A Beginner s Guide to Modern Set Theory

Transcription

1 A Beginner s Guide to Modern Set Theory Martin Dowd Product of Hyperon Software PO Box 4161 Costa Mesa, CA Copyright c 2010 by Martin Dowd

2 1. Introduction Formal logic Axioms of equality The integers Informal set theory Structures and models Models of Peano arithmetic The real numbers Computability Independence ZFC Proper classes Ordinals and cardinals The real numbers (II) The continuum hypothesis Absoluteness Admissible sets Formalization of syntax Constructible sets CH is true in L Forcing CH is consistent Clubs, stationary sets, and diamond Trees The Suslin hypothesis Diamond implies SH Iterated forcing Martin s axiom SH is consistent Inaccessible cardinals Mahlo cardinals Greatly Mahlo cardinals Reflection principles Indescribable cardinals Ultrapowers Measurable cardinals Indiscernibles # Relative constructibility Direct limits L[U] and iterated ultrapowers The sharp operator i

3 43. Cardinals larger than measurable Kunen s theorem Rudimentary functions The Jensen hierarchy Fine structure Upward extension Fine structural ultrapowers The covering lemma Cardinal arithmetic Square Independence of AC Proper forcing Core models Consistency strength Descriptive set theory Determinacy Determinacy and descriptive set theory Determinacy and 0# Determinacy and large cardinals Forcing axioms Some observations Appendix 1. Axioms for plane geometry Appendix 2. Computability (II) References Index Index of symbols ii

4 1. Introduction. As the title suggests, this book is intended to provide an introduction to modern set theory, to readers with little or no knowledge of mathematical logic. As such, it should be useful to anyone interested in learning about modern set theory, without having to wade through an entire text such as the Millennium Edition [Jech2]. Readers might fall in to two categories, those who are not interested in reading further, and those who are. For the latter, this book hopefully provides useful orientation. It is hoped that advanced high school students will find this book useful. Admittedly only the most intrepid student would finish it in high school; but the first 15 chapters, and the two appendices, are hopefully fairly accessible. Resources for advanced high school mathematics are mainly in calculus and linear algebra, with some resources in other areas. Resources in mathematical logic have typically been scarce, one example being a 1958 book on Godel s proof [NagNew]. The website [Wiki, Mathematical logic] has overviews of various topics, and links to additional resources. The present book contains an introduction to mathematical logic sufficient for its purposes, and thus should serve as a useful introduction for other purposes. Various other topics are covered for the same reason, so that the book is fairly self-contained. Set theory, like any branch of contemporary mathematics, consists of an overwhelming volume of technical definitions and arguments. On the other hand, non-technical introductions sometimes engage in circumlocutions intended to avoid technical detail, so convoluted that they become confusing. The present book pursues an intermediate course, covering technical details in outline and giving references, so that the main content can be given with some discussion of technical details. The book consists of a series of sections, each covering a particular topic. The table of contents gives a list of the sections. The end of a proof is denoted using the symbol. The author thanks Dr. Herbert Enderton for reading a draft of the manuscript. 2. Formal logic. It is a discovery of late 19th and early 20th century mathematics, that mathematical theorems can be stated and proved in formal logic. This discovery did not change the way mathematics is done; theorems are proved by working mathematicians using informal logic, which other mathematicians can follow, and which may refer to extensive amounts of material already accepted as fact. Rather, formal logic brought complete precision to the analysis of mathematical reasoning, clarified various issues which had been under debate, and produced formal logic as itself 1

5 a branch of mathematics. Formal logic relies on the fact that statements of mathematics can be specified in a formal language. Indeed, this observation holds in other areas, and formal logic has found uses in addition to its use in mathematics. Statements are finite strings of symbols, each symbol being chosen from an alphabet of symbols. For this reason, formal logic is also called symbolic logic. The alphabet of the formal language of mathematics is divided into groups of symbols, as follows. Logical symbols Punctuation marks (), Propositional connectives Quantifiers Variables x 0, x 1,... Non-logical symbols Predicate symbols P0 n, P 1 n,... Function symbols f0 n, f1 n,... Constant symbols c 0, c 1,... The superscript n in predicate and function symbols is an integer giving its valency, i.e., the number of arguments it applies to; it will invariably be omitted. Not every string of symbols is legal ; those that are, are called formulas. These may be defined by giving rules for building them up, as follows. A term is either a variable, a constant symbol, or f(t 1,..., t n ) where f is a function symbol of valency n and t 1,...,t n are terms. An atomic formula is a formula P(t 1,..., t n ) where P is a predicate symbol of valency n and t 1,...,t n are terms. A formula is either an atomic formula, F, F 1 F 2, F 1 F 2, F 1 F 2, F 1 F 2, xf, or xf, where F, F 1, and F 2 are formulas and x is a variable. The preceding style of definition, where objects which are already built up can be use to build up new objects, is called recursive. A shortcut has been taken; the subformulas in the definition of a formula should be enclosed in parentheses, to avoid ambiguity, although some of the parentheses can be made optional (requiring a more laborious recursive definition). The notion of the free and bound occurrences of variables in a formula is an important one, and may be defined recursively as follows. In an atomic formula, all occurrences of variables are free. In a propositional combination of formulas, all occurrences of variables are free or bound as they are in the constituent subformulas. In xf or xf, any free occurrence of x in F becomes bound; all other occurrences are free or bound as they are in F. A sentence is a formula in which all 2

