Introduction to Model Theory and to the Metamathematics of Algebra (Studies in Logic and the Foundations of Mathematics, 32)

I N T R O D U C T I O N TO M O D E L THEORY A N D TO T H E METAMATHEMATICS O F ALGEBRA ABRAHAM ROBINSON University of ...

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I N T R O D U C T I O N TO M O D E L THEORY A N D TO T H E

METAMATHEMATICS O F ALGEBRA

ABRAHAM ROBINSON University of California, Los Angeles

1963

NORTH-H 0L L A N D PUBLISH I N G COMPANY AMSTERDAM

No part of this book may be reproduced in any form by print, microfilm or any other means without written permission from the publisher

PRINTED I N THE NETHERLANDS

PREFACE

Souvienne vous de celuy B qui, mmme on demanda B quoi faire il se peinoit si fort en un art qui ne pouvoit venir B la cognoissance de gueres de gents: ”Yen ay assez de peu, respondit-il,fen ay assez d’un, j’en ay assez de pas un.”

MONTAIGNE, De la Solitude

The author’s book “On the Metamathematics of Algebra” which was published in this series in 1951 has now been out of print for some time. The book was concerned with the logical analysis of the methods of Abstract Algebra and beyond that, set out “to make a positive contribution to Algebra using the methods and results of Symbolic logic.” This involved among other things the development of certain topics in what is now known as Model Theory. In the years since the publication of “On the Metamathematics of Algebra” the subject has developed vigorously. Accordingly, it was decided to replace the book by an entirely new work. The result is the present volume. At an estimate, less than half of its material was given already in the earlier book, and much of this material is presented here in an different way and with a simplified terminology. With one or two exceptions, the remainder of this volume is concerned with more recent developments. The general character of the work has been changed to some extent and it should now be suitable as a textbook for a first year graduate course on the subject. Many of the topics included in “On the Metamathematics of Algebra” - such as the development of a non-countable language, the properties of classes of structures which are closed under extension or under intersection, the method of diagrams, the completeness of the notion of an algebraically closed field of given characteristic, particular applications to Algebra - are by now well - established and there is no need to justify their inclusion in the present book. On the other hand, the suggestion V

VI

PREFACE

that numerous important concepts of Algebra possess natural generalizations within the framework of the Theory of Models has met with a rather less lively response. Nevertheless, it is still the author’s belief that investigations in this direction are both interesting and valuable. Indeed, the theory of algebraic ideals and varieties, up to the existence of the generic point for irreducible varieties, the notion of an algebraically closed extension, the notion of a system of resultants to a given set of equations, - all these can be discussed profitably in a metamathematical setting. Apart from providing a certain unity of outlook, this approach occasionally also produces new algebraic results, e.g. in the case of the concept of a differentially closed field. In the last three sections of the book, we present an introduction to Non-standard Analysis. This is a new application of Model theory which provides an effective calculus of infinitesimals and which appears to have considerable potentialities. Both the author’s working energy and the measure of patience that can be expected of a prospective reader imposed a limit on the scope of the book. Accordingly, it was not found possible to include a number of relevant items such as the theory of undecidable algebraic systems and problems (Tarski, Mostowski, R. M. Robinson, J. Robinson, Novikov, Boone, Markov, Rabin), the theory of computable algebraic systems (Frohlich-Shepherdson, Rabin, Higman), the proof-theoretic investigation of Arithmetic and Algebra (Kreisel), the properties of direct products of given structures (Mostowski, McKinsey, Horn, Fefermann-Vaught, Bing, Overschelp), the solution of the homomorphism problem (Lyndon), the model-theoretic approach to set theory (Godel, Shepherdson,Vaught, Montague, Mendelsson, Lkvy), the algebraization of quantification theory (Tarski, Henkin, Halmos), and various other topics which are mentioned in the text. The author takes this opportunity to emphasize the importance of these subjects, for which the reader should consult the original papers. Once again, my thanks are due to the editors of “Studies in Logic” and to the North-Holland Publishing Company for having invited me to contribute this volume to the series. In particular, I wish to put on record my indebtedness to Arend Heyting who read my books and papers before publication on several previous occasions. I am also grateful to Mr. D. Louvish for his help in reading proofs. ABRAHAM ROBINSON Hebrew University, Jerusalem, December, 1961

CHAPTER I

THE LOWER PREDICATE CALCULUS

1.1. General Introduction. The Metamathematics of Algebra is concerned with the analysis and development of Algebra by the methods of Mathematical Logic. Model theory deals with the relations between the properties of sentences or sets of sentences specified in a formal language on one hand, and of the mathematical structures or sets of structures which satisfy these sentences, on the other hand. Since the methods employed in Model theory are frequently algebraic in spirit if not in detail and since the algebraic theories of fields, rings, groups, etc. are in view of their transparent structure well suited to a detailed logical analysis, Model theory and the Metamathematics of Algebra supplement one another in a natural way. Indeed, in many cases it is hard to decide whether a particular topic belongs properly to the former subject or to the latter. The algebraic approach has penetrated into many other branches of Mathematics, and it was inevitable that the Metamathematics of Algebra should follow suit. Again, since the language of Topology has proved equally ubiquitous it can be used also in connection with the matters dealt with in this book. In fact, the task of drawing precise boundary lines within Mathematics has become notoriously difficult, and the above definitions are intended only to give a general indication of our purpose. However, in order to keep this book within reasonable proportions, we shall choose most of our examples from Algebra and we shall dispense with the use of topological language. The plan of the book is as follows. In the present chapter, we set up the formal language of the Lower predicate calculus, which is fundamental for the sequel. We then discuss the relations between the sentences of our formal language and the mathematical structures in which they hold. Finally, we establish the extended completeness theorem of the Lower predicate calculus and we derive some of its immediate consequences. The second chapter begins with a discussion of some fundamental concepts which are relevant to algebraic systems such as the notions of equality, of extension, of isomorphism, of homomorphism. This is 1

2

THE LOWER PREDICATE CALCULUS

11.1.

followed by the specification of axiomatic systems of some common algebraic concepts. The chapter concludes with a number of fairly direct applications of the extended completeness theorem, including Malcev’s theory of normal chains. In the third chapter, we discuss a number of typical problems of Model theory. These include the set-theoretic characterizations of systems of structures which are given by axioms in prenex normal form with certain special types of prefixes (e.g. with existential quantifiers only). Systems of structures which are closed under intersection are also considered. The fourth chapter deals with various notions of completeness, including model-completeness and relative model-completeness. Some of the most important instances of these notions are considered in detail. The first two sections of the fifth chapter are concerned with Beth’s theorem on definability in the Lower predicate calculus and with related results. This is followed by the discussion of another kind of definability problem which arises in Algebra, and by a metamathematical analysis of the notion of an algebraically closed extension and of related concepts. The chapter concludes with applications to Differential Algebra and to Hilbert’s theorem on the zeros of polynomials, the Nullstellensatz. Chapter VI contains logical analyses and generalizations of various standard algebraic concepts such as the notion of polynomial extension and of separability. The seventh chapter deals with the metamathematical theory of ideals, and the eighth chapter is concerned with the varieties of metamathematical ideals. It is shown that this includes the theory of algebraic varieties as a special case. Other applications deal with differential ideals and with a generalization of Artin’s theorem on Hilbert’s seventeenth problem concerning the representation of definite functions as sums of squares. The use of function symbols and their application to the elimination of quantifiers is discussed in the ninth and last chapter. This is followed by a brief description of the ultraproduct construction and of its fundamental properties. The chapter concludes with an introduction to Non-standard analysis. It is shown that the metamathematical approach provides a tool for the development of function theory in certain nonarchimedean fields, and that the resulting methods can be used for the proof of theorems in classical Analysis. The reader is expected to be familiar with the elements of the Lower predicate calculus up to the transformation of a sentence into prenex

1.2.1

3

RULES OF FORMATION

normal form. On the mathematical side, we shall make use of some standard results concerning groups, rings, and fields. We shall use the following set-theoretic notation - a E A denotes the membership relation, a is an element of A . A = B and A # B state that the sets A and B are equal (co-extensive, contain the same elements) or unequal, respectively. A C B and B 2 A indicate that A is a subset of B, including the possibility that A equals B. The union, intersection, and difference of two sets A and B will be denoted by A u B, A n B, and A - B respectively. The union and intersection of a set of sets {A,] will be denoted by {Ad

u

and by

V

nM . V

The cardinal number of a set A will be denoted by I A I. The sign of equality will be used also in order to define one symbol in terms of another symbol, or in terms of a group of symbols, e.g., X = [ [ A ] 3 [ B ] ] .

1.2. Rules of Formation. We set up a formal language L by means of the following rules : The atomic symbols of L are The (individual) object symbols, usually denoted by a, b, c, .. ., small italics, near the beginning of the alphabet, with or without subscripts, occasionally by other symbols such as numerals, 0, 1, 2, . .. to conform with common usage. They constitute a well-defined set of arbitrary transfinite cardinal number. The dummy symbols or variables, denoted by u, v, w, x , . . . ; these symbols are supposed to constitute a countable set. The relative symbols, divided into disjoint classes, R,, n = 0, 1,2, .. . (relative symbols of order n, or n-place relative symbols. Relative symbols of order n 2 1 will be denoted by A ( , , .. .), B ( ,), . . . (capital italics followed by n empty spaces, separated by commas, in round brackets). Relative symbols of order 0 will be denoted simply by A , B, C, . .. The classes R, constitute well-defined sets whose cardinals are specified but arbitrary. The connectives. These are the five symbols (negation), V(disjunction), A (conjunction), 2 (implication), and (equivalence). The quanti3ers. These are - the universal quantifier, denoted by (V ) and the existential quantifier, denoted by (3 ). N

4

THE LOWER PREDICATE CALCULUS

p.2.

The separation symbols, denoted by [ (left square bracket) and ] (right square bracket). This completes the list of atomic symbols. Atomic formulae are obtained by filling the empty spaces of a relative symbol of order n, n = 1, 2, . . . with object symbols or variables. For instance if A ( ,,) is a 3-place relation, then A (a, b, a) and A (b, a, z) are both atomic formulae. 0-place relative symbols are atomic formulae by themselves, by definition. Well-formed formulae, briefly ‘wff’ or ‘formulae’ are now defined inductively. They will be denoted by capital italics taken from the end of the alphabet, V, W, X, .. . Observe that these symbols do not belong to the formal language L. 1.2.1. Atomic formulae bracketed by square brackets are wff.

X and Y are wff, then [X V Y ] , [ X A Y ] , [ X 3 Y ] ,and [ X = Y ] are all wff.

1.2.2. If X is a wff then [- XI also is a wff. If

1.2.3. If X is a wff then [ (Vy) X ] and [ (3y) X ] are both wff, provided X does not already contain one of (Vy) or (3y). Thus, [ (3y) [ A(a, x ) ] ] and [ (3y) [ A(a, y ) ] ] are both wff while [ (Vy) [ (3y) [AQ)]]]is not a wff. It will be understood without detailed explanation what is meant by the phrase “X contains (Vy)”, etc. We shall also say that a wff X contains a wiT Y if X is constructed from Y and other wff by the (repeated) use of 1.2.2. and 1.2.3. We shall assume throughout that the language with which we are concerned contains at least one wff. WE are divided into complete formulae, or sentences, and incomplete formulae, or predicates, as follows. A wff X is called complete if whenever a variable is contained in it, say y , then y is contained in a wff Z such that 2 occurs in X in one of the forms [ (Vy)Z] or [ ( 3 y ) Z l . If y occurs in X more than once - excepting the cases where it occurs within the brackets of a quantifier - then the above condition is supposed to be satisfied whenever y occurs. For example, [ (Vy) [ [ A(y ) ] A [B (y ) ]] ] and [ [ (Vy) [ A( Y ) ] ] [ ($9 [ C ( r , a ) ] ] ]are both complete, while [ [ (Vy) [ A( y ) ] ]A A [ B ( y ) ] ]is incomplete. We may express this definition in a different way by introducing the notion of the scope of (an individual occurence of) a quantifier, e.g. (3y) within a wff X. This is the wff contained in X which begins with the left bracket following immediately upon (3y) and ends with the corresponding right bracket. X is then complete if every variable contained in a relation within Xis within the scope of a quantifier

1.2.1

5

RULES OF FORMATION

with the same variable. Any variable in X which does not have this property is calledfree (or more precisely, the occurence of the variable in question is free). We define the order of a wff as the number of pairs of square brackets contained in it. Thus, the order of [ A @ ) ] is 1 and the order of [[ (Vx) [ B ( x ) ] ]V [ C ( u ) ] ]is 4. The order of a wff constructed by negation or quantification (see 1.2.2. and 1.2.3. above) exceeds by 1 the order of the formula from which it is constructed. The order of a formula constructed by disjunction, conjunction, implication, or equivalence exceeds by 1 the sum of the orders of the two formulae from which it is constructed. The above is a version of the language of the Lower (restricted) predicate calculus which is distinguished by its straightforwardness at the cost of some lack of economy. Thus, it is known that three of the connectives (conjunction, implication, and equivalence) and one of the two quantifiers can be expressed in terms of the remaining symbols. Moreover, in our formulation a large number of brackets is required even for relatively simple expressions. An advantage of this notation is that it indicates quite clearly the mode of construction of a formula. However, later we shall simplify our notation by adopting the following rules. In the successive construction of a wff from a set of atomic formulae, the following square brackets may be omitted. Square brackets enclosing atomic formulae. Square brackets following a negation provided they enclose a negation. For example, [ [ X I ] may be replaced by [ XI. Square brackets preceding or following the symbol of conjunction provided they enclose a negation; square brackets preceding or following the symbol of disjunction, provided they enclose a conjunction or a negation. For example, [ [X A Y ] V [- Z ] ] may be replaced by [XA Y V ZJ. Square brackets preceding or following the sign of implication or of equivalence, provided they enclose a disjunction or a conjunction or a negation. In any sequence of quantifications such that both the quantifiers on the left and the brackets on the right follow immediately upon one another, all the square brackets may be omitted, with the exception of the innermost and the outermost pairs (which may however be removable by virtue of another rule). For example, [ (Vx) [ (3y) [ (Vz) [ X I ] ] ]becomes [ (W (3Y) (Vd [ X ] ] Finally, the outermost brackets in a wff may also be dropped. For

--

-

N

-

-

-

-

6

THE LOWER PREDICATE CALCULUS

example

[ (W [ (9) [ (V.4 [ [ [ A(x, Y ) ] A [ A(v, 411 may be written in simplified form as

t1.3.

