come to realize some “obviously true” set-theoretic principles which do not follow from ZFC. Conceivably, CH could be then settled using those principles. Even at the present, there are several ways besides ZFC for handling the axiomatization of currently accepted set-theoretic principles (see 9 12). The 1 2 The foundations of set theory [Ch. I, 42 methods of this book are easily modified to handle those systems as well, although the technical details are slightly simpler for ZFC. 92. Why formal logic? The idea of setting down one’s axioms harks back to Euclid, and is hardly revolutionary. Usually in mathematics the axioms are stated in an informal language, such as Greek or English. But here we shall state our axioms in a formal, or artificial, language, called the first-order predicate calculus. The feature of a formal language is that there are precise rules of formation for linguistic objects. There are two main reasons for this approach. Reason 1. Formal logic is needed to state the axioms of set theory precisely. For example, ZFC has a Comprehension Axiom asserting that sets of the form {x€A:P(x)} exist, where A is a given set and P(x) is any property of x. But what is a property? Intuitively, it is any well-defined English assertion about the variable x. Is ‘‘x is happy” a property? It is clear that we need a rigorous definition of which properties we are to admit. We shall require that P(x) be expressible in our formal language, which will be capable of expressing mathematical notions, but not non-mathematical ones. The fact that an imprecise notion of property can lead to trouble is illustrated by the follow- ing “paradox” in ordinary reasoning: Let n be the least positive integer not definable by an English expression using forty words or less. But I have just defined it in forty words or less. Reason 2. Even after we have defined ZFC, what does it mean to say that CH is not provable from ZFC? Intuitively, it means that there is no way of deriving CH from ZFC using legitimate rules of inference. This intuitive notion can be made precise using the concept of a formal deduction. We shall only sketch here the development of formal logic, referring the reader to a text on the subject, such as [Enderton 19723, [Kleene 19521, or [Shoenfield 19671, for a more detailed treatment. We shall give a precise definition of the formal language, as this is easy to do and is necessary for stating the axioms of ZFC. We shall only hint at the rules of formal deduc- tion; these are also not hard to define, but it then takes some work to see that the standard mathematical arguments can all be formalized within the prescribed rules. The basic symbols of our formal language are A, 1,3, (, ), E, =, and‘ Ch. I, $21 Why formal logic? 3 uj for each natural number j. Intuitively, A means “and,” imeans “not,” 3 means “there exists,” E denotes membership, = denotes equality, uo, ul, ... are variables, and the parentheses are used for phrasing. An expression is any finite sequence of basic symbols, such as )33i). The intuitive interpreta- tion of the symbols indicates which expressions are meaningful; these are called formulas. More precisely, we define a formula to be any expression constructed by the rules: (1) ui E uj, ui = vj are formulas for any i, j. (2) If 4 and tj are formulas, so are (4) A ($), i(4),and 3ui(4)for any i. So, for example, 3uo (3ul ((u, E ul) A (ul E uo))) is a formula. Our formal definition departs somewhat from intuition in that, in an effort to make the definition simple, the use of parentheses was prescribed very restrictively. For example, Uo E U1 A 1(U1 E Uo) is not a formula. Another seeming drawback of our formal language is that it seems to lack the ability to express certain very basic logical notions, like, e.g. V (for all). However, this is not really a problem, since i(3ui( i (4))) expresses Vui(4). Similar remarks hold for v (or), + (implies) and c, (iff), which may all be expressed using A and 1.To save ourselves the work of always writing these longer expressions, we agree at the outset to use the following abbreviations. (1) Vui(4) abbreviates i(3ui( i (4))). (2) (4) ” (II/) abbreviates 1(( 1 (4)) A (1(44 1. (3) (4)+ abbreviates (1(41) ” ($1. (4) (4) - ($1 abbreviates ((4) + ($4) A ((4v + (4)). (5) Parentheses are dropped if it is clear from the context how to put them in. (6) ui # uj abbreviates i(ui = uj) and ui Pf uj abbreviates i(ui E uj). (7) Other letters and subscripted letters from the English, Greek, and Hebrew alphabet are used for variables. We shall explain (7) in more detail later. There are many abbreviations other than these seven. Actually, in this book we shall very rarely see a formula. We shall follow standard mathe- matical usage of writing expressions mostly in English, augmented by logical symbols when this seems useful. For example, we might say “there are sets x, y, z such that x E y A y E z”, rather than The Comprehension Axiom (see Reason (1) above) will be made into a precise statement by requiring that properties P(x) occurring in it be ex- pressible in the formal language, but it will not be necessary to write out the formula expressing P(x) each time the Comprehension Axiom is used. 4 The foundations of set theory [Ch. I, 52 A subformula of 4 is a consecutive sequence of symbols of 4 which form a formula. For example, the 5 subformulae of (3uo (uo E 01)) A (+, (u2 01)) (1) are uo E u,, 3uo(uo E ul), u2 E u,, 3u, (u2 E u,), and the formula (1) itself. The scope of an occurrence of a quantifier 3ui is the (unique) subformula be- ginning with that 3ui. For example, the scope of the 3uo in (1) is 3u, (uo E ul). An occurrence of a variable in a formula is called bound iff it lies in the scope of a quantifier acting on that variable, otherwise it is called free. For example, in (1) the first occurrence of u, is free, but the second is bound, whereas uo is bound at its occurrence/ and u2 is free at its occurrence. Intuitively, a formula expresses a property of its free variables, whereas the bound or dummy variables are used just to make existential statements which could be made equally well with different bound variables. Thus, formula (1) means the same as (3u4@4 E 01)) A (3U,(U2 E d). Note that since Vui is an abbreviation for i3uii, it also binds its variable ui, whereas the abbreviations v, -+, c-) are defined in terms of other pro- positional connections and do not bind variables. Often in a discussion, we present a formula and call it @(x,, ... ,x,) to emphasize its dependence on xl, ... , x,. Then, later, if y,, ... , y, are other variables, $(y,, . .. , y,) will denote the formula resulting from substituting a yi for each free occurrence of xi. Such a substitution is called free, or legitimate iff no free occurrence of an xi is in the scope of a quantifier 3yi. The idea is that 4(yl, ..., y,) says about y,, ... ,y, what ..., x,) said about x,, .. ,x,, but this will not be the case if the substitution is not free and some yi gets bound by a quantifier of 4. In general, we shall always as- sume that our substitutions are free. The use of the notation &(x,, .. ,x,) does not imply that each xi actually occurs free in 4(x1, ..., x,); also, 4(x1, . .. ,x,) may have other free variables which in the particular dis- cussion we are not emphasizing.