6 occurrences of variables are bound. Later it will be seen that, given an interpretation in a mathematical setting of the non-logical symbols, a meaning can be assigned to any formula. Some discussion is useful here. In general, a formula defines a predicate on the universe of discourse : if values from the universe are assigned to the free variables, the formula takes on the value of either true or false. In particular, a sentence is a statement which is either true or false. A brief statement of the meaning of the propositional connectives and quantifiers can be given, as follows. F means not F (negation) F 1 F 2 means F 1 and F 2 (conjunction) F 1 F 2 means F 1 or F 2 (disjunction) F 1 F 2 means if F 1 then F 2 (implication) F 1 F 2 means F 1 if and only if F 2 (bi-implication) xf means for all x, F (universal quantification) xf means there exists x, F (existential quantification) Having a formal definition of a mathematical statement, a formal definition can now be given of a proof. Certain formulas are specified as axioms, and rules are given for deducing formulas from formulas already deduced. Some axioms are axioms of formal logic, and are called logical. Other axioms are specific to a particular setting, and are called non-logical. The rules are all logical. The logical axioms of formal logic are chosen so that they are true in any setting, and in any setting the rules produce true statements from statements already known to be true. The non-logical axioms are true in settings of interest. Even though the principles are clear without giving one, an example of a system of logical axioms and rules will be given. Such will be given for a variation of the alphabet, namely a smaller one. A larger alphabet is more expressive, but a smaller alphabet results in fewer axioms and rules. Needless to say, the variation is inessential; in particular, the larger alphabet can be expressed in terms of the smaller one. The alphabet of the axioms and rules will be. In the following, let F, G, H be formulas. If F is a formula, x a variable, and t a term, F t/x will denote the formula obtained from F by replacing each free occurrence of x by t. There are three propositional logical axioms. F (G F) (H (F G)) ((H F) (H G)) ( F G) (( F G) F) There is one propositional rule. 3

7 From F and F G, deduce G. There is one quantifier axiom. F xg = F G t/x, provided no occurrence of a variable of t becomes bound. There is one quantifier rule. From F G deduce F xg, provided x does not occur free in F. Note that arbitrary formulas may occur in a proof, and not just sentences. This is an artifact of the method; quantifiers get introduced as the formulas of the proof become more complex. A formula is considered to be true if it is true, regardless of the values assigned to the free variables (if its universal closure is a true sentence). As has been seen, the syntax of formal (or mathematical) logic consists of an alphabet, and rules for building formulas. Statements of mathematics are proved to be true using the axioms and rules of a formal system for making deductions. The semantics of mathematical logic consists of assigning in a rigorous manner a meaning to each formula; this requires some additional concepts, and is left to section 6. Once all this is specified, theorems may be proved about mathematical logic itself, which delineate the way in which it captures mathematical reasoning. There are a number of introductions to mathematical logic, among them [Belaniuk], [Enderton], [Mendelson], [Magnus], and chapter 11 of the author s self-published advanced undergraduate algebra text [Dowd1]. As will be seen in section 11, formal logic is an essential ingredient of modern set theory. Historically, early developments in mathematical logic and set theory overlapped and influenced each other. A relatively recent development in mathematical logic is the use of computers to produce formal proofs of mathematical theorems, using a known informal proof as a starting point. The December 2008 issue of the Notices of the American Mathematical Society contains several articles on the subject. 3. Axioms of equality. The equality predicate, for which the symbol = is used, has a special status in formal logic. It is a binary (valency 2) predicate. As for many common binary predicates, the notation x = y is used in mathematical writing rather than =(x, y). In settings where equality is present, it is meant to be interpreted as equality, that is, x = y holds only when x and y are assigned the same value. There are some subtleties in handling the special status of the equality predicate; and some variations in how this is done. More will be said in section 6. 4

8 If equality is present, the axioms for it may be considered to be added as quasi-logical (standardized non-logical) axioms. These are as follows. x = x x = y y = x x = y y = z x = z x 1 = y 1 x n = y n P(x 1,..., x n ) P(y 1,..., y n ), for any valency n predicate symbol P. x 1 = y 1 x n = y n f(x 1,...,x n ) = f(y 1,..., y n ), for any valency n function symbol f. In the foregoing, x, y, etc. denote variables. Also, the abbreviation F 1 F k is used for F 1 ( F k ); this may also be written as (F 1 F k 1 ) F k, or just F 1 F k 1 F k. The axioms of equality are written without quantifiers, all variables being implicitly universally quantified. This is a serendipitous coincidence between common use in mathematical writing, and a convention of formal logic. 4. The integers. The integers are fundamental mathematical objects, which are familiar from everyday life. With modern machinery, a theory of the integers can be given either for all the integers, including negative integers; or for only the non-negative integers. Historically, the theory of the non-negative integers has been important in the development of mathematical logic, and it continues to play a significant role. The non-negative integers 0,1,2,... comprise a universe of discourse concerning which mathematical statements can be made. A set of nonlogical symbols which turns out to be satisfactory as those of the formal language for such statements is as follows: a constant 0; a valency 1 function s, the successor function; a valency 2 function +, addition; a valency 2 function, multiplication; and the equality predicate =. The notation x s will be used for the successor function; x s equals x+1. Even though it is not ordinarily used in mathematical writing, it is convenient and traditional to have it as one of the symbols of the formal language in this setting. The above symbols comprise the language of Peano arithmetic. Let F denote a formula in this language, and let x, y, etc., denote variables. The following formulas are known as Peano s axioms. 1. x s = y s x = y 2. x s = 0 5