= [ A(x,.>I ] ] ] ]

(W (VY) (W [ A (XY Y ) A A (YY 4 = A (x,41 * The above rules are framed in such a way that any formula which has been simplified by the use of some or all of them can be restored to its fully bracketed form without ambiguity. Rules for the omission of brackets in successive conjunctions or disjunctions are not included, since they involve the validity of the associative law for these operations. However, later on we shall introduce the further simplification of denoting the conjunction and disjunciton of any number of wff taken in any order of association by [XI A X2 A . . . A Xn] and [XI V X2 V . .. V Xn], respectively. Any statement or argument involving such expressions will then be meant to apply whenever the latter are replaced by corresponding fully bracketed expressions. Thus, [ X I V X2 V X3] will be replaceable by [XI V [X2 V X3]] or by [[XI V X Z ]V X3] where the particular choice of the fully bracketed expression is irrelevant. Moreover, different fully bracketed expressions may be selected for the same simplified formula if the latter occurs more than once. Finally, we may mention that some of the subsequent work is simplified if we rule out wfT in which the same variable appears more than once within the brackets of a quantifier (e.g. xin (3x)). This can be done without limiting the scope of our language, but the practice will not be adopted here. 1.3. Rules of deduction. From the set of sentences in L we now select a subset - to be called the set of theorems of L - by a purely formal procedure. Given any sentences X , Y, Z in L, the following are theorems. 1.3.1.

‘ X = [ Y = XI1 [ X = Y ] ]3 [ X = Y ] ] “ X D Y ] = [ [ Y DZ ] = [ X = Z ] ] ‘[XA Y ] = X ] . [ X AY ] 3 Y ] - [ X = Y ] = “X=, Z ] = [ X I [ Y V Z ] ] ] ] 2 3 [XV Y]] ‘Y = [XV Y ] ] & [ X IZ ] = “ Y = Z ] = “ X V Y ] = z ] ] ] [X- Y ] 3 [X = Y ] ]

x=

1.3.1

RULES OF DEDUCTION

7

[[I=Y ] = [ Y = XI]

Y ] = [ [ Y = XI = [ X E Y ] ] ] [ [ X I YI = [ [- YI = [- X I ] ] [ X = [- [- X I ] ] [- X I ] = X ]

[ [X =

"-

In simplified notation, the sixth of the above sentences, for example, becomes [ X I Y ] =) [ [ X = 21 I [ X = Y A Z ] ] We shall sometimes single out an object symbol or free variable with reference to a wff X by writing X(u) or X ( y ) in place of X . In the same way we may display several object symbols or variables, without necessarily mentioning all object symbols or free variables contained in X. Given X(u) we then mean by X ( b ) the result of substituting b for a wherever a occurs in X, with a similar notation for the substitution of variables. With these conventions, the following are supposed to be theorems, for arbitrary X , a, y.

1.3.2.

[ [ (VY) [ X ( Y ) l ] = X ( 4 3 [ [X(a)l = [(W X(Y)l]]

provided these are sentences in accordance with the rules of formation. Further theorems are obtained by the application of the following three rules of inference :

1.3.3. If X and [X 3 Y ] are theorems, then Y is a theorem. This is the rule of modus ponens. If [ X 3 Y(u)] is a theorem, where X does not contain a, then [X 2 [ (Vz) Y ( z ) ] ]is a theorem, provided it is a sentence. If [Y(u) X ] is a theorem, where X does not contain a, then [ [ (32) Y ( z ) ] 3 X ] is a theorem, provided it is a sentence. It should be understood that the symbols a and z mentioned in these rules signify arbitrary object symbols or variables, respectively. All rules of substitution are then deducible and need not be introduced as postulates. Two of these which are of frequent use are as follows

1.3.4. From any theorem X which contains a quantifier with variable y, another theorem is obtained by replacing y by any other variable z both in the quantifier and in its scope, provided that scope is not part of the scope of a quantifier which contains z.

8

r1.3.

THE LOWER PREDICATE CALCULUS

1.3.5. In any theorem, we may replace a formula which is obtained by bracketing a relative symbol of order 0 by an arbitrary sentence. The result is a theorem provided it is a wff at all. Two more standard results of the Lower predicate calculus which will be required in the sequel are as follows:

1.3.6. If

[[[ ...[X l A X z I A X 3 1 A ... AXn] 3 and

[[[ ... [ Y i A Yz] Y3]

...

Ym], m = l , 2 Yk] 3

,..., k

z]

are all theorems then

is a theorem.

[ [ [ ... [XiAXz]AXs]A ... A X n ] 321

1.3.7. To every sentence X there exists a sentence X ' which contains the same object and relative symbols as X such that X s X' is a theorem, and such that x' is in prenex normal form, i.e. where the qk denote quantifiers while Y, the matrix of the sentence, does not contain any further quantifiers. We include the possibility n = 0, in which case X' contains neither quantifiers nor variables. Let K be a set of sentences in L. We say that a sentence Y is deducible from K, in symbols K F Y, if there exists a finite sequence of sentences XI, XZ,. .., X,, in K such that

[ [ [ ...[X I A X ~ I A X ... ~ ] AX,,]

3

Y]

is a theorem in L. By a natural convention we include the possibility = 0, in which case it is understood that Y itself is a theorem. The set of sentences in L which are deducible from K will be denoted by S ( K ) . S (K) includes K as well as all theorems of L. Using 1.3.6. we may show that for all K, S ( S ( K ) ) = S ( K ) . If K c S ( K ) then K' is said to be deducible from K, in symbols K F K . A set of sentences Kis called contradictory if S ( K ) includes all sentences of L, otherwise K is consistent. K is contradictory if and only if S ( K ) contains a sentence of the form [X A [ X I ] . In addition to the theorems which were obtained above as a certain subclass of the sentences of L, we shall also have occasion to formulate

n

N

1.4.1

SEMANTIC INTERPRETATION

9

theorems outside L, in the usual sense of the term. By way of distinction these are sometimes called meta-theorems. We shall not use this term and shall instead relay on the context for elucidation. 1.4. Semantic interpretation. We now come to the semantic, or descriptive, interpretation of the sentences of the given language. A mathematical StructureMwhichcan be described by sentencesof Lis of the following type. It consists of a set of (individual) objects or individuals which (like the object symbols) will be denoted by small italics a, b, . . . and of sets of relations of order n, (e.g. A ( ), B ( , ), ...) such that for every relation A of order n defined in M and for every ordered n-ple al, ..., a n of different or identical constants of M, the instance A (al, .. .,a n ) of the relation either holds or does not hold (in M). We shall indicate the situation also by saying that A holds, or does not hold, at (al, . . ., an). We do not identify the relations as such with sets of ordered n-ples of individual objects of M so that two relations may hold at the same n-ples. We also include the possibility that relations of order 0 belong to M. Such a relation then holds, or does not hold in M, without reference to the individual objects of M. Relations of order 0 do not appear in the familiar mathematical structures which will be considered later. However, they do appear as the elements of structures defined in connection with the propositional calculus. On the other hand such structures do not normally contain any objects, or relations of positive order. Let C be a one-to-one correspondence which maps the individual objects of M on a subset of the set of object symbols of a language L and which, at the same time, maps the relations of M on relative symbols of L of the same order. Let K be the set of wff of L whose object and relative symbols all appear in C. We say that these wff are dejhed in M under C. To every atomic formula X in K which does not contain any variables there corresponds, by C, the expression X' of a relation between certain individuals of M , which either holds or does not hold in M. The following rules then determine, by definition, whether a sentence of K holds or does not hold in M (under the correspondence C). 1.4.1. Let Y be a sentence of order 1 in K, Y = [XI where X is an atomic formula. Then Y holds in A4 if and only if the expression X', which corresponds to X under C, holds in M. 1.4.2. Given two sentences Y and Z in K,

[YV 21 holds in M if and only if at least one of the two sentences Y

10

T€i@LOWER PREDICATE CALCULUS

[1.4.

or 2 holds in M; [ Y A Z ] holds in M if and only if both Y and 2 hold in M, [ Y 3 Z ] holds in M if Z holds in M , and also if neither Y nor Z holds in M ; [ Y 3 21 holds in M if both Y and 2 hold in M and also if neither Y nor Z holds in M ; and finally, [ Y]holds in M if and only if Y does not hold in it. N

1.4.3. Given a wff Y = Y(x)in which z is not quantified (does not appear within a quantifier) and no other variable is free, [ (Vz) Y (z)] holds in M if and only if Y(a) holds in M for all object symbols a in L which correspond to objects in M; [ (32) Y (z)] holds in M if and only if Y(a)holds in M for at least one a in L which corresponds to an object in M. Since each of the above rules 1.4.2., 1.4.3. bases the decision whether a sentence does or does not hold in M on a sentence of lower order while by 1.4.1. the decision whether or not a sentence of order 1 holds in M depends directly on M, it follows that these rules determine uniquely for any Y in K whether Y holds or does not hold in M (under C). However, the rule 1.4.3. is not effective in any constructive sense since it leaves the decision whether or not Y holds in M on the corresponding question for a set of sentences which may be infinite. Now let K be any set of sentences in L and let C be a one-to-one correspondence which maps the object symbols occuring in sentences of K (if any) on objects of M, and the relative symbols of K on relations of M of corresponding order. Extend C in some way to a correspondence C' as considered above, i.e. such that all objects and relations of M are mapped in one-to-one correspondence on object and relative symbols of L. Then it is not difficult to see that the answer to the question whether or not such a sentence of Kholds in L depends only on the correspondence C and not on the particular choice of the extension C'. However, since the cardinal number of the set of object symbols of L is limited it may happen that the number of objects in M is so large that no correspondence C' of the required type exists. This can be avoided by considering, in connection with any given structure M, only languages which possess a sufficiently large pool of object symbols from the outset. Alternatively, L as given may be embedded in a more comprehensive language L', the particular choice of L' being again irrelevant. If all sentences of a set K hold in a structure M under a correspondence C then we say that M is a model of K (under C). If K contains only a single sentence Y, then we shall say also that M is a model of Y.

1.5.1

RELATION BETWEEN DEDUCTIVE A N D DESCRIPTIVE CONCEPTS

11

So far, we have distinguished strictly between object and relative symbols on one hand, and individual objects and relations on the other. However, it frequently simplifiesmatters a great deal to suppose that the objects and relations of a structure M coincide with object and relative symbols of a language L in which M is described. In that case it is usual to suppose that these objects and relations coincide individually with the corresponding object and relative symbols of L, i.e., C is the identity. For certain purposes, it is useful to consider a language which is based on the same types of atomic symbols as the language considered above and whose set of wff is extended by the introduction of certain infinitary rules of formation. There exists a considerable body of information on such languages, but we shall make use of them only in some special cases. Thus, if XI, X Z ,Xs,. . . is a countable sequence of wff we may include among the wff also the infinite disjunction [XI V X ZV X3 V . . .] and the infinite conjunction [XI A X2 A X3 A . . .I. If all object and relative symbols of the Xi are reIated to the objects and relations of a structure M by a correspondence C as before, then we say that the above infinite disjunction holds in M if at least one Xt holds in M, and the infinite conjunction holds in M if all Xt hold in M . 1.5. The Relation between Deductive and Descriptive Concepts. Let K be the set of sentences of L which is defined in a structure M under a correspondence C. Then 1.5.1. Every theorem of L which is included in K holds in

M.

We shall omit the proof of this theorem, which is obtained by checking through the rules of deduction, (1.3). Some care is required in dealing with the rules of inference 1.3.3. Next 1.5.2. Every sentence

Y of K that is deducible from a set of sentences which hold in M must itself hold in M. Indeed, suppose that Y is deducible from a set of sentences X I , XZ,..., X, which hold in M.The sentence

[[[ ... [XiAXzIAXs]... AX”]3 Y] then is a theorem and so must hold in M. But if so, then Y must hold in M, by 1.4.2. A contradictory set K cannot possess a model. In fact, by assumption, we consider only languages which contain at least one statement. Since K is contradictory it cannot be empty. Let Y be an element of K and let

12

THE LOWER PREDICATE CALCULUS

[IS.

Z = [ Y A [- Y]]. If K is defined and holds in a structure M under a correspondence C then Z holds in M under the same correspondence. But if so then Y both holds and does not hold in M, which is impossible. From now on we shall usually omit references to the correspondence C which establishes the connection between a set of sentences, K, and a structure M. Thus, when stating that the sentences of K are (or, K is) defined in M, we shall take it for granted that this involves a particular correspondence C .We are going to prove 1.5.3. THEOREM. If a sentence X of L holds in every structure in which it is defined, then Xis a theorem. This is, essentially, Godel's completeness theorem for the Lower predicate calculus. 1.5.4. THEOREM (EXTENDEDCOMPLETENESS THEOREMOF THE LOWER

PREDICATE CALCULUS). Every consistent set of sentences K in a language L possesses a model.

1.5.5. THEOREM. If a set of sentences K and a sentence Yare such that Yis defined and holds in any structure which is a model of K then Y is deducible from K. Hence, Y is deducible from a finite subset of K.