9 3. x + 0 = x 4. x + y s = (x + y) s 5. x 0 = 0 6. x y s = (x y) + x 7 F. F 0/x x(f F xs /x) xf. Again, axioms 1 to 6 are written without quantifiers, and all variables are implicitly universally quantified. Peano s axioms are clearly basic facts about the non-negative integers. In accordance with the axiomatic method, they are taken as true, and more complex statements deduced to be true by mathematical reasoning. Axiom 7 is an infinite family of axioms, one for each formula F (and variable x). Such a system of axioms is called an axiom scheme, and these occur frequently in mathematical logic. Note that x is not required to occur free in F; some authors do require this, but it is unnecessary to do so. This axiom scheme is a formal statement of the principle of mathematical induction. Mathematical induction may be stated in a version using sets of integers; but the formal machinery given so far does not provide for this, and Peano s axioms provide a method for giving axioms for the non-negative integers within the confines of basic formal logic. Historically, this was a reason for their introduction. They remain a topic of considerable interest in mathematical logic, even though they are subsumed by formal set theory, as will be seen. In particular, the logical strength of Peano s axioms is of great interest. As will be noted in section 10, not every true statement about the integers can be proved using them (this is in fact the case for any formal system for arithmetic which proves only true statements); but stronger systems can be given. Whether a particular true statement about the non-negative integers can be proved using Peano s axioms is a topic of interest in mathematical logic. Of course, Peano s axioms are of interest because they are strong enough that a wide variety of basic facts about the non-negative integers can be proved using them. Treatments of this topic can be found in [Mendelson] and [Shoenfield1]. Among these facts are the following. - The basic properties of + and are provable. - There is a formula defining the order relation (indeed, x y if and only if w(y = x + w)), and its basic properties are provable. - The division law states that for any nonnegative integer x and positive integer d there are unique nonnegative integers q and r such that x = q d + r; this is provable. - The exponential function is definable, that is, there is a formula E(x, y, z) which is true if and only if z = x y. The basic properties of the exponential function are provable. 6

10 - More generally, any of the class of functions known as the primitive recursive functions (see appendix 2) is definable. Another result of mathematical logic of interest concerning Peano s axioms, is that there is no finite set of axioms from which the statements provable are exactly those provable in Peano arithmetic. [Shoenfield1] has a proof of this. 5. Informal set theory. Informal set theory has become so indispensable to mathematical discourse that it is now taught early in mathematical education. Like the integers, the sets are mathematical objects which comprise a mathematical universe of discourse. Indeed, they comprise a single universe of discourse for all of mathematics. This is a more advanced topic, but in view of the fact, it should not be surprising that the notion of a set is useful throughout mathematics. Basic set theory and logic are both tools used throughout mathematics, in particular in the consideration of each other. This results in the need for forward references in the presentation of the two topics, which various authors handle in various ways. A formal definition of the meaning of formulas has been deferred to section 6, and until then the reader s existing knowledge will be relied on, indeed already has been in the preceding section. The language of set theory has a single binary predicate symbol, called membership and denoted. The fact that x y is stated variously as, x is a member of y, x is an element of y, or x belongs to y. The notation x / y is used to abbreviate (x y). The equality predicate will also be considered a basic symbol, although in set theory it can be defined. The formula x = y w(w x w y) is called the extensionality axiom. It is assumed as an axiom of set theory if equality is considered to be a predicate symbol; or it may be taken as the definition of equality. The concepts of informal set theory can all be defined in terms of membership and equality. However, it is necessary to posit that certain construction operations can be carried out to obtain new sets from already known sets. The axioms of set theory give formal rules for these constructions. For example, if objects x 1,...,x k are given then there is a set {x 1,..., x k } whose elements are exactly these objects. In set theory there is no distinction between an object and a set; but in specific settings it may be convenient to make such a distinction. For example, one can consider the integers as objects, and then consider sets of integers. 7

11 The integers can be defined within set theory as specific sets, in a way which by now is standard; this will be discussed further in section 13. The set containing no elements is called the empty set and denoted. The axioms of set theory ensure that it exists and is unique. It plays a role in set theory analogous to 0 in arithmetic. The main topics of informal set theory can be organized into the following areas. - Subsets, the power set, and operations on the power set. - Ordered ntuples and the Cartesian product. - Relations. - Functions. Each of these will be considered in turn. The website [Wiki, Naive set theory] is one of numerous references covering these topics, and has links to additional resources. Introductory set theory books such as [Monk1] cover them also, deriving basic facts from the axioms. Textbooks in other areas of mathematics frequently review informal set theory in introductory material, [Dowd1] for example. A set x is said to be a subset of a set y, written x y, if w x w y. By the extensionality axiom, x = y if and only if x y and y x. If x y but x y then x is said to be a proper subset of y, and this is written x y. It should be noted that, as usual, the foregoing is just one of various notational conventions in use. If x is a set then the collection of all its subsets comprises a set, called the power set of x, and denoted Pow(x). This is one of the construction principles provided in the axioms of set theory (indeed, it is the power set axiom). Note that x (the defining formula holds vacuously, since there are no w satisfying w ); and hence Pow(x) for any set x. Suppose U is a set; then the following operations may be defined on Pow(U). - union: w x y if and only if w x or w y. - intersection: w x y if and only if w x and w y. - complement: w x c if and only if w U and w / x. The following formulas are the axioms for the structures known as Boolean algebras, with the binary functions and, the unary function c, and the constants and U (structures are defined in section 6). - x y = y x, x y = y x - x (y z) = (x y) z, x (y z) = (x y) z - x (y z) = (x y) (x z), x (y z) = (x y) (x z) - x = x, x U = x - x x c = U, x x c = 8

12 It is easy to verify that Pow(U) forms a Boolean algebra with the operations given above. Further identities involving these operations may be proved from the axioms, with that advantage that they then have been shown not only for Pow(U), but for any Boolean algebra. Such identities may be found in various references, including [Dowd1]. The operations x y and x y are in fact defined for any pair of sets. A generalization of the union operation is important in the development of formal set theory. The complementation operation however is only defined on the subsets of a given set. The relative complement, or difference, x y may be defined for any sets x and y: w x y if and only if w x and w / y. The use of the minus sign for both subtraction of real numbers and relative complement causes no confusion. The context makes clear which is intended, with rare exceptions which can be clarified explicitly. For readers familiar with the concept of overloading from programming languages, the minus sign is overloaded, and may have arguments which are real numbers (or more generally elements of a commutative group); or sets. Additional terminology includes the following. A set y is said to be a superset of x, written y x, if x is a subset of y. Sets x and y are said to be disjoint if x y =. A set z is the disjoint union of sets x and y if z = x y and x y =. The symmetric difference x y of two sets equals (x y) (y x). As noted above, if x and y are objects there is a set {x, y} such that w {x, y} if and only if w = x or w = y. This in fact is the axiom of pairing. If x and y are the same object than {x, y} only contains a single object, otherwise it contains two objects. Also, {x, y} and {y, x} are the same set. One of the basic constructions of set theory is that of the ordered pair x, y of two objects x and y. This is designed to have the property that x 1, y 1 = x 2, y 2 if and only if x 1 = x 2 and y 1 = y 2. It is not necessary to add this as a basic construction principle; x, y may be defined to be {{x}, {x, y}}. It follows using the axioms of extensionality and pairing that with this definition x, y has the desired property. A history of the notion of ordered pair can be found in [Kanamori1]; the modern definition is therein credited to Kuratowski. The Cartesian product x y of two sets x and y is defined to be the set such that w x y if and only if w = w 1, w 2 where w 1 x and w 2 y. In a more convenient notation, the definition may be written as x y = { w 1, w 2 : w 1 x, w 2 y}. From hereon such notation will be used without further comment. In 9

13 formal set theory the Cartesian product is proved to exist from the axioms. In informal set theory the existence may be accepted as intuitively obvious; note, however, that x y Pow(Pow(x y)), and this fact is part of the formal existence proof. The Cartesian product x 1 x n of n sets may be defined recursively to be x 1 (x 2 x n ). There is an obvious correspondence between w 1, w 2, w 3 and w 1, w 2, w 3, which can usually be ignored, and the triple written as w 1, w 2, w 3, which in tedious formality is the first version. Similar remarks hold for other nested Cartesian products. An nary relation on a set x is defined to be a subset of x x, where there are n factors of x. If n = 1 the relation is called unary; a unary relation is the same thing as a subset. If n = 2 the relation is called binary. A function f from a set x to a set y is a subset of x y, such that for all u x there exists a unique v y, such that u, v f. A function assigns an element of y to each element of x. [Kanamori1] notes that the definition of a function in this generality was an early triumph of set theory, with Felix Hausdorff being a major contributer. Having a definition such as this, a function may be considered as an object, as is done in calculus for example. The notation f : x y is used to denote that f is a function from x to y. Basic definitions concerning such a function include the following. - f(u) = v may be written, rather than u, v f; similarly f(u) may be used for v in formulas. - In mathematical writing, the terminology graph of f is used for the relation f, although in formal set theory f as an object is the relation. - The domain of f is x; Dom(f) will be used to denote it. - For x x, f[x ] denotes {v : u x (f(u) = v}. - The range of f equals f[x]; Ran(f) will be used to denote it. - If x x the restriction of f to x is the set { u, v f : u x }. This is a function from x to y, which is denoted f x. - f is said to be injective, or 1-1, if f(u 1 ) = f(u 2 ) implies u 1 = u 2. - f is said to be surjective, or onto, if its range is y. - f is said to be bijective, or a 1-1 correspondence, if it is both injective and surjective. - If f : x y and g : y z then there is a function g f : x z, defined by the formula (g f)(u) = g(f(u)). This function is called the composition of g and f. - An nary function on a set x is just a function from x x to x, where there are n factors of x in the domain. A function is also called a mapping or map, emphasizing the fact that, 10