Theorems 1.5.5. and 1.5.3. can both be reduced to 1.5.4. Suppose that the sentence Xmentioned in 1.5.3. is not a theorem of L. In that case the sentence [- XI - or, more precisely, the unit set containing [- X I cannot be contradictory. For if it were, then [ [- XI 3 Y] would be a theorem for all Yin L, hence [ [- XI 3 XI, [- XI] V X ] ,[XVXI, and finally X would all be theorems in L, by the rules of the calculus of propositions. This is contrary to assumption and so [- X I is consistent and possesses a model M , by 1.5.4. But if so then X i s defined, but does not hold in M,contrary to the hypothesis of 1.5.3. This proves 1.5.3. Coming to 1.5.5., suppose that Y is not deducible from K. It follows that for any finite number'of sentences XI, .. .,Xn of K, the conjunction

[I-

[[ . . . [XiAXz]AX3]... A [ - Y ] ] is not contradictory (otherwise [ [ [. . . [XI A X23 A X3] . . . A Xn]3

Y] would be a theorem, i.e. Y would be deducible from K). But this means that the set {XI, . . .,X,, Y} is consistent, and hence that K u { Y} is consistent. If so, then by 1.5.4. there exists a model M of K u {- Y > . M is a model of K i n which Y holds, contrary to assumption. This proves 1.5.5.

-

-

-

1.5.1

RELATION BETWEEN DEDUCTIVE AND DESCRIPTIVE CONCEPTS

13

To prove 1.5.4., suppose first that the sentences of K do not include any quantifiers or object symbols. That is to say, K consists of sentences built up from relative symbols of order 0 by means of connectives (and brackets). A model M of K then is a set of relations of order 0 in one-toone correspondence with the relative symbols of K such that any relation of the set either holds or does not hold in M y and such that the rules of 1.4.2. then imply that the sentences of K all hold in M. To simplify the argument we shall identify these relations with the corresponding relative symbols. Let S be the set of relative symbols of K. Then the question whether any A E S holds or does not hold in a structure M as indicated may be expressed by a valuation function a, ( A ) defined in S and ranging over the ‘truth values’ 0 (holds) and 1 (does not hold). Reference to 1.4.2. shows that the question whether or not a sentence which is built up from elements of S holds in Mycan then be decided by the standard truth table evaluation if we identify 0 with ‘true’ and 1 with ‘false’. Thus, if K contains only a single sentence X , the fact that we may assign values a, ( A ) = 0 or = 1 to the relative symbols of X so as to ensure that X holds may be accepted as a well-known result of the calculus of propositions. If K contains a h i t e number of sentences only, this may be reduced to the case of a single sentence by taking the conjunction of the elements of K in any arbitrary order. In order to prove the theorem for K of arbitrary infinite cardinal (countable or non-countable) we require the following auxiliary consideration. Without changing our notation we may, temporarily, regard S as an abstract set the character of whose elements, A , By C, . . . is irrelevant. By a partial valuation of S we mean a function of one variable which is defined on a subset V of S and takes values in the two-element set (0, l}. The valuation a, is called total if its domain of definition is the entire set S. We write Da, for the domain of definition of a partial valuation a,. Also, if U is any subset of S, we write a, I U for the restriction of a, to the set Da, n U. With this terminology, we propose to prove the following VALUATION LEMMA.Let @ = { p v } be a set of partial 1.5.6. SPECIAL valuations of S with index set I = { v } , such that for every finite subset U of S there exists a a, E @ which includes U in its domain of definition. Then there exists a total valuation JU, of S such that for every finite subset U of S there exists a yV E @ which includes U in its domain of definition and such that yI U = pvl U, i.e. v/ coincides with qv on U.

14

THE LOWER PREDICATE CALCULUS

[1.5.

For the proof, we shall call a partial valuation yl of S admissible if for any finite subset U of S there exists a pvE @ such that U c Dp, and yl U = p,lDy/ n U,i.e. yl coincides with pv on the intersection of U and Dyl. Given two partial valuations p and yl we shall call cy an extension of p if D p c Dyl and yllDp = p, in symbols p < cy. Let Y be the set of admissible partial valuations of S. Yis not empty for, by the hypothesis of the lemma it contains the empty partial valuation, i.e. the partial valuation whose domain of definition is empty. Also, Y is partially ordered by the relation of extension, <, defined above. Any non-empty totally ordered subset Y' of Y possesses an upper bound y'. Indeed, if Y = {wp) then we may define the domain of definition of w' as the union of the domains of definition of the yo, and the value of yl' for any argument within its domain of definition as the joint value of all ylh which are defined for that argument. Accordingly, by Zorn's lemma, Y contains at least one maximal element, V/O say. We wish to show that ylo is a total valuation of S. Suppose on the contrary that S - Dylo is not empty, and let A be an element of that set. Define a partial valuation yl1 of S, with domain of definition Dylo U (A}, by y l 1 = ylo on Dylo and yll(A) = 0. Since ylo is maximal, yl1 cannot be admissible. That is to say, there exists a finite subset V of S such that the conditions V c Dp, and ~ 1 V1 = VvlDyl1 n V are not satisfied for any pp.E @. Define a partial valuation y2 of S, with domain of definition Dylo U {A}, by V / Z = cyo on Dylo and ylz(A) = 1. Again, ylz cannot be admissible, and so there exists a finite subset W of S such that the conditions W c Dp, and ylzl W = q V ( D v zn W are not satisfied for any pvE @. We note that Y must contain A, otherwise the existence of a pv as required for admissibility follows from the fact that ylo is admissible. Similarly W must contain A. Now consider the set U = V U W. Since ylo is admissible there exists a pp E @ which, contains U in its domain of definition such that cyol U = prrlDylo n U. A belongs to the domain of definition of pp, since A E U and so either p p ( A ) = 0 or p,(A) = 1. But in the former case we then have y l i I V = pplDyll n V and V = D p p while in the latter case yl2l W = pplDy/z n W and W = Dp,. This contradicts our earlier conclusion that no element of @ with these properties can exist. Accordingly S - Dylo is empty, cyo is a total valuation, and since yo is admissible it follows that yo satisfies the conditions stated in 1.5.6. This completes the proof of the lemma. We return to the proof of 1.5.4. for the case that K is infinite and

1.5.1

RELATION BETWEEN DEDUCTIVE A N D DESCRIPTIVE CONCEPTS

15

consists of sentences without quantifiers or object symbols. Thus, K contains relative symbols of order 0 only. Denoting by S the set of relative symbols which appear in K, let S’ be any finite subset of S, and let K‘ be the set of sentences of K whose relative symbols belong entirely to S’. Since the number of elements of S’ is finite it follows that while the number of sentences of K’ may be infinite they must all be equivalent, in the sense of the calculus of propositions, to the sentences of a finite subset of K . We conclude that there exists a total valuation rpv of S‘ (i.e. ampartial valuation of S ) such that pvattributes the value 0 (‘true’ or ‘holds’) to all elements of K‘ according to the familiar truth table evaluation. For elements of S‘ which are not contained in,any sentences of K the value of rpv may be chosen arbitrarily, e.g. qv = 0. Let @ = {qp} be the set of all partial valuation of S obtained in this way, and let yo be a total valuation of S with the properties stated in 1.5.6. Let X be any sentence of K. We maintain that the values which are attributed by yo to the relative symbols of X yield the value 0 (‘true’ or ‘holds’) for X.Indeed if V is the set of relative symbols which occur in X then for some qv E @, V c Dqv = S’ and V / O = vvon V. It follows that X belongs to the set K defined above for the given finite set S’. And since qv yields 0 for X the same applies to yo, which is equal to vv on V. We conclude that yo yields 0 (‘holds’) for all sentences of K. This establishes 1.5.4. for the case that the sentences of K are built up from relative symbols of order 0 only. Suppose next that K may include relative symbols of arbitrary order, while still not containing any quantifiers or variables. As before, we shall use the relative and object symbols of K as the relations and objects of the structure M which satisfies K (i.e. in which K holds). Let S be the set of atomic formulae which occur in the sentences of K. Choose S’as a set of relative symbols of order 0 which are in one-to-one correspondence C with the elements of S. Two atomic formulae are regarded as different if they contain different relative symbols, or if they contain the same relative symbol, but differ at some point in the object symbols or variables that occupy corresponding places. By our definition of a language L, it is in fact not certain a priori that there are enough relative symbols of order 0 available for S’. However, in that case we may replace L by a more extensive language. We agree once and for all to carry out such extensions tacitly whenever a problem of this kind arises. Now let K be the set of sentences X‘ obtained from the sentences X of K by replacing all atomic formulae of S by the corresponding relative

16

THE LOWER PREDICATE CALCULUS

r1.5.

symbols of S‘. We claim that K‘ is consistent. This is true, trivially, if K and K‘ are empty. In the general case, let A be an element of S and let A‘ be the corresponding element of S’. If K‘ is contradictory, then it contains a finite number of sentences X i , . . . X i , corresponding to sentences XI, . . .,X , in K such that the sentence

is a theorem, where Y‘ = [ [ A ’ ]A substitution 1.3.5., the sentence

[-

[ A ’ ] ] ] .Hence, by the rule of

is a theorem where Y = [ [ A ] A [ [ A ] ] ] ,A being the atomic formula that corresponds to A ’ . But this shows that K is contradictory, contrary to assumption. We conclude that K‘ also is consistent. It follows from what has been proved already that K‘ possesses a model M‘, i.e. there exists a valuation of S‘ which makes all elements of K‘ ‘true’. For any atomic formula R(a1, . . ., a,) of S, we now define that R(m, . . ., an) holds in M if and only if the corresponding relative symbol of S‘ holds in M’. For any R (al, . . ., a,) of M which does not belong to S and hence, does not occur in K,we define arbitrarily that R (a,. ..,a,) holds in M. Since the decision whether a sentence X of K holds in M depends only on the question which of the atomic formulae of K hold in M, and since this connection is the same in M’ as it is in Myit follows that all sentences of K hold in M y M is a model of K. We now drop all restrictions on the form of the sentences of K. However, in view of the cases disposed of previously, we may suppose that K is not empty and that it contains at least one sentence that involves a quantifier. Replace every sentence X of K by a sentence X’ in prenex normal form, such as exists by 1.3.7., and let K be the set of all sentences X’ which are obtained in this way. Since X X’ is a theorem for all X E K it follows that every model of K‘ is also a model of K. And since X 3 X’ is a theorem for all X E K, and K is consistent, it follows (compare 5.3.6.) that K‘ also is consistent. Accordingly, we shall from now on confine ourselves to the case that all sentences of K are in prenex normal form from the outset. We count the quantifiers in any sentence in prenex normal form in their “natural” order, from left to right. In order to exemplify our procedure, we shall suppose that K includes a sentence of the form

1.5.1

17

RELATION BETWEEN DEDUCTIVE A N D DESCRIPTIVE CONCEPTS

where Q does not contain any further quantifiers. We shall use the relative symbols of K as the relations of a model M. The set of individual objects of M will be introduced in due course. A sequence of sets of sentences {KO,K I , Kz, , .] and a sequence of sets of object symbols {PO,P I , Pz, . . } are now defined by induction as follows. KO= K, while POis the set of object symbols contained in the sentences of K. However, if K does not contain any object symbols then we define that PO contains the single object symbol a, picked at random. KI contains all sentences of KO. Moreover, if X is a sentence of KO which begins with an existential quantifier then K1 contains a sentence X* which is obtained from X by deleting the existentialquantifier (and the brackets which belong to it) and by replacing the corresponding variable everywhere in the remaining wff by a new object symbol. Thus, if X is of the form 1.5.7. then we may select as X * the sentence

.

.

1.5.8.

provided b does not occur in PO.Moreover, it will be understood that we introduce distinct new object symbols for distinct sentences of KO.This completes the definition of K I .P I is obtained as the union of Po and of the new object symbols introduced just now. KZcontains all sentences of K I .Moreover, if Xis a sentence of K1 which begins with a universal quantifier, then KZ contains all sentences which are obtained from X by deleting the quantifier and by replacing the corresponding variable in the remaining wff by elements of Pi. Thus, we obtain from 1.5.8. the sentences

[@.I[(W[ (WQ (b, a,

2,

u,

41 . . .]

where a varies over all elements of P I .This completes the definition of K2. PZ is simply given by PZ = P I . In general, we obtain Kn and Pn from Kn-l and Pn-1 for odd n just the same way as K1 and PI was obtained from KOand PO,and we obtain Kn and Pn from Kn-1 and Pn-1 for even n in the way in which we obtained Kz and Pz from KI and P I . Let K = U {Kn} = KO U Ki U Kz U . . and P' = U {Pn} = PO u PI u PZ u n

n

.

. .. The set P' will serve as the set

of individual objects of the required structure M .

18

I1.5.

THE LOWER PREDICATE CALCULUS

We propose to show that K' is consistent. Since K can be contradictory only if a finite subset of K is already contradictory, and since K is the union of an ascending chain of sets Kn, it will be sufficient to prove that all Kn are consistent. If not, then there exists a first n, n = m say, such that Km is contradictory. Also, m 1 since KO = K is consistent, by assumption. Suppose first that m is even. Then there exist sentences Y1, . .., Y k E Km, and Z1, . . ., Zz E Km - Km-1 such that the set {Yi,. . ., Yk, Z1,. ..,Zz} is contradictory. 1 must be positive, otherwise Km-1 would be readily contradictory. By the definition of Km there exist sentences VI, . . ., Vi E Km-1 which begin with universal quantifiers,

1.5.9. such that

Vr = [ ( v x ) Sr(X)],

i = 1, . ..,I ,

zr = &(at)

where the at are certain object symbols which belong to Pm. Then the sentences [Vg 3 Zt] are theorems, by 1.3.2. Now if { Y17 . . Y k , Z1, . .., Z Z }is contradictory, then there exists a sentence W = [ [ A ]A [A]]], A a relative symbol of order 0, which is deducible from { Y I , ..., Yk, Z1, . . ., Z Z }and this is equivalent to the statement that

.

1.5.10.

[Yl A [Yz A [.

. .,

. . A Yk]

3

1-

W]

is deducible from (21, . Z Z }(or, if k = 0, that W is itself deducible from { Z l , . . . Z Z } ) But . since the sentences [ Vt 3 Zt] are theorems, this implies that the sentence 1.5.10. is deducible from { V1, .. . VZ},and hence that W is deducible from { YI , ..., Yk, V1, ... V Z ]= H, say. Then H is contradictory, although it is a subset of Km-1, which is consistent. This is impossible and shows that Km cannot be contradictory. Suppose next that m is odd. Then there exist sentences Y I ,. . .,Yk E Km-l, 21, . . . Zz E Km - Km-i, I > 0, such that { Yi7 . .., Yk, 2, ..., Zz} is contradictory. By the definition of Km there now exist sentences V1, . , ., Vz E Km-1 which begin with existential quantifiers,

1.5.11.