14 in addition to constituting an object itself, it has an active aspect. The function from X 1 X 2 to X i where i is 1 or 2, which maps x 1, x 2 to x i, is called a projection function. These functions are quite convenient, and will be denoted as π 1 and π 2. Note that, for example, Dom(f) = π 1 [f]. In formal set theory, a notion of the size, or cardinality, of an arbitrary set may be defined; this was an early triumph of set theory, due to Cantor. A treatment will be given in section 13; here a few facts are noted which will be needed before section 13. Given two cardinalities, one is greater than or equal to the other; and given any cardinality there are larger ones. For a nonnegative integer n, let N n be the set {0,...,n 1}; N 0 is the empty set (it will be seen in section 13 that in set theory the notation N n is unnecessary). A set x is said to be finite if there is a bijection from N n to x for some n. It may be shown by induction on k that if f : N k N l is a bijection then l = k. It follows that n is unique; this unique n is said to be the cardinality of x. A set is said to be infinite if it is not finite. Letting N denote the set of all natural numbers, a set x is said to be countably infinite if there is a bijection f : N x. Such a set is infinite. Rather than attempting to be encyclopedic in this section, additional definitions of basic set theory will be introduced as needed. 6. Structures and models. As already noted, set theory is a tool required in the development of mathematical logic. The notion of a universe of discourse referred to in earlier sections can be formalized using it. A first-order language is defined to be a set of predicate, function, and constant symbols. Each predicate or function symbol has a valency associated with it. For many purposes, the set may be finite; however there are contexts where infinite sets are used, and the definition may easily be given in this generality. In a special case of frequent interest, there may be an infinite set of constants, while the predicates and functions are a fixed finite set. Given a first-order language L, a structure for L consists of a nonempty set D, called the domain or universe of the structure, together with the following. - For each nary predicate symbol P of L, an nary relation ˆP on D. - For each nary function symbol f of L, an nary function ˆf on D. - For each constant symbol c of L, a element ĉ of D. The relation, function, or constant assigned to a symbol is called its interpretation. Predicate symbols are also called relation symbols. In this section, if = is in L, initially no restriction is placed on its interpretation. 11

15 Set-theoretically, a structure is a domain D, together with a function assigning to each symbol of L its interpretation. A frequently used notational abbreviation is to let D denote the structure, with the function understood, and let ˆP, etc. denote the interpretation of P according to the structure. The interpretation of a valency n predicate symbol is an nary relation. From hereon a valency n predicate symbol will be called nary. Likewise, a valency n function symbol will be called nary. A formal definition of the meaning of a formula F in a first order language, in a structure D for the language, will now be given. Typically of mathematical logic, the definition is a tedious and long-winded formalization of a fact which is completely obvious. To begin with, the semantics of the propositional connectives must be specified. Let {t, f} be the two element set of truth values true and false. A propositional connective denotes a function on this set; the same symbol will be used to denote this function as the connective itself. For the function is unary, with t = f and f = t, For the other connectives the function is binary, as follows. X Y X Y X Y X Y X Y t t t t t t t f f t f f f t f t t f f f f f t t Given a structure D and a set of variables V, an assignment to V is defined to be a function α which assigns to each x V an element of D. For a term t, let V t be the variables which occur in t. Similarly for a formula F let V F be the variables which occur free in F. Given a structure D, the interpretation ˆt of a term t is a function from assignments to V t, to D. It is defined recursively as follows. - If t is a variable x then ˆt is the function which assigns to the assignment α to {x}, the value α(x). - If t is a constant c then ˆt is the function which assigns to the empty assignment, the value ĉ of c in the interpretation. The ambiguity of the notation causes no confusion. - If t = f(t 1,...,t n ) and α is an assignment to V t, for 1 i n let α i be the assignment to V ti induced by α, i.e., α V ti. Then ˆt(α) = ˆf(ˆt 1 (α 1 ),..., ˆt n (α n )). Similarly, given a structure D, the interpretation ˆF of a formula F is a function from assignments to V F, to {t, f}. It is defined recursively as follows. - If F is an atomic formula P(t 1,...,t n ) then ˆF(α) = ˆP(ˆt 1 (α 1 ),..., ˆt n (α n )), where α i is as for terms. 12

16 For the remaining cases let α i = α V Fi. - If F is F 1 then ˆF(α) = ˆF 1 (α 1 ). - If F is F 1 F 2 then ˆF(α) = ˆF 1 (α 1 ) ˆF 2 (α 2 ). - If F is F 1 F 2 then ˆF(α) = ˆF 1 (α 1 ) ˆF 2 (α 2 ). - If F is F 1 F 2 then ˆF(α) = ˆF 1 (α 1 ) ˆF 2 (α 2 ). - If F is F 1 F 2 then ˆF(α) = ˆF 1 (α 1 ) ˆF 2 (α 2 ). - If F is xf 1 then ˆF(α) = t if and only if ˆF 1 (β) = t for all assignments β to V F1 such that β V F = α. - If F is xf 1 then ˆF(α) = t if and only if ˆF1 (β) = t for some assignment β to V F1 such that β V F = α. Some basic definition from mathematical logic are as follows. Fix a first order language L. - A formula is said to be a formula in (or over) L if its non-logical symbols are all in L. - If A is a set of formulas in L, and F is a formula in L, the notation A F is used to denote the fact that there is a proof of F in formal logic, using axioms from A, where all formulas of the proof are in L. - Given a structure D for L, and a formula F in L, = D F is used to denote the fact that F is true in D. - Given a set A of formulas in L, the fact that = F holds for every F A is denoted = D A, and D is said to be a model of (or for) A. - A set of formulas A is said to be consistent if for no sentence F do both F and F have proofs. Suppose = D A, and A F. It is straightforward (if tedious) to show that = D F. This fact is called the soundness of formal logic; it states that the logical axioms and rules are sound. A proof of this fact may be found in any of various introductory logic texts, including [Enderton], [Mendelson], and chapter 11 of [Dowd1]. Note that extra symbols may be allowed in a proof; this follows by simply enlarging (the technical term is expanding ) L. Suppose for any D, if = D A then = D F; then A F. This fact is called the completeness of formal logic. Not only does formal logic prove only true statements, it proves all statements which follow by logic alone from the non-logical axioms. That is, either a formula is true in some models and false in others (so additional axioms are needed); or it follows from the axioms by formal logic. The completeness theorem was first proved by Kurt Godel in 1929; a proof may be found in any of the above cited references. Given a proof of F from A, let A 0 be the formulas of A which occur in the proof; this set is finite. Let L 0 be the symbols of L which occur in A 0 or F. A model D for A 0 in L 0 may be considered a model 13