V C = [ @ y ) ~ t ( x ) ] , i = 1, ..., 2 ,

such that Zr = Si (at) where the ad are certain object symbols which belong to P, but not to Pm-1. Moreover, we may suppose that the 2, are distinct and so the Vt and the at are distinct. Since {YI, .. ., Yk, Z1, . . .Zz} is contradictory there exists a sentence W = [ [ A ] A [ [A]

-

1

1.5.1

19

RELATION BETWEEN DEDUCTIVE AND DESCRIPTIVE CONCEPTS

as before such that W is deducible from { Y1, ..., Yn,2, ..., Zr) and hence, by the rules of the calculus of propositions, such that 1.5.12.

[ Y i A [ Y z A [... A [ Y n A [ Z z A

... AZz] . . . I =

W]=U

is deducible from 21. Thus Z1 U is a theorem and hence, by the third rule of 1.3.3., V1 =I U is a theorem. It follows that { Y1, . . ., Yk, V1, 2 2 , . . . Z } is contradictory. Applying the same argument to 22,. . ., Z Z in turn, we conclude that { Y I , ..., Yk, VI, . . ., VZ} is contradictory. But this set is included in Km-l, and so our assumption was wrong, Km must be consistent. This concludes the proof of our assertion that K is consistent. Let K* be the set of sentences of K which do not contain any quantifiers. K* is consistent since it is a subset of K', and the relative symbols in K* are the same as in K' and in K while the set of object symbols in K* is P'. It therefore follows from an earlier argument that K* possesses a model M whose relations are the relative symbolsjust mentioned and whose set of objects is P'. We propose to show that Mis a model of K,rnoreprecisely, that M is a model of K' = K. This will establish 1.5.4. in its entirety. In order to prove that the sentences of K' all hold in M we use induction on the number of quantifiers in the prefix. For sentences without quantifiers the assertion is clearly valid for such sentences are included in K*, and M is a model of K*. Suppose then that it is true for all sentences of K' with k < n quantifiers, n 2 1, and let X be a sentence of K' that contains exactly n quantifiers. Consider first the possibility that X begins with an existential quantifier, X = [ (3y) S ( y ) ] . Since X belongs to K it belongs to all Km from some m onward, and in particular it belongs to some Km with even subscript m. But if so then Km+l contains a sentence S(u) which holds in M since it has only n - 1 quantifiers. It then follows from the semantic interpretation of X (1.4.3,) that [ (3y) S (y)], which is X,also holds in M. Suppose next that X contains exactly n quantifiers, of which the first is universal, X = [ (Vy) S ( y ) ] . Then we have to show only that S(u) holds in M for any u E P'. But it follows from the construction of K' and P' that there exists an odd subscript m such that X E Km and u E Pm (and the same is then true for all greater subscripts). Hence, by the definition of Km+l, S(u) is contained in that set and hence in K'. But S(u) contains n - 1 quantifiers only and so S(u) holds in Myas required. This completes the proof of 1.5.4. We shall now estimate the number of objects required for the model M

20

THE LOWER PREDICATE CALCULUS

11.6.

of K,in relation to the number of sentences in K. If K is finite then all Kn and all Pn are finite and so P' is, at most, countable. If K is infinite, of cardinal k, then all Kn, P n have at most k elements, and so P' has at most Hok = k elements. Hence if we define the cardinal number of a model as the cardinal of its set of objects, we have the following 1.5.13. THEOREM. If the consistent set of sentences K is finite then it possesses a finite or countable model M. If K is infinite then it possesses a model whose cardinal does not exceed the cardinal of k. For finite or countable Kthis is the theorem of Lowenheim and Skolem. The notion of a structure can be defined in a slightly different way which appears to be somewhat more natural from a set-theoreticalpoint of view. In this definition a relation of order n within a structure is identified with a subset of the set of ordered n-ples of individuals of the structure. It is then no longer possible that two distinct relations hold for the same n-ples (for in that case the relations are not distinct set-theoretically) but on the other hand it may happen that the correspondence between relative symbols and relations is many-one. While we may still use object symbols as individual objects it is now impossible for a relative symbol to take the place of a relation. In order to base our own approach on set-theoreticalconcepts we have to think of objects and of relations of different orders as the elements of distinct sets of individuals in some (absolute or axiomatic) Set Theory. The statement that R(a1, ..., an) holds in a structure M then signifies that the sequence of n 1 individuals R, U I , . . . an, belongs to a preassigned set.

+

1.6. Sets of Sentences and Their Varieties. Unless an explicit remark is made to the contrary, we shall suppose from now on that the relative and object symbols coincide with the relations and individual objects (constants) which they denote, and we shall refer to them simply as relations and individuals, or constants. Given a consistent set of sentences K in a language L we may, as we have seen, suppose that L has a model M but we may not, without making special assumptions on L, suppose that all the individuals of M are contained in L. However, it is clear that if we choose a 'universe' of individuals D which is large enough (e.g. that contains as many individuals in addition to the individuals of L as there are sentences in L) then to any given consistent set of sentences K in L we may find a model M such that all individuals of M belong to D. This

1.6.1

21

SETS OF SENTENCES A N D THEIR VARIETIES

ensures the existence of models to all consistent sets in a single universe of individuals and saves us from the embarassment of ill-defined totalities. A set of sentences K in a given language L will be called a T-system (T for Tarski) if it includes all sentences which can be deduced from it, in symbols if S ( K ) = K. For example, the set of all theorems in L is a T-system. A set of structures V (based on a definite universe of individuals, as explained above) will be called a variety (of structures) if Vconsists of all structures which are models of a set of sentences K in L. We shall also say in this case that Vis the variety of K, in symbols K + V.The theory of systems of sentences and of varieties of structures will be developed later (Chapter VII and VIII) in a more elaborate setting. In the present section we wish to establish a particular result on varieties which is closely related to 1.5.4. 1.6.1. COMPACTNESS THEOREMFOR VARIETIES OF STRUCTURES. Let { VV}be a set of varieties of structures such that the intersection of any

finite number of elements of { V y } is not empty. Then the intersection { VV}is not empty. V

Since the VVare varieties there exist sets of sentences KV such that for every v, KV+ VV,i.e. VV is the variety of Kp. Let K = (J {KV.}Then V

every model of K is a model of every Kv and hence belongs to every VV. Thus, in order to prove 1.6.1. we only have to show that K possesses a model. This, by 1.5.4. requires only that K is consistent, i.e. that every finite subset of K is consistent. Let {XI, . . ., X n } be any finite subset of K. Then there exist sets Kv, e.g. KI, . . .,Kn, such that Xt E Kt, i = 1, . ..,n. Now let M be an element of V I n . . . n Vnwhere Kt -+ VC,i = 1, . . .,n. Such a structure M exists by the assumption of 1.6.1. Then the sentences Xg hold in M, i = 1, ...,n. It follows that {XI, ...,Xn} is consistent. Thus, K is consistent and the theorem is proved. Closely related to 1.6.1. is the following

1.6.2. PRINCIPLEOF LOCALIZATION. Let K be a set of sentences such that every finite subset of K possesses a model. Then K possesses a model. Indeed, every finite subset of K is consistent since it possesses a model. Hence K is consistent. Hence K possesses a model, by 1.5.4. This proves 1.6.2. There is an essential difference between Theorem 1.5.4. on one hand and

22

[1.7.

THE LOWER PREDICATE CALCULUS

Theorems 1.6.1. and 1.6.2. on the other. Whereas 1.5.4. establishes a connection between the deductive or syntactical properties of a set of sentences on one hand, and the semantic or model theoretic properties of the set on the other, 1.6.1. and 1.6.2. refer only to the models of a set and do not contain any mention of its deductive properties. It turns out that for many applications 1.6.2. may be used in place of 1.5.4. The question arises whether the former theorems cannot be proved directly without reference to the rules of deduction. This is indeed possible, e.g. on the basis of the special valuation lemma 1.5.6. However, even in cases when the use of the rules of deduction is not essential, it may still be instructive. Since the very nature of the calculus of deduction shows that any deduction can involve a finite number of sentences only, the consistency of any set must be determined by the consistency of its finite subsets. Thus, the equivalence of consistency and of the existence of a model provides a natural explanation of 1.6.2. and 1.6.1. 1.7. Problems 1.7.1. Show how to modify the theory of this chapter if functions (function symbols) are included, such as p(x, y) for x y. (Compare section 9.1, below). 1.7.2. Show how to modify the theory of this chapter if object symbols are excluded. 1.7.3. Derive 1.6.2. and 1.6.1. (in that order) without using the calculus of deduction.

+

1.7.4. Modify the theory of semantic interpretation (section 1.4.) so as to suit the set-theoretic definition of a structure (end of section 1.5.).

References. The calculus introduced in sections 1.2, and 1.3. is a modified version of Hilbert-Bernays 1934/1939. 1.5.3. is proved in Godel 1930. Theorems 1.5.4., 1.6.1., 1.6.2. are equivalent if 1.5.3. is taken for granted. First came 1.6.2., which is due to Malcev (stated in Malcev 1941). Proofs of 1.5.4. will be found in Henkin 1949,A. Robinson 1951, compare also Rasiowa-Sikorski 1950, Beth 1951. 1.6.1. is stated in Tarski 1952. For the theorem of Lowenheim-Skolem (with a semantic definition of consistency) see Lowenheim 1915, Skolem 1920. The set-theoretic definition of a structure is given in Tarski 1952, where a structure is called a relational system. 1.5.6. is a special case of 9.3.2., below. It follows immediately from a lemma in Rado 1949. Deductively closed systems are considered in Tarski 193511936.

C H A P T E R I1

ALGEBRAIC THEORIES

2.1. Equality. There is scarcely a mathematical theory which does not

involve the notion of equality. Frequently, this notion is awarded a status which puts it outside the ordinary Lower predicate calculus as described in the preceding chapter. We shall not adopt this procedure and instead shall regard equality as a relation E ( x , y ) in the sense used previously, which satisfies the following conditions.

2.1.1.

w-9E (4 w 4 (VY) [m,Y ) = E ( Y , XI]

(reflexivity) (symmetry) (Vx)(Vy)(Vz) [E(x,y ) A E ( y , z) 3 E (x, z ] (transitivity)

2.1.2. For any relation under consideration (for example, contained in a

given structure), E satisfies the condition of substitutivity. Thus, if the relation in question is of order n, A (xi, . ..,xn), say, then the condition is

(VXI)

.. (Vxn) (Vyi) -

-

---

(VYI) [E(xi,YI) A E ( ~ z , ~Az ) A E(xn, ~ AA(x1, . - . , x n ) =A(.YI,. . - , ~ n ) l

n )

(Observe that for A = E, 2.1.2. is deducible from 2.1.1.). It will be seen that we have adopted the simplified bracket notation explained in section 1.2. above. Also, when mentioning a relation we shall from now on either fill its empty places with variables, or else omit the brackets, and the commas and spaces included in them. Any relation satisfying 2.1.1. is called an equivalence. There may well be two relations of equivalence contained in a given structure without being co-extensive. In general two relations A (xi, ...,xn) and B(x1, .,., xn) are co-extensive if for any ai, ... ,an in the structure under consideration A (al, ...,an) fB ( m , . . .,an). Two relations of equality must be co-extensive. For let Ei ( x , y ) and EZ( x , y ) be two such relations. Then we wish to show that El (a, b) E (a, b) holds for all individuals a, b in the structure. Indeed, by 2.1.2. with E = El, A = Ez, and using 1.3.2., Ei (u, a) A Ei (a, b) A Ez (a, a) 3 E2 (a, b) 23

24

ALGEBRAIC THEORIES

[2.1.

But E1(a, a) and Ez(a, a) hold by the first sentence of 2.1.1. and so El (a, b) 3 EZ(a, b) holds, and similarly EZ(a, b) 3 El (a, b), and hence E, (a, b) EZ(a,b) as asserted.We observe that while the use of the rules of deduction of 1.3. may help to ensure the rigor of an argument, such as that just given, we may also argue intuitively, by virtue of 1.5.3. In general, it will be convenient to suppose that there is only one relation of equality. It may also be convenient to suppose that E (a, b) does not hold between any two distinct individuals. In such a case we call the structure normal. Any structure M , which is not normal can be reduced to a normal structure M‘, by replacing its individuals by the equivalence classes that correspond to the relation of equality. Thus, the elements a and b belong to the same equivalence class, a, if E(a, b) holds in M. Any relation A (XI, ...,xn) is defined to hold between individuals al, . .., ag,of M’ (i.e. equivalence classes of M ) if for some, and hence for all elements al, ...,anof mi,. . .,an,respectively, A (al, ...,an)holds in M. It is then not difficult to verify that every sentence X that holds in M holds also in M‘. Two structures are called similar if they contain the same relations. Let M and M’ be two similar structures. Then M is called a partial structure or a substructure of M y and M’ is called an extension of M if all individuals of M belong to M’ and if for any relation A ( X I , . . .,xn) and individuals al, . . .,an which belong to M y A (al, . .., an) holds in M’ if and only if it holds in M . We then write M c M’, as for ordinary inclusion. Now suppose that a relation of equality E(x, y) is given in M and M’ and that M c M’. The above definition does not exclude the possibility that there exists an individual a which belongs to M’ but not to M such that E (a, b) holds in M’ for some b which belongs to M. If this does not happen then we say that the relation of inclusion between M and M’ is normal. Let M be any structure. Consider the set of all formulae A (al, ...,an) which hold in M , for all A and al, . . ., an in that structure. The set of these formulae is called the positive diagram of M yDf( M ) . Furthermore, the set of all formulae A (al, . . ., an) which hold in M (i.e. A (a, ..,an) does not hold in M , although A, ai, ..., an, belong to M) will be called the negative diagram of M , D- ( M ) .Finally, D ( M ) = D+ ( M ) U D - ( M ) is called the diagram of M . We have the simple but important

.