17 in L; and since there is a proof of F, it is true in D considered as a model in L, whence it is true in D considered as a model in L 0. By completeness, then, there is a proof of F from A 0 which uses only symbols from L 0. There are syntactic proofs of facts such as this, using Gentzen systems for example; see [Smullyan]. If a set A of formulas has a model then it is consistent, since for a sentence F only one of F and F can be true in the model, so only one can be provable. It follows by completeness that if a set A of formulas is consistent then it has a model. In fact, this is usually proved first, and completeness deduced from it. In some cases, a system of axioms A is intended to be used to prove theorems about a particular structure; Peano s axioms are an example. It is a fact of mathematical logic, however, that such systems will generally have other models than the intended one. Indeed, it follows from the Lowenheim-Skolem theorem that if A has infinite models then it has a model, of any infinite cardinality greater than or equal to the cardinality of the language. A proof of this may be found in the above cited references, and a version is given in section 20; see [Wiki, Lowenheim-Skolem theorem] for some historical comments. In the next section, a few comments will be made on models of Peano s axioms. On the other hand, some systems of axioms A are intended to be used to prove theorems about any of a variety of structures, namely those which are models of the axioms. This is a basic tool of abstract algebra; a system of axioms for structures of a certain type is specified, and the theory of these developed by deducing facts from the axioms. An example has already been seen, namely Boolean algebras in section 5; additional examples will be seen in section 8. If the language contains the equality predicate, say that a model is an E-model if = is interpreted as equality. By completeness, a consistent set A of formulas, which includes the axioms of equality, has a model M. M need not be an E-model; however an E-model can be constructed from M. It follows that, in considering systems of axioms where = is in the language and the axioms of equality are assumed, only E-models need be considered. An outline of the construction of an E-model will be given; see for example [Dowd1] for details. A binary relation satisfying the first three axioms of equality is called an equivalence relation. Given an equivalence relation, let [x] = {y : y x}; [x] is called the equivalence class of x. By the axioms, x [x], and two equivalence classes are either disjoint or equal. A binary relation on the domain of a structure D which satisfies all the axioms of equality is called a congruence relation. A structure D/ may be constructed, called the quotient of D by. This has as the ele- 14

18 ments of its domain, the equivalence classes. The value P([x 1 ],, [x n ]) for a predicate symbol P may be defined as P(x 1,...,x n ); the axioms ensure that the value depends only on the equivalence classes, and not the particular choice x 1,..., x n of representatives of the classes. Similarly f([x 1 ],, [x n ]) may be defined as [f(x 1,, x n )]. If α is an assignment in D, let α be the assignment in D/ which assigns to x the value [α(x)]. A straightforward induction shows that for any formula F, ˆF(α) in D equals ˆF(α ) in D/. In particular, if M is a model of A and is the interpretation of =, then M/ is a model of A. Clearly, it is an E-model. Assignments are somewhat cumbersome, and are used in mathematical logic for the definition of the semantics of formulas, etc. There is a more convenient method of referring to the semantics of a formula, which is in common use and will be used in this text (assignments will be used occasionally also). Suppose F is a formula, and v = v 1,...,v k is a list of variables which includes the free variables of F. Given elements x = x 1,..., x k in a structure S, let F v ( x) be ˆF(a) where a assigns x i to v i for 1 i k. It is common practice to use F( x) as an abbreviation for F v ( x), when the explicit list of the variables is not needed. Another variation in use is F( x 1,..., x k ); the variables are x i,...,x k, and x i is assigned to x i for 1 i k. k will frequently be used to denote the length of a list v. Thus, F v is a kary predicate on S. A predicate P which is F v for some F and v is said to be definable; the formula F defines P in S. For a formula to define a predicate, a correspondence must be given between the argument places of the predicate and the free variables of the formula. The value of the predicate depends only on the values assigned to the free variables; additional variables are allowed for convenience. 7. Models of Peano arithmetic. Models of Peano arithmetic have become a topic of interest in mathematical logic, [Kaye] being one reference on the subject. Let L A denote the language 0 s + =. Let N denote the structure of the non-negative integers over this language. This may be defined in set theory; facts to be given here provide some description of it. For a nonnegative integer n, let n be the term, 0 followed by n s s; this is called the numeral for n. Given a structure D in a language L, let Th(D) be the set of formulas in L which are true in D. Th(D) is called the theory of the structure D. Let PA denote the formulas which are provable from Peano s axioms. Let Q denote the formulas which are provable from the first 6 of Peano s axioms, and the formula x 0 y(x = y s ). Let D 1 and D 2 be structures for a language L. Let ˆ denote 15