N

2.1.3. THEOREM. Let M and M’ be two similar structures. Then M’ is an extension of M if and only if it is a model of the diagram of M , D ( M ) .

2.2.J

SPECIFICATION OF AXIOMATIC SYSTEMS

25

In the formulation of this theorem it is understood that the correspondence C which occurs in the explicit description of the relation between M' and D ( M ) reduces to the identity for the elements of D ( M ) - i.e. the individuals of D ( M ) denote themselves. The proof is immediate and will be omitted. A one-to-one correspondence C between the individuals and relations of a structure M and the individuals and relations of a structure M' will be called an isomorphism, if the relations of any given order correspond to one another, and if, whenever a relation holds between certain individuals of M , the corresponding relation holds between the corresponding individuals of M ' , and vice versa. If M and M' are similar it will be taken for granted that C maps every relation on itself. If M = M' then the isomorphism is said to be an automorphism. Structures between which there exists an isomorphism are called isomorphic. A correspondence C between the relations and individuals of a structure M and the relations and some of the individuals of a structure M', respectively, is called a homomorphism if the correspondenceis one-toone between the relations while every individual of M is mapped on an individual of M' in such a way that whenever a relation holds between individuals of M , the corresponding relation holds between individuals of M'. The homomorphism is onto if all individuals of M' appear in the mapping, and M' is then said to be homomorphic to M. Let K be a set of sentences, M a model of K and M' a structure which is similar and isomorphic to M by an isomorphism under which all individuals of M which appear in K correspond to themselves. Then it is not difficult to see that M' also is a model of K .

2.2. Specification of axiomatic systems. We shall now detail axiomatic systems for a number of familiar concepts in Algebra. It should be understood that within the terminology of the present book the terms axiom and sentence are synonymous, so that the word axiom is used only for convenience. In many cases the detailed formulation of an axiom is irrelevant, the main point being the fact that it can (or cannot) be formulated within the language L. Accordingly, it is important that the reader should acquire some facility in deciding whether or not a given informal statement or property can in fact be formalized within the language of the Lower predicate calculus. However, it may happen that a statement or property which are not a priori given in this way possess equivalent formulations within the required framework.

26

r2.2.

ALGEBRAIC THM)RIES

All the sets of axioms which will be specified in the sequel will include a relation of equality E(x, y). By the term axioms of equality we shall mean 2.1.1. and the axiom of substitutivity 2.1.2. applied in connection with all the remaining relations of the set under consideration. In the formulation of the axioms no attention has been paid to the requirement of extreme economy (independence). First we axiomatize the notion of a group in terms of the two-place relation of equality E(x, y ) (read ‘x equals y’) and the three place relation S(x, y, z) (read ‘z is the product of x and y’). No individuals will be used in the axiomatization. 2.2.1. Axioms of equality (see above, 2.1.1. and 2.1.2.). 2.2.2.

(W VY)(3.4 s (x, Y , 4 (W VY)(VZ) W W ) 1s(x, Y , 4 A s (x, Y , w) = E (2, 4 1 (W (W (W (W V’U) (W P ( X , Y , 4 A s (z, t, 4 A s0.1,t, 0) = s (x, U , U ) 1 (3x1(YY) (34 [ S (x, Y , r) A s (2, Y , 41

The axioms of 2.2.2. ensure, in turn, the existence of the product, the uniqueness of the product, the associativity of the group operation, and the existence of a (left) unit with respect to which every element possesses a (left) inverse. The set of axioms 2.2.1. and 2.2.2. will be denoted by KG. An alternative to KG is obtained by introducing an individual constant e and by replacing the last axiom of 2.2. by 2.2.3.

(try> (W rs (e7 Y7 u) A

s

(z7

Y, 4 1

The resulting set will be denoted by Kk. In order to obtain a set of axioms for the concept of an abelian (commutative) group, we add to K G or Kk the axiom 2.2.4.

(W (try) (tr4 [ S(X7Y , z) = S(Y, x7 41

and we call the result KAGor KLG,respectively. In order to formulate the notion of an ordered abelian group, we introduce an additional two-place relation Q (x, y ) (read ‘x is smaller than or equal to y’) and we add the following axioms to KAGor KiG 2.2.5. 2.2.6.

CW VY)0’2) (W[E(x719A E(z7w) A Q (x, z> 3 Q (v,w)] (WVY)V 4 [Q (x, Y ) A Q ( Y , z) 3 Q ( x , z)] (W VY)[Q (4 r) V Q (u, 41 (W VY)[ [Q (x, Y ) A Q ( Y , XI] = E(x, Y ) ] (W (W (W (V0‘4 [s (x, Y , 4 A S (x, u, w)A Q 0, 0 ) 1 Q (z,~)]

2.2.1

SPECIFICATION OF

Axiomnc SYSTEMS

27

2.2.5. is the axiom of substitutivity for the relation Q ( x , y). The reader will have no difficulty in interpreting the axioms of 2.2.6. The resulting sets of axioms will be denoted by KOAGand KLAC,respectively. While it is possible in the present case to eliminate the relation of equality as a fundamental relation and to define it in terms of Q (x, y), we do not find it convenient to adopt this procedure. Next, we introduce a set of axioms KR for the notion of a general ring. There will be no individuals, one two-place relation, which is the relation of equality, E(x, y ) , and two three-place relations S (x,y, z)and P (x,y, z) (where the latter relation stands for ‘xy = z’while the former relation will now be interpreted as ‘ x y = z’). KR shall consist of the axioms of equality for the relations of KR, together with 2.2.2., 2.2.4., and the following list of axioms

+

2.2-7. Wx) (VY) (32)P (x,Y , 2)

[w,

(vx~~vY~(vz)(tJ~) YY z>v P(x, Y , 4 = E(z,w)] ~ ‘ x > ( ~ y ) ~ z ) ( v ~ ) ( v ~ ) Y( ,v4 ~ )A[ PP(z, ( x ,f, 4 A P(Y, t, 0) = = P(x, u, 41 (Vx) ~ y ) ( v z ) ( V t ) ( V u ) ( V v ) ( v w[)S @ ,Y , z> A Wz,I, 4 A P ( x , t, v) A P ( Y , t, w) = S(U, w, 4 1 ( V x > ( V y ) ( ~ z ) ( V ~ ) ( V u ) ( V ~ ) ([S(x, V w ) y, z) A W , z,4 A W , x , u) A

m y , w)

= S(0, w, 41 -

Of these, the first three ensure the existence, uniqueness, and associativity of the product, and the last two constitute the distributive law. Aset of sentences for the notion of a commutative ring, KCR is obtained by adding to K R the commutative law of multiplication, 2.2.8.

(W (VY) (VZ) [ p(x, Y , z) = P ( A x, 41

A set of axioms KF for the notion of a (general, skew) field is obtained as the union of KR and of 2.2.9.

(3x1(3v)[ E(XY v)] (VX) (VYX3Z)[s (XY x, XI v p ( x , z,v)] (VX)W Y )(34 [ S(XY x, x> v P ( z , x , u)] *

The first of these axioms requires that the structure contain at least two different individuals. The second axiom ensures the solvability of the equation xz = y for given y and for given x # 0. Observe that the predicate S(x, x, x) has been used as a characteristic property of 0. It is of course possible to avoid this somewhat artificial formulation by

28

r2.2.

ALGEBRAIC THEORIES

introducing a symbol for 0 within L. The third axiom of 2.2.9. ensures the solvability of zx = y for given y and x, x # 0. By adding 2.2.8. to KF we obtain a set of axioms KCFfor the concept of a commutative field. We now wish to formalize the notion of the characteristic of a field. For this purpose, we define predicates Sn(x, y), n = 0, 1,2, .. . by the following recursive definition 2.2.10.

So (x, y ) = S ( y , y, y ) S ~ ( Xy ), = (32) [S,-I(X, Z) A S(Z,X, y ) ] ,

n = 1,2,

...

In the first line of this definition, S ( y , y, y) is again used in order to characterize the neutral element with respect to addition, 0. Note that we write So (x, y ) even though x does not appear in the predicate on the right hand side. If it is desired to introduce x explicitly we may do so by adding E(x, x ) in conjunction. The predicate Sn (x, y ) is equivalent to y = nx (i.e. y is the result of adding x to itself, n times). Let X , be defined by 2.2.11.

Xn

= (VX)

(try) [s,(x, y )

3

s@,y, y ) ] , n = 1,2,3, ...

i.e. X states that nx = 0 for all x. A set of axioms for the notion of a commutative field of characteristic p, p a prime, is now obtained by adding X , to KCF.The result will be denoted by K;. On the other hand, in order to obtain a set of axioms K j for the notion of a commutative field of characteristic 0 we add to KCFthe set of sentences 2.2.12.

{- X2,

- x3,

x5,

...,

-

X,, ...)

where p varies over all prime numbers, K j is the first of the sets of axioms, considered so far which involves an infinite number of sentences. The natural question arises whether it might not be possible to find an equivalent set of sentences which is finite. The answer to this question will be given by theorem 2.4.5., below. Again, we may modify the axioms for commutative fields (of given characteristic) by introducing two individuals, 0 and 1 (which, as usual, will serve as the neutral elements with respect to addition and with respect to multiplication). Using these symbols, we replace the last sentence of 2.2.2. by 2.2.13.

(W (3-4 [S (0,YY v) A S (ZY YY 011

and the sentences of 2.2.9. by

2.2.1

-

29

SPECIFICATION OP AXIOMATIC SYSTEMS

E(0, 1) (VX)(3Y) [E(x, 0) v 4 %Y, 01 where we have taken into account that 2.2.8. is included. Also, the definition of SO(x, y ) in 2.2.10. may now be taken as 2.2.14.

So(& v) = EO,0)

2.2.15.

The resulting sets of sentences which replace KCF,KS, p = 0,2, 3, 5,

...,

will be denoted by K&, K(;). From now on, we shall use these setsin

preference to KCFand Kg. A set of axioms for the notion of an ordered field KOFis obtained, for example, by adding to KdF a number of axioms, involving the relation Q ( x , y ) (read ‘x smaller than or equal to y’, as before), more particularly the axioms 2.2.5.,2.2.6., and 2-2-16.

(Vx)WY)(Vz) [ p(x, Y , 2) A Q (0,~) A Q (0,r) 3 Q (0,~ I]

.

We also wish to formalize the notion of an archimedean ordered field. Archimedes’ axiom requires that for any positive number x and for any y there exists a positive integer n 2 1 such that nx 3 y. In this connection, nx is supposed to denote the sum x x ... x (n times). Using 2.2.10., we may write the inequality nx 2 y for any particular n as

+ +

2.2.17.

Q n k

U) = (32) [ S n k

2)

+

A Q(Y,

211

a

Introducing the notion of an infinite disjunction as indicated at the end of section 1.4., it is natural to express Archimedes’ axiom by 2.2.18-

(Vx)(Vy)[E(O,x ) V Qn<x,y)V . .]

-

-

Q(0,x ) V Q n ( x , y ) V Q ~ ( x , JV J)

.

V

Adding 2.2.18. to KOFwe then obtain a set KAFfor the concept of an archimedean ordered field. However, 2.2.18. is not a sentence of the Lower predicate calculus as defined in Chapter I, and the question arises whether it might not be possible to replace KAFby a set of axioms formulated entirely within that language. This question is settled by Theorem 2.4.22., below. Next, we turn to the axiomatization of the concept of an algebraically closed field. First, we define the predicates Pn(x, y), n = 0, 1,2, ... by 2.2.19.

Po(x, y )

= E ( y , 1)

- -

Pn (x, y) = (32) [pn-~ ( x , 2) V ~ ( 2x ,, u)], n = 1,2, Thus, Pn(x, y) corresponds to the operation of exponentiation y = xn.

30

12.2.

ALGEBRAIC THEORIES

The following predicate then expresses the equation 2.2.20.

xo

+ x1 y + . . . + xnyn =

22-21.

2

.. .,X n , y , and z Tn (XO, ...,X n , y , 2) = (3UO) (3241).- . ( h n ) (300) (3U1) . .. (3%) ( 3 ~ 1 ) .. . (3wn-1) [Po ( y , UO) A P l ( y , UI) A P z ( y , ~ 2 A ) ... A

regarded as a predicate of

XO,Xn,

Pn (Y, un) A P ( X O , YO,UO) A P (XI,UI,UI) A . . . A Pn (xn,Un, un) A ~ ( V O V, I , WI) A S(W, UZ, ~ 2 A) . . . A S(wn-Z, Un-1, Wn-1) A S(wn-1, Un, Z) ]

-

2.2.21. may be regarded as the correct formalization of 2.2.20. except for some obvious formal modifications for n Q 2. In order to obtain a set of

axioms for the concept of an algebraically closed field, RCFwe now add to KAF the sequence of axioms

22-22.

WXO)(vx1) . . . (Vxn-1) (3.~)T n (XO,X I ,...,~ n - 1 ,~y1, O),

n=2,3,

... .

2.2.22. states that every monk polynomial (polynomial with leading coefficient 1) of degree n possesses a root. In order to obtain sets of

axioms for the concepts of algebraically closed fields of characteristic p ,

Kc$?we add 2.2.22. to K ( $ ) , p > 0 .

A formally real field may be defined as a commutative field in which a sum of squares is always different from 0 unless all the (bases of the) squares are equal to 0 individually. This is expressed by the sequence of axioms

.

..

. . ( V X ~ )(V~I)(Vy2) . . . (VJJ~)(VZZ) . (VZn-1) [P2 (XI,YI) A P Z( ~ 2 YZ) , A . . A Pn (xn, yn) A S ( ~ 1 YZ, , ZZ) A S(ZZ,~ 3 ~ , 3 A ) A S ( Z ~ - 2yn-1, , zn-1) A S(zn-1, yn, 0) 3 E ( x i , 0) A E ( x z ,0)A A E ( x ~0,) ]yt = 2,3, .. .

2-2-23, (VXI) ( V X ~ )

-.

-

.. .

.

Adding 2.2.23. to KhFwe obtain the required set of axioms, KRF,say. An ordered field is by necessity formally real. A field is real-closed if it is formally real and if it possesses no algebraic extension which is formally real. As it stands, this definition cannot be expressed within the Lower predicate calculus. However, it is known that an ordered field F is realclosed if every monic polynomial of odd degree with coefficients in F possesses a root in F and if every positive element of F possesses a square root in F. This can be formalized by 2.2.21., for n = 3, 5, 7, . . . and by 2.2.24.

Wx) ( 3 ~ )Q[ (0,~) 3 P (Y, Y , 4 1

.

2.3.1

31

RELATED SETS OF SENTENCES

Adding the sentences just mentioned to KOFwe obtain a set of axioms ROFfor the notion of a real-closed ordered field. In a real-closed ordered field, an element is non-negativeif and only if it possesses a square root. Accordingly the following sentence holds 2.2.25.

W 4 WY) [Q (x, Y ) [ (32) (34 [S (x, z, v) A P (w,w,z) ] ] ]

.

Thus, we may if we wish replace Q ( x , y ) in all axioms of &IF by the predicate (3z) (3w) [S(x, z, y ) A P (w,w,x ) ] . The result is a set of axioms RRFfor the notion of a real-closed field which does not contain the order relation as a basic concept. In future, we shall not consider it necessary in all cases to give a detailed formulation of the axioms which correspond to a specified notion. The reader will have no difficulty in filling in the missing formulae. Instead of referring to a particular set of axioms for a given concept, it will sometimes be convenient to deal arbitrarily with ‘any set of axioms for’ that concept. Since the phrase which has just been given in quotes is vague, the reader may interpret it as meaning ‘any set of axioms mentioned previously for’ the concept in question. We shall then say that the concept as such has a certain property, e.g. ‘the concept (or the theory) under consideration is consistent’. In this way the meaning of some of our results (e.g. Theorems 4.2.14., 4.2.25. below) will become more intuitive. We shall also refer to the individual objects of a structure sometimes simply as its elements, again in agreement with common usage. 2.3. Related Sets of Sentences. In the preceding section we introduced, on several occasions, different sets of axioms for the same intuitive notion. In the cases in which this was done, the sets in question are therefore interchangeable in an intuitive but definite sense. The following framework makes this interrelation precise. By the vocabulary of a set of sentences K we mean the totality of individuals and relations which occur in sentences of K, with a similar definition for a set of wff or for a single wff. We shall say that a wff Xis defined in a set Kif the vocabulary of Xis included in the vocabulary of K. Two sets of sentences, K and K , we called equivalent if the sentences of K are deducible from Kand vice-versa, i.e. in terms of a symbol introduced earlier (section 1.3.) K k K’ and K‘ t K. In particular if K and K‘ possess the same vocabulary then they are equivalent if and only if their varieties of sentences coincide. TWOsets of sentences K and K’ will be said to be related if there exists a

32

[2.3.

ALGEBRAIC THFDRlFS

one-to-one correspondence between the individualsa of K and the individuals a' of K , a e,a', and the relations A of K and certain predicates R' which are defined in K', A e,R', and the relations A' of K' and certain predicates R which are defined in K, A ' e , R such that relations of order n correspond to predicates with exactly n free variables, and such that the following conditions (2.3.1., 2.3.2.) are satisfied.

2.3.1. If in any sentence X E Kwe replace the individualsby the corresponding individuals of K' and the relations by the corresponding predicates of K (more precisely - which are defined in K') then the resulting sentence X' is deducible from K';if in any sentence X' E K we replace individuals and relations by the corresponding individuals and predicates of K there results a sentence X which is deducible from K.

.

. .,xn) be any relation in the vocabulary of K and let R'(x1, ..., x,) be the corresponding predicate of K'. Let R(x1, ..,xn) be obtained by replacing the individuals and relations which appear in R' by the corresponding individuals and predicates of K. Then 2.3.2. Let A (XI,

.

.

K t- (Vxi) . . ( V X ~ )[A(xi,

. . .,xn)

R(xi,

. ..,x,)] .

A corresponding condition is supposed to hold for the relations of K. Consider for example the set of axioms K O A Gwhich expresses the concept of abelian ordered group in terms of the relations E (equality), S (addition, or multiplication), Q (order) and in terms of e (neutral element with respect to the group operation). In order to obtain a set K* which is related to K'OAGwe delete the axioms which contain Q (2.2.5, 2.2.6.) and we introduce a one-place relation P ( x ) (real ' x is non-negative, x 2 0') which satisfies the following axioms 2-3.3.

(x, Y ) A p (4 = p ( Y ) ] (W(W S(X,Y,O) = P ( X ) V P ( Y ) ] (W VY) S(X, Y , 0) A p (4 A P ( Y ) = E ( x , 011

w 4 (VY) [

I

w]

(vx)wY)(tJz)[S(x,y,z)hP(x)hP(y) * The set K* is obtained by adding 2.3.3. to K A G .A correspondence which establishes that K* is related to K O A Gis obtained by mapping E, S, e on themselves while Q (x, y) in K O AG is mapped on the predicate

[w,

(34 z, Y ) A (z)] which is defined in K* and P ( x ) is mapped on Q (e, x). We may verify that the conditions of 2.3.1. and 2.3.2. are satisfied. For example, the sentence of 2.3.2. becomes, for A = Q, fJ

2.3.1

RELATED SETS OF SENTENCES

33

and this is indeed deducible from K'OAG. Let X be any sentence which is defined in a set K and let K' be related to K. In X replace the individuals and relations by the corresponding individuals and relations of K', obtaining a sentence X' which is defined in K . Apply the corresponding procedure to X'. This yields a sentence X" which is defined in K. While X" does not in general coincide with X it is not difficult to see that X X" is deducible from K. Moreover, K k X if and only if K' t X'. It follows that two related sets are either both consistent or both contradictory. The notion just defined is not as yet sufficiently general to account for certain possibilities which arise in this connection in a natural way. Thus, the example of KG and K b shows that two sets may describe the same concept although one of them does, and the other does not, include an individual (object symbol). Accordingly, we introduce the following definition. Let K and K be two sets of sentences containing a relation of equality E(x, y) such that the vocabulary of K is included in the vocabulary of K' and such that all relations of K' are contained also in K. Then K will be called a contraction of K' if K t K and if for every individual a which is contained in K' but not in K there exists a predicate Q a ( x ) which is defined in K such that Qa(u) is deducible from K while the sentence

[ (3x1 Qa ( x ) ] A [ cJ4 (VY) [Q a (4A Qa 0 )=) E ( x , Y ) ]] is deducible from K. Moreover, if X = X ( a , b, ..., c) is in K and a, b, . . c are the individuals which appear in X but do not belong to the vocabulary of K,it is required that 2.3.4-

.

--

( 3 ~(32) ) .( 3 ~[X(Y, ) 2, . ..,W ) A Qa (.Y)A Q b (2)A - . . A Qc (w)] is deducible from K. (If no such individuals appear in X , this sentence is supposed to reduce simply to X). It can be shown that every model of K is a model of K. Conversely, a model M of K is a model of K if we permit a general correspondence between the objects of M and the object symbols of K', i.e. if we return to our original assumption that the object symbols of K' and the corresponding objects of M do not necessarily coincide. The set of axioms for the concept of a general group K Gis a contraction of the set Kb. The predicate Qe (x) which corresponds to the individual e in K b may be taken to be S(x, x, x).

23.5.

34

ALGEBRAIC THEORIES

[2.4.

We call the sets K and K' potentially equivalent if they possess related contractions. While these definitions do not exhaust the problem of this section they provide a formal framework which should be sufficient for most situations which are likely to arise in this connection. 2.4. Embedding Theorems and Transfer Principles. The extended com-

pleteness theorem of the Lower predicate calculus (1.5.4.) and its equivalents provide a powerful tool for the solution of certain classes of problems in Algebra. Let M be a structure and let K be a set of sentences such that the vocabulary of K and the vocabulary (sets of relations and individuals) of M may overlap but need not coincide. We enquire whether M can be embedded in a model M' of K. By this we mean that M' is a model of K if we ignore the relations of M' which do not occur in K, and M' is an extension of M if we ignore the relations of M' which do not occur in M. Then we have the following simple and fundamental result.

THEOREM. The structure M can be embedded in a model of the set of sentencesKif and only if every finite substructure of M can be embedded in a model of K.

2.4.1.

A finite substructure of M is obtained by taking any finite subset M' of the set of individuals of M and by defining that any relation of M holds between elements of M' if and only if it holds between them as elements of M (compare section 2.1. above). This will be expressed concisely by saying that the structure so obtained is the restriction of M to M'. Now any structure is an extension of M if and only if it is a model of the diagram D(M) of M. Hence, by 1.5.4., a structure of the required kind exists provided K U D(M)is consistent. Suppose K U D(M) is contradictory, then a finite subset of K u D (M) is contradictory. It follows that K U D* is contradictory for some finite subset D* of D (M). D* contains only a finite number of elements of M and these give rise to a finite substructure M* of M in the sense detailed above. Also, D* is included in the diagram D ( M * ) of M*, and so K U D(M*) is contradictory. But this is incompatible with the assumption that every finite substructure of M possesses an extension which is a model of K. Hence K U D(M) is consistent and the theorem is proved. Applying 2.4.1., we obtain for example, 2.4.2. THEOREM. Let R be a ring. If every finitely generated subring of R

can be embedded in a skew field then R can be embedded in a skew field.

2.4.1

EMBEDDING THEOREMS AND TRANSFER PRINCIPLES

35

PROOF.Let R* be a finite set of elements of R and consider the restriction of R* to R in the sense defined above. This structure will, without risk of misunderstanding, be denoted again by R*. R* is not necessarily a ring, but the intersection of all subrings of R that include R* is a ring R1,which is finitely generated since it is generated by the elements of R*. Hence, by the assumption of the theorem, RI can be embedded in a skew field and the same evidently applies to R* (which is included in R1). The notion of a skew field can be formalized in the Lower predicate calculus, e.g. by the set of sentences, KF. The truth of the theorem now follows directly from 2.4.1. An application with some different features is provided by 2.4.3. THEOREM. Let G be a group such that every finitely generated subgroup of G can be totally ordered. Then G can be totally ordered.

PROOF.We first determine a set of axioms for the notion of a totally ordered (not necessarily abelian) group. A set KOGas required is obtained by omitting 2.2.4. from KOAGand by replacing the fourth axiom of 2.2.6. by

W)WY)( V 4

OW ( V 4 [S(xyY , 4 A S (u, u, 4 A Q (x, 4 A Q (uy Qkw)] .

The argument used in the proof of the preceding theorem now shows that G can be embedded in a model of KoG, i.e. in a totally ordered group G'. But the total ordering of G' induces a total ordering in G and so G can itself be totally ordered. This proves 2.4.3. Leading up to our next group of results is the remark that the use of a formal language enables us to make certain statements about entire classes of theorems where 'conventional' Mathematics enables us to deal only with individual theorems. In particular, we may be able to show that any theorem of a certain class which is true for one type of mathematical structure is true also for another type of mathematical structure. A result of this kind will be called a transfer principle. In fact, the classical principle of duality in Projective Geometry may be regarded as a metatheorem of this kind. However, the logical character of this principle is so simple that its discovery did not require any detailed formalization. We shall say that an infinite sequence of sentences Yn,n = 1, 2, ... formulated within a language L (of the Lower predicate calculus, as usual) constitutes an increasing chain if Yn 3 Ym is a theorem for all n 2 m. This may be indicated by the expression

36

[2.4.

ALGEBRAIC THEORIES

which is not, however, included in L. An increasing chain will be called strictly increasing if Yn 3 Ym is a theorem only if n 2 m. 2.4.4. THEOREM. Let K be a set of sentences Y I , Yz,

.. .

which constitute a strictly increasing chain. Then there does not exist any sentence Y which is equivalent to K,i.e. such that K t Y and Y t K.

PROOF. Suppose, contrary to the conclusion of the theorem that K k Y, and Y k K for some Y. Then Y is deducible from a finite subset K of K, K = {Ys, Yj, . . ., Y z } ,say, where the subscripts i, j , . ., 1 have been written in their natural order. Since YZ3 Ym is a theorem for all m < Z, we have in particular, YZ3 Ys, YZ2 Yj, etc. Hence Y is deducible from YZalone, YZ3 Y is a theorem. (If K is empty, so that Y is a theorem, we take I = 1, arbitrarily). On the other hand Y t K and so in particular Y k Y z+I,i.e. Y 3 Yz+1 is a theorem of L. But if so then YZ2 Yz.;1 is a theorem, contrary to the assumption that the Yn constitute a strictly increasing chain. This proves 2.4.4.

.

2.4.5. THEOREM. The concept of a commutative field of characteristic 0 cannot be formalized by a finite number of axioms within the Lower predicate calculus. More precisely, there is no finite set equivalent to PCF or KAg.

PROOF.If it is possible to formalize the concept under consideration by a finite number of axioms, we may replace these by a single axiom, Y, i.e. their conjunction. Referring to 2.2.11, we define the sentence Y I as the conjunction. of the sentences of KCF or of KbF and we then define Yz = YI A X2, Y3 = YZA X3 and, in general, Y , = Yn-l A x p n - 1 where pn is the nth prime number, p l = 2, pz = 3, p3 = 5, etc. Yn states that the structure in question is a commutative field whose characteristic p is 2 Pn, or equal to 0. The sequence Y1, Y2, ... is an increasing chain since Yn 3 Yn-1 or, which is the same, 1Yn-l A Xpn-,] 3 Yn-1 is a theorem for n = 2, 3, . . . Moreover, the sequence is strictly increasing since for any n 2 1 the fields of characteristic pn satisfy Y , but not any Ymfor m > n. It follows that Y , 3 Ywzcannot be a theorem for m > n. Let K = { YI , Yz, . . . } then the models of K are precisely the commutative fields of characteristic 0, i.e. K is equivalent to K t F or to K&$ as the case may be. It then follows that KEF or K&$ cannot be

-

-

-

-

2.4.1

37

EMBEDDING THEOREMS AND TRANSFER PRINCIPLES

equivalent to a single sentence, otherwise the same would apply to K , contrary to 2.4.4. A slightly simpler argument leads to the following transfer principle: 2.4.6. THEOREM. Let X be a sentence of the Lower predicate calculus formulated in terms of equality, addition and multiplication, which holds in all commutative fields of characteristic 0. Then X holds also in all commutative fields of characteristic p > po where po is a positive integer which depends on X .