19 the interpretation in D 1, and the interpretation in D 2. D 2 is said to be a substructure of D 1 if the following requirements hold, where x 1,...,x n D 1. - For each predicate P, P(x 1,..., x n ) if and only if ˆP(x 1,..., x n ). - For each function f, f(x 1,...,x n ) = ˆf(x 1,...,x n ). - For each constant c, c = ĉ. A function h : D 1 D 2 is said to be a homomorphism if the following requirements hold, where x 1,...,x n D 1. - For each predicate P, P(h(x 1 ),..., h(x n )) if and only if ˆP(x 1,...,x n ). - For each function f, f(h(x 1 ),..., h(x n )) = h( ˆf(x 1,..., x n )). - For each constant c, c = h(ĉ). The third requirement is redundant, since a constant is a 0-ary function symbol. Some authors (such as [Dowd1]) weaken the requirement for predicates, and call a homomorphism as above a strong homomorphism; others (such as [Sacks1]) give the above definition. It is readily seen that if h is a homomorphism then h[d 1 ] may be made into a substructure of D 2 in a unique way (or see [Dowd1]). If h is an injection then it is called an isomorphic embedding of D 1 in D 2. If h is a bijection then it is called an isomorphism of D 1 with D 2. If D is any structure for L A, the predicate x y is defined by the formula w(y = x + w). The following are some basic facts concerning the above defined concepts. Let M denote a model of Q. 1. Th(N) has models other than N; such models are called nonstandard. 2. Q PA Th(N). 3. The map h defined by the formula h(n) = ˆn is an isomorphic embedding of N in M. 4. If y M and y h(n) for some n N then y = h(m) for some m N (h[n] is said to be an initial segment of M). 5. Suppose M satisfies the second order induction axiom, that is, for any subset S M, if 0 S, and x(x S x s S), then x(x S). Then h is an isomorphism. Fact 1 was first observed by T. Skolem in 1933; a proof is as follows. Let be a new non-logical symbol, and add to Th(N) the formulas n < for each integer n. If the enlarged set of formulas were inconsistent, there would be some finite set of the added formulas which, when added to Th(N), would result in an inconsistent system. But this is impossible, because the ordinary integers with a large enough value assigned to would be a model. Thus, the enlarged set has a model, 16

20 and this is a model of Th(N) which contains an element greater than every standard integer. To prove fact 2 it is only necessary to give a proof in PA of x 0 y(x = y s ); this is an easy exercise, or may be found in [Yasuhara]. Fact 3 follows from the following facts, where denotes provability in Q. - If k + l = m then k + l = m. - If k l = m then k l = m. - If k l then k l. Fact 4 follows from the additional fact - x k (x = 0 x = k). Proofs of these facts can be found in [Yasuhara]. To prove fact 5, let S be h[n]. The axiom of fact 5 is called second order because it involves the use of subsets of the universe of discourse, and must be formalized within set theory (or at least an adequate fragment of it). Together with the preceding facts, it may be seen that second order methods are stronger than strict first order methods. 8. The real numbers. Like the integers, the real numbers are fundamental mathematical objects, which are familiar from everyday life, and form a mathematical universe of discourse. The real numbers may be constructed from the non-negative integers N in informal set theory, and second order axioms can be given which completely characterize the structure. It is valuable to first construct some substructures which are themselves fundamental mathematical objects. The structures to be constructed are the integers Z, the rational numbers Q, and the real numbers R. Some families of structures will be defined, of which the preceding structures are important examples. The language of commutative rings is =. The axioms for commutative rings are C1 (x + y) + z = x + (y + z) C2 x + y = y + x C3 x + 0 = x C4 For all x there exists y such that x + y = 0 C5 (x y) z = x (y z) C6 x y=y x C7 x 1 = x C8 x (y + z) = x y + x z Various additional facts can be shown readily from the axioms; these may be found in any of numerous introductions to abstract algebra, 17

21 including [Dowd1]. In particular, subtraction may be defined, and its basic laws proved. N is not a commutative ring, because axiom C4 does not hold. N can easily be enlarged to a structure which is a commutative ring, by adding the negative integers. One method of doing this is as follows. On N N, define the binary functions - m 1, n 1 + m 2, n 2 = m 1 + m 2, n 1 + n 2 and - m 1, n 1 m 2, n 2 = m 1 m 2 + n 1 n 2, m 1 n 2 + m 2 n 1 ; and the binary predicate - m 1, n 1 m 2, n 2 if and only if m 1 + n 2 = n 1 + m 2. By straightforward if tedious calculation is verified to be a congruence relation on N N with +. The equivalence class [ m, n ] will represent m n. In the quotient (N N)/, + and are defined by the above equations. Another straightforward calculation shows that the quotient is a commutative ring, with [ 0, 0 ] as 0, [ 1, 0 ] as 1, and [ m, n ] + [ n, m ] = 0. This is the ring Z. The function h where h(n) = [ n, 0 ] is an isomorphic embedding of N in Z. A binary predicate on a set D is said to be a partial order if the following hold. 1. x x (reflexive law) 2. x y and y z imply x z (transitive law) 3. x y and y x imply x = y (antisymmetry law) A partial order is a linear order if the following also holds. 4. x y or y x The subset order on Pow(U) for a set U is an example of a partial order which is not a linear order (provided U has at least two elements). The relation on N, defined to hold if w(y = x + w), is a linear order. Given a partial order, the predicate x < y may defined by the formula x y x y. This relation is called the strict part of the partial order, and satisfies the transitive law and x x. On the other hand, given such a predicate the relation x < y x = y is a partial order. If is a partial order on D and S D then x S is said to be a least element of S if x y for all y S. An element x D is said to be an upper bound for S if y x for all y S. An upper bound x for S is a least upper bound if x x whenever x is an upper bound. An ordered commutative ring is one where a unary predicate P (positive) has been added to the language, and satisfying the following additional axioms. O1 P(0). O2 if x 0, exactly one of P(x) or P( x) holds. 18