-

- -

PROOF.Since X is deducible from KOCF, i.e. from KCF U { XZ, X3, X5, . . .} it must be deducible from KCFtogether with a finite subset K‘ of the set {- X2, X3, X5, . . .} Let X,,, be the sentence with highest subscript po that belongs to K‘ (or if K‘ is empty, put po = 1). Then all fields of characteristic p > PO satisfy KCF and K and, hence, satisfy X . This proves the theorem. We shall now give a purely algebraic application of 2.4.6. For any polynomial q (XI, ...,x,) with rational integer coefficients we shall denote by q ( p ) (XI, . . .,x,) the corresponding polynomial with coefficients in the prime field of characteristic p , i.e., the polynomial which is obtained from q (XI, ...,x,) by reducing its coefficientsmodulop.

- -

-

2.4.7. THEOREM. If a set of polynomial equations with integer coefficients, 4, (XI, . . .,x,) = 0,q 2 (XI, . . .,xW) = 0, . ..,q k (XI, . . .,x,) = 0 does not possess a solution in any extension of the field of rational numbers then there exists a positive integer po (which depends on the given system) such that the set of equations

2.4.8.

2.4.9.

q(P)(X1,

. . ., x,)

...,

= 0,@‘)(XI, xn)

=o

=O,

.. .

q(P’(~1,

. . ., xn)

does not possess a solution in any field of characteristic p for p

> PO.

In order to reduce 2.4.7. to 2.4.6. we first formulate, for any polynomial with integer coefficients q(x1, . ., x n ) a predicate Q g ( x i , . . ., X n , y ) which contains the relations E, S, P, and does not contain any individuals and that &(XI, ...,X n , y) expresses the fact that y = q(x1, .. ., x,) in any field of characteristic 0. It is not difficult to construct such a predicate although it is by no means unique. Moreover, we may ensure by a natural choice of Q, that the same predicate expresses the equation y = q ( p ) (XI, . ..,x,) in any field of characteristicp , p an arbitrary prime.

.

38

12.4.

ALGEBRAIC THEORlEs

For example y = 3x2 may be expressed by Q q ( x , y) = ( 3 z ) [ P ( x ,x , z ) A Ss (z, y ) ] and the same Q q can be interpreted in any field of characteristic 2 and then states that y = 3x2 or, which is the same, that y = x2. The hypothesis of 2.4.7. is that for particular polynomials 41, . . ., qk: the sentence

X = (Vxn)

.. . (Vxn)[-

[Qql ( X I ,

.. , ~ n 0), A .- - A Qqk ( X I ,.. .,xn,O>]]

holds in all fields of characteristic 0. It follows, by 2.4.6., that X holds in any field of characteristic p > P O where PO is hence positive integer which depends on X. But in such a field, X expresses the fact that 2.4.9. does not possess any solution. This proves 2.4.7. A mathematical proof of 2.4.7. is as follows. If 2.4.8. has no solution in any extension of the field of rational numbers then there exist polynomialsfl (XI, . . .,xn) . .fk: (XI, . .., xn) with rational coefficients such that

.

f1q1

+f2q2 + .. .

+fk@

=1

.

Multiplying this equation by a suitable integer, a, we see that there exist polynomials g l , . . .,gk: with integer coefficients such that g l q l + g2q2

+ ... + gkqk

=a .

Let PO be the greatest prime divisor of a. Then if we take p > PO we have glql

+ 8242 + - - - + grqk + 0 (PI

and this proves that 2.4.9. cannot have any solution in a field of characteristic p. Thus, while it is not difficult to provide a mathematical proof for 2.4.7., the proof does make use of certain algebraic results which are not required for the metamathematical proof. However, it is not difficult to replace 2.4.7. by a somewhat more general result for which a purely mathematical proof does not seem to readily available, as follows. We shall say that two solutions of 2.4.8. are different if they differ with respect to at least one of the xt, i = 1,2, . . .,n. We then have 2.4.10. THEOREM.If the system of polynomial equations 2.4.8., does not

possess more than m different solutions in any (commutative) field of characteristic 0, then it cannot possess more than m different solutions in any field of characteristic p > PO where PO depends on the set of polynomials. The proof is similar to that of 2.4.7. It depends on the formalization if the statement ‘2.4.8. does not possess more than m different solutions’.

2.4.1

39

EMBEDDING THEOREMS AND TRANSFER PRINCIPLES

2.4.6. may be varied in several ways, e.g. by replacing ‘commutative’ in the two instances in which it occurs in 2.4.6. by ‘skew’ or by ‘algebraically closed‘. When interpreting the result for skew fields we may use polynomials whose variables do not necessarily commute. Consider next an integral domain R with unit (1) which satisfies the condition that every element of R is contained only in a finite number of prime ideals of R. Examples of such rings are provided by the algebraic integers in any finite extension of the rationals. Let X be any sentence which is formulated in terms of the relation of equality and in terms of addition andmultiplication, and in terms of the elementsof R. We then have 2.4.11. THEOREM. Suppose X as described, holds in all (commutative) fields which are extensions of R. Then for all except a finite number of prime ideals J in R, X holds in all fields which are extensions of the quotient ring R/J.

PROOF.Let D(R) be the diagram of R. Then the assumption of the theorem is that X holds in all models of K& U D (R), and hence, that X is deducible from that set. Thus, there exists a finite subset of elements of D ( R ) , Y1, . . ., Yk,say, such that YI . . . Yk =I X i s deducible from K&. Now the sentences Yj are of one of the following six forms

E(a, b), S(a, b, c), P(a, b, c),

N

E(a, b),

N

S(a, b, c),

-

N

P(a, b, c )

.

We may eliminate the sentences which are of the form S(a, b, c) by introducing the element d which satisfies S (a, b, d ) in R, and by replacing S(a, b, c ) by S(a, b, c) together with E(c, a). Sentences of the form P(a, b, c) may be eliminated in a similar way. Again, any sentence of the form E(a, b) may be replaced by a sentence of the term E(c, 0) together with S(a, c, b), where c = b - a in R. After all these reductions, we conclude that the Yj may be supposed to be of one of the following four forms. E(a, b), s(a, b, c), P(a, b, c), E(c, 0) .

-

--

N

N

-

Since the number of the Y5 is finite, we have only a finite number of individuals CI .. . cmsuch that the sentences E(ct, 0) appear among the Yj.By the assumptions of 2.4.1 1, there exist only finitely many prime ideals in R which include at least one of the ci. We propose to show that for any prime ideal J which does not belong to this set, X holds any field F which is an extension of RIJ. Indeed, since YIA . . . A Yk = X is deducible from KAF we only have to show that the Y I , . . ., Yk hold in F and since these sentences do not contain any variables or quantifiers N

40

[2.4.

ALGEBRAIC THEORIES

they hold in F if and only if they hold already in RIJ. But RIJ is a homomorphic image of R and so all Yj which are of one of the forms E(u, b), S(a, byc), P (a, b, c), hold in RIJ since they hold in R. On the other hand any of the form E(Q, 0) holds in RIJ since E(c4, 0) in RIJ would imply that ct belongs to J and such J have been excluded explicitly. This completes the proof of 2.4.1 1. As an important application, we have

-

2.4.12. THEOREM. Let R be an integral domain such that every element of R is contained only in a finite number of prime ideals of R. Let p(x1, ...,x,) be a polynomial with coefficients in R which is absolutely irreducible over R. That is to say, p (XI, . . .,x,) is irreducible in all fields which are extensions of R. Then XI, . . ., x,) is irreducible also in all extensions of the integral domain RIJ where J is any prime ideal of R, excepting only a certain finite number of such ideals.

PROOF.Let al, .. ., ar be the coefficients of p(x1, . . ., x,) ranged in some arbitrary but definite order. If p (XI, ...,x,) is reducible then the degrees of the factors are by necessity bounded by the degree of p. Thus, we may paraphrase the statement )(XI, . . ., x,) is reducible’ by the statement ‘there exist polynomials q (XI, . . ., x,) and r (XI, . . ., x,) of degrees not exceeding d - 1 such that, q(x1,

..., x,)

r(x1, .

..,x,)

= p(x1,

.. * ,

x,)

.’

It will be seen that this statement is essentially equivalent to the assertion that a certain system of equations, involving the coefficients of q and r as unknowns possesses a solution. For example, the statement ‘mx21 ~ 2 x 1 ~ 2~ 2 3 x 2 ~ U4x1 U5X2 a6 is reducible’ is equivalent to the statement that this polynomial is identically equal to a product ( ~ 1 x 1 ~ 2 x 2 y3)(y4x1 y5X2 Y6) and this in turn amounts to the assertion that the set of equations

+ +

2.4.13.

+

+ +

+

yly4 = al,Yly5

+

+ +

+

YZy4 = a2,Y2Y5 = a3, YlY6

y2y3

+

y3y5 = a4, y3y6 = a6

+

y3~4 = a4,

possesses a solution. Now this latter fact can be expressed without difficulty in terms of E, S,P,and the individuals al, ...,a k , by a sentence Y which expresses the solvability of 2.4.15 not only for any given field which is an extension of R but also for any field which is an extension of a homomorphic image of R. Taking X = Y in 2.4.11. we may therefore reduce 2.4.12. to 2.4.11. N

2.4.1

41

EMBEDDING THEOREMS AND TRANSFER PRINCIPLES

For number fields, it is customary to say > ( X I , . . ., Xn) is absolutely irreducible’ instead of ‘p is irreducible in all extensions (of the given integral domain)’. Supposing that R is the ring of algebraic integers in finite extension of the ring of rational numbers, we may then reword 2.4.12. as follows. 2.4.14. If a polynomial p(x1, . . .,xn) whose coefficients belong to the ring R of algebraic integers in a finite extension of the rationals, is absolutely irreducible, then p (XI, . . ., X n ) remains irreducible over the quotient ring R/J for all except (possibly) a finite number of ideals in R. 2.4.14. has been discovered repeatedly in classical Algebra. The reader will find it instructive to compare the classical proofs, which are by no means trivial, with the argument given above. If R is the ring of rational integers, then 2.4.11. reduces to 2.4.6., i.e. the rings R/J become simply the prime fields of prime characteristics. It is therefore not difficult to reformulate 2.4.14. for this particular case, in terms of fields of prime characteristic. Next, we consider briefly some properties of infinite conjunctions and disjunctions as introduced at the end of section 1.4. above. Let Y be an infinite conjunction in the sense of that section, Y = XI A XZA X3 A ... such that the Xi belong to a language L of the Lower predicate calculus defined in the usual way while Y itself is a sentence only in some extended language. We shall call Y ‘eflectivelyfinite’ if one of its partial conjunctions, Ym= [ X l A [ X z A [... AX,]

...I]

is such that Ym 3 Y holds in all models in which Y is defined. We then have as a consequence of 2.4.4. 2.4.15. THEOREM. An infinite conjunction Y is equivaIent to a sentence 2 of L (in the sense that Y EE 2 holds in all models in which it is defined) if and only if it is effectively finite.

.

In fact, Y =) Yk holds in all models in which it is defined, k = 1,2, .. If in addition Ym 3 Y holds in all models in which it is defined, for some particular m, then Ym Y, showing that the condition of the theorem is sufficient. To prove that the condition is necessary, consider the sequence YI, Yz, Y3,... This sequence constitutes an increasing chain, Yn 3 Ym is a theorem for n 2 m. Suppose now that for every positive integer rn there exists an n > m such that Ym 2 Yn is not a theorem. In that case we may select a strictly increasing chain from the sequence YI, Yz, Y3, . . ,,so

42

[2.4.

ALGEBRAIC THEORIES

... Ykn = ... Y k s = Ykz = Yk,, The set K of the elements of this sequence holds if and only if Y holds. But according to 2.4.4. no sentence Z of L has that property. Hence, if such a sentence exists, then for some m, Ym Yn for all n > m, and hence for all n, n = 1,2, ... This proves the theorem. Now let Y be an infinite disjunction, Y = [XI V X2 V X3 V .. .] such that the Xn are contained in L, and let Ym be the corresponding partial disjunctions Ym = [ X I V [ ~ V2 [. . . V X ~ . I..]] Then Y, = Y holds in all in all models in which Y is defined. Similarly, Ym Yn is a theorem for m < n. We call Y effectivelyfinite if for some rn, Y 2 Y, holds in all models in which it is defined. In that case, Y = Y, holds in all models in which it is defined, for all n 2 m. It follows that Yk 2 Yn holds for all k and for n 2 m. These considerations show that the condition of the following theorem is necessary. 2.4.16. THEOREM. The infinite disjunction Y is effectively finite if and only if there exists a positive integer m such that Yn = Ym is a theorem for all n(n = 1,2,3, ...).

In order to prove that the condition of the theorem is also sufficient, we shall show that if it is satisfied then Y = Ym holds in all models in which Y is defined. The contrary assumption implies that there exists a structure M in which Y holds and Ym does not hold. It follows that the sentences XI, . . ., Xm do not hold in M while XZholds in M for some 1 > m. But if so then YZholds in M while Ym does not, and so YZ 2 Y, cannot be a theorem. This is contrary to the hypothesis and proves 2.4.16. Now let Z be an expression of the form

.

. Z) V 22 (u,U , W , . .. Z) V . . .]

2.4.17. Z = (VU)(VV)(VW) . . (Vz)[Zi (u, U, W, . . v . . . v Z n ( U , u, w, . . . z) v

such that the components Z n of the infinite disjunction are themselves wff in the ordinary sense. We introduce the partial disjunctions Wm (u, U, w, ...,z) by Wm (u, U, W, . .,Z) =

[z,(u, U, W , .. .,Z) V [ z Z (u,U, W , . . .,Z) V 1. .. V Z m (u, U, W , and we define

z~T,

= (VU)(VU)(VW)

.. .,z)] .. . ]

. . . (VZ) Wm (u, U, W , . . ., z), rn = I ,2, .. . .