22 O3 P(x) P(y) P(x + y). O4 P(x) P(y) P(x y). Properties which follow immediately include the following: - 1 is positive (unless 0=1 and the ring is trivial); - the relation P(x y) is the strict part x > y of a linear order x y on the ring; - if x < y then x + z < y + z, and if x y then x + z y + z; and - if x < y then y < x, and if x y then y x. The absolute value x is defined to be x if x is positive or 0, else x. This satisfies the triangle inequality x + y x + y. Axioms can be given using the order predicate; using positivity results in a slightly simpler set of axioms. Z is an ordered commutative ring; the elements [ n, 0 ] for n 0 constitute a set of positive elements. If M is any ordered commutative ring, mapping 0 and 1 to 0 and 1 induces a unique isomorphic embedding of Z in M. The following second order axiom ensures that the embedding is in fact an isomorphism. - If S M is nonempty and bounded below then S has a least element. A proof will be outlined. Call elements of the image of the embedding integers. There can be no element greater than every integer. If not, let S be the set of such, and let a be the least element of S. Then a 1 m for some integer m, whence a m + 1, a contradiction. There can be no element less than every integer; if a is such then a is greater than every integer. Suppose m < a < m + 1 where m is an integer. Then 0 < b < 1 where b = a m. The set {b j : j N } is a set which is bounded below but has no least element. A field is a commutative ring which satisfies the following additional axioms. F1 For all x, if x 0 then there exists y such that x y = 1 F2 0 1 An ordered field is an ordered commutative ring satisfying F1 and F2. Z is not a field, because there is no x such that 2 x = 1, as may be easily verified. Z may be enlarged, to construct a field, as follows (in fact this construction may be carried out in any integral domain, which is a commutative ring satisfying some additional axioms). Let Z denote the nonzero elements of Z. On Z Z, define the binary functions - m 1, n 1 + m 2, n 2 = m 1 n 2 + m 2 n 1, n 1 n 2 and - m 1, n 1 m 2, n 2 = m 1 m 2, n 1 n 2 ; and the binary predicate 19

23 - m 1, n 1 m 2, n 2 if and only if m 1 n 2 = m 2 n 1. By straightforward calculation is verified to be a congruence relation on Z Z with +. The equivalence class [ m, n ] will represent m/n. In the quotient (Z Z )/, + and are defined by the above equations. Another straightforward calculation shows that the quotient is a field, with [ 0, 1 ] as 0, [ 1, 1 ] as 1, and, provided m 0, [ m, n ] [ n, m ] = 1. This is the field Q. The function h where h(n) = [ n, 1 ] is an isomorphic embedding of Z in Q. Q is an ordered field; the elements [ m, n ] where m, n > 0 constitute a set of positive elements. If M is any ordered field, mapping 0 and 1 to 0 and 1 induces a unique isomorphic embedding of Q in M. Clearly Q is the unique ordered field which is isomorphically embedded in any ordered field; this seems to be the best uniqueness property for Q. The rational numbers suffer from a deficiency. Let S = {q Q : q 2 < 2}; it is not difficult to show that if S has a least upper bound r then r 2 = 2; and there is no r Q such that r 2 = 2 (this is proved in the ancient Greek text Euclid s Elements ). Thus, S does not have a least upper bound in Q. Q can be enlarged, so that the deficiency just mentioned is eliminated. This was an important issue in the history of mathematics, and its resolution was important to early set theory. See [MacTutor, Real numbers] for remarks on the history of the subject. The construction to be outlined below can be found in numerous references, [Rudin] for example. A linearly ordered set D is said to have the least upper bound property if, whenever S D is nonempty and has an upper bound, then S has a least upper bound. Q does not have this property. One method of constructing the real numbers is to enlarge Q to a linearly ordered set which does have the property. It turns out that there is exactly one way to do this. If D is a set with a partial order on it, say that a subset S D is -closed if x S w x w S. Considering Q with its usual order, define a cut to be a set of rationals which is nonempty, bounded above, -closed, and has no greatest element. Let R be the set of cuts; R will be equipped with interpretations for = P, to produce a structure for this language. For q Q let q < denote {r Q : r < q}; this is readily seen to be a cut. To begin with, some facts about R will be proved using only the order on Q; these facts are of interest in themselves. A linear order is said to be a dense linear order without endpoints if it satisfies the additional axioms 20