2.4.1

43

EMBEDDING THEOREMS A N D TRANSFER PRINCIPLES

Then Z: ZJ Z holds in all models in which Z is defined. We call Z effectively finite if, conversely, there exists a positive integer m such that Z =I Z: holds in all models in which Z is defined. It will be seen that, quite generally, Z: 2 Z: is a theorem for m < n, and so if Z 2 Z: holds in all models in which Z is defined, so does Z 2 Z: for n < m. 2.4.18. THEOREM. A sentence Z (in the wider sense, as defined by 2.4.17) is effectively finite if and only if there exists a set K of sentences of L, whose vocabulary is contained in the vocabulary of 2, such that Y holds in a structure M if and only if M is a model of K.

PROOF. The condition of the theorem is necessary. For if Z is effectively finite and Z; is the sentence used above in the definition of that concept then the set K = { Z i }has the required properties. Conversely, if a set K with the required properties exists then we have to show that Z is effectively finite. If this is not true then there exists, for every positive integer m a structure M in which Z is defined but such that Z 3 Z, does not hold in it. Such a structure does not satisfy Z:, but satisfies Z, and hence satisfies all sentences of K. Thus, K u {- Z:} is consistent for all m. Now let a, b, c, . . .,d be a set of individuals in one-to-one correspondence with the variables u, u, w, . . . z in Z, and which are not contained in 2. Then we maintain that 2-4-19. K U {

N

2 1 (a,

. ..)d),. .

byC, . . .,d),

Z2 (a, byC,

.}

. . .,d),...,

-

Zn (a, byC,

is consistent. For if this were not the case then, for some positive integer n

K U {- Zl(a, byC, .. . d),. . .,

N

Z~(U by,C,

. .., d)}

.

would be contradictory, hence Z1 (a, byc, . . . d ) V . . . V Zn (a, byc, . . d) would be deducible from K, and hence (since a, by c, . . . d do not appear in K ) the sentence

V

(Vu)(Vu)(Vw) [zZ

. . . (VZ) [Zl(U,u, w, .. .)z) v .. . V Zn (u,U, W, . ..,Z)1 .. .]

(u,U, W,Z) V

would be deducible from K. But this sentence is Z i , and so we have shown on one hand that that K U {- Z,} is consistent for all m and on the other hand, that K t Zn. This is impossible, showing that 2.4.19. is consistent. Let M * be a model of 2.4.19. Then M * satisfies K but it cannot satisfy Z, since this would imply that at least one of the formulae Zn(a, b, c, . .d )

.

44

r2.4.

ALGEBRAIC THEORIES

holds in M*. This contradicts the definition of M*. However, by the condition of 2.4.18. Z holds in every model of K. Thus we have again arrived at a contradiction, and we conclude that Z is effectively finite. This completes the proof of 2.4.18. Let K be a (finite or infinite) set of sentences within L and let K' be obtained from K by the inclusion of a sentence Z where Z is given by 2.4.17, and such that the sentences Z: can be deduced from K for m = 1,2, . . . Then

-

-

2.4.20. THEOREM. If the sentence X of L holds in all structures in which

K' holds then X can be deduced from K alone.

Indeed, the set K' U {- X } does not hold in any model, by assumption. It follows that the set

K U { - X y ~ Z i ( a Y b,...d),... ,c ,-Z~(U,~~C,...,~)~...} is contradictory, where we suppose that the individuals a, by c, . . . d do not occur in either K or X. We conclude that for some positive integer m y the set

-

. .. d)A [. .. A Z ~ ( Uby,C, .. .,d)] . . .]} is contradictory. Hence, as before, the set K u {- X , Z ; } is contraK U {- X,

[ Z i ( a ,by C,

--

dictory, i.e. X V Z: is deducible from K. But since Z: is deducible from K by assumption, it follows that X must be deducible from K. This proves the theorem. Let K, Z and Z: be defined as before and suppose that Z: can be deduced from K for all m. Then 2.4.20. THEOREM. If the set K is consistent then

model M of K.

-

-

-

Z holds in some

PROOF.Put X = A A A in 2.4.20., for some arbitrary relation A of order 0. If K u {- Z } does not possess any model then X holds in the models of K U {- Z } and hence is deducible from K alone. But this implies that K is contradictory, proving the theorem. We apply the above results in order to obtain some information regarding the position of Archimedes' axiom. 2.4.22. THEOREM. The set of axioms KAF for the concept of an archi-

medean ordered field cannot be replaced by any set of sentences K within L (finite or infinite) which are formulated in terms of the vocabulary of KAF.

2.5.1

45

MALCEV'S THEORY OF NORMAL CHAMS

PROOF. We first replace KAFby a single sentence of the form 2.4.17, as follows. Let V be the conjunction of the sentences of KOF,which are finite in number. Referring to 2.2.18., introduce the sentence of the extended language -

- ..

-

2.4.23. Z = (VX) (Vy) [ [ V h E(0, x)] V [ V h Q (0, x)] V [ V h V [ V A Q ~ ( x , Y )V] .]

N

QI(x,

v)]

Z is the required single axiom for the notion of an archimedean ordered field. Now, by 2.4.18., the existence of a set K as described in 2.4.22 would imply that 2.4.23. is effectively finite, i.e. that for some m ythe sentence 2 3 Z m holds in all models in which 2 is defined, where 2; = (Vx)(Vy)[ [ V h E(0, x)] V [ V h Qm-1

N

Q (0,x)] V

... V [V h

(w)]].

But this would signify that in every archimedean ordered field, with 1 for x, the number m - 1 = (m - 1) 1 is greater than all other elements of the field. This is impossible and proves the assertion. 2.4.24. THEOREM. If a sentence X formulated in the Lower predicate calculus in terms of the relations of equality, addition, multiplication, and order (and in terms of 0 and 1) holds for all non-archimedean ordered fields then it holds for all ordered fields.

We may put this in a slightly different way be saying that if a sentence (i.e. X) is consistent with KOFtogether with the sentence 2.2.18., then Xis consistent with KOFtogether with the negation of 2.2.18. Indeed, by 2.4.21., with 2 for 2.2.18., we only have to show the set KOF U {- X) is consistent, and this certainly follows from the assumption of the theorem to be proved. The present book is dedicated chiefly to the Lower predicate calculus and we shall therefore not pursue the subject of an extended language as above, any further. However, we mention that this subject has in fact been developed more systematically and has led to a number of important results. N

2.5. Malcev's Theory of Normal Chains. We shall now consider certain

questions in general Group theory amenable to treatment by metamathematical methods. A survey of Group theory shows that most interesting problems in that field go beyond the language of the Lower predicate calculus, particularly since they refer frequently to the existence and

44

[2S.

ALGEBRAIC THEoRlES

interrelations of the set of all subgroups, or of the set of normal subgroups, of a given group and these notions belong in a natural way to languages of a higher order. However, it was shown by Malcev in a pioneer paper that this difficulty can be overcome in certain cases. We shall use K & which is formulated in terms of E, S and e as a standard set of axioms for the concept of a group. Suppose that we wish to express the fact that a group G contains subgroups J and H, H a normal subgroup of J such that the factor group J / H satisfies a given sentence X within the language L of the Lower predicate calculus. In order to do so we first introduce two new one-place relations, R ( x ) and T ( x ) , which are supposed to indicate membership in J and H respectively. The sentences 2.5.1.

( V X ) WY)%[Y

(W (try> [J%,

39 A R (4 = R (Y)] Y) A T(x)= T(Y)]

then ensure that the relation of equality possesses the property of substitutivity with respect to R and T. In the terminology used at the beginning of section 2.1. this means that the relations of inclusion between G and J and between G and H are normal. The following sentences now express the fact that the elements of G which satisfy R constitute a subgroup of G 2.5.2.

( V x ) ( v Y ) w [ R ( xA ) R O A S(XY YY 4 R (4 ( V X ) (VY) [ R(4A S(X, Y,4 = R ]

= R(4]

We denote this set by KGRand we denote the correspondingset for Tby KGT. Thenext two sentences state that H (which is the set of elements of G that satisfy T ) is a normal subgroup of J (which is the set of elements of G that satisfy R)

2.5.3.

(W [W)= R (41 ( V X ) WY)V (4P O(‘d4[s (x, YY 4 A s (x, 2, 0 A s 0,YY 4 A R(x) A T(z)

T(u)].

Now let X be any sentence formulated in terms of E, S and e. We wish to state that X holds in J/H. Let the relation of equality in J/H be E* (x, y) and let the relation which represents the group operation be S* (x, y , 2). Instead of regarding J / H as a set of equivalence classes of J we shall supposethat it is made up of the same elements as J but that the definitions of equality and of the group operation have been changed so that

2.5.1

47

MALCEV’S THBORY OF NORMAL CHAINS

equality in J/H means equivalence in J with respect to the normal subgroup H of J. Thus, E* ( x , y ) is defined by 2.5.4.

E* (4Y ) = (W

[wA T(x,

z 7

Y)]9

Also, S* (x,y, z) has to hold whenever (lw)[S(x, y , w) A E* (w,z)] holds, so that the appropriate definition of S* ( x , y , z) is 2.5.5.

( W W [ S ( X ,Y , w) A S(W, 1, y ) A

W)]

We now introduce a sentence X * which is obtained from X by the following transformation. We first replace E and S everywhere in X by E* and S* respectively, and we replace E* and S* in turn by the right hand sides of 2.5.4. and 2.5.5. Then it will be seen that X * holds in G if and only if X holds in J/H. This is a particular example of the reduction of a statement which is, to begin with, outside the Lower predicate calculus, to a sentence within that calculus. A quasi-elementary property of a group is a property which can be expressed by a set of sentences P formulated in the Lower predicate calculus in terms of E, S, and e, and such that whenever P holds in a group G then P holds also in every subgroup of G. We shall identify P with the property described by it. By a normaI chain of a group G we mean, as usual, a finite sequence of subgroups of G, {GO,GI, Gz,. . ., GL) such that 2.5.6.

G = Go

3

GI

3

G2

3

... Gk = (e)

such that Gj is a normal subgroup of G j - ~ , j= 1,2, . . . k. k is called the length of the chain. Given a finite sequence of quasi-elementary properties 17 = {Pr, ...,P k ) we shall say that a group G is of type 17 if C possesses a normal chain of length k as in 2.5.6. such that the factor group Gj = Gj-I/Gj has the property Pj, j = 1, .. ., k. If a group G is of a given type 17 = {Pz, ... P k ) , then every subgroup H of G is of type 17. Indeed, G possesses a normal chain {Go, GI, . . .,Gk} as required. Now consider the sequence of subgroups of H, {Ho,HI, ...)Hk) where Hj = Gj n H. This is a normal chain since HO= H, H k = (e), and Hj = Gj rl H is a normal subgroup of Hj-1 = G5-1 (I H for j = 1, . . .,k. Moreover Hj-lIHj is isomorphic to a subgroup of Gj-l/Gj and so, by one of the conditions satisfied by an elementary property, Hj-IIHj like Gj-l/Gj satisfies Pj. Hence H also is of type IT. For given 17,we construct a set of sentencesKnas follows. We introduce one-place relations Rj, j = 1, ...,k and we include in Kn,(i) the set K i ,

48

AWEBRAIC THEORIES

j2.5.

(ii) sentences which state that the relation of equality is substitutive with respect to all the Rj (compare 2.5.1.), (iii) sentences which state that for every j, the set Gj of elements which satisfy Rj is a subgroup of the given group (see 2.5.2.), (iv) sentences which state that Gj is a normal subgroup of GI-1, j = 1, . . ., k (compare 2.5.3.), (v) a sentence which states that Gk contains only e and elements equal to e, e.g. the sentence (Vx)[Rk(x)3 E ( x , e ) ] ,(vi) sentences which state that the factor group G f = Gj-I/GJ possesses the property Pj,j = 1, . . ., k. In order to express this fact we replace every sentence X which belongs to P5 by a sentence X* which is obtained from X by means of the relations Rj-1 and Rj just as in the example given above (see 2.5.4. and 2.5.5.) we obtained X * from X by the use of the relations E and T. All sentences that are obtained in this way will be included in Kn. This concludes the construction of Kn. It follows immediately from the construction that every model of Kn is a group of type 27 (the subgroups Gj being determined by the predicates RJ.)A given group G of type 17 is not in the usual sense a model of Kn since the relations Rj which we obtained in the vocabulary of Kn do not belong to G. However, we have the theorem 2.5.7. THEOREM. For given h ' , a group G is of type I7 if and only if the set D ( G ) U Kn is consistent where D(G) is the diagram of G.

PROOF.If the group G is of type l?, then it includes a normal chain {Go, . . ., Gk} with the required properties. We introduce relations Rj

into the structure G by defining that Rj(a) holds for any element a of G if and only if a belongs to Gj,j = 1, . . .,k. With this supplementary definition G becomes a model both of D (G) and of K, so that D (G) U Kn must be consistent. Conversely, if D(G) u Kn is consistent, let G' be a model of that set. Then G' is a group of type 17 and an extension of G, and so, by a previous remark, G also is of type 17. This proves the theorem. We now come to the main theorem of this section. 2.5.8. THEOREM. Suppose that every finitely generated subgroup of a

group G is of type 17. Then G is of type 17.

PROOF.Construct Kn as above and consider the set D(G) u Kn.By 2.5.7. we only have to show that this set is consistent. We know from the usual argument that this will be the case if every finite subset of D ( G ) u K, is consistent. Now this will certainly be true if for every finite subset of elements of G, G' say, D(G)' U Kn is consistent where

2.6.1

PROBLEMS

49

D (G') is the diagram of the restriction of the relations of G to G'. But if G" is the group which is generated by the elements of G', then on one hand D (G") U Kn is consistent by 2.5.7., and on the other hand D (G') c c D(G"). Hence D(G') U Kn also is consistent. This proves 2.5.8. From among the many interesting applications that Malcev made of his theorem (2.5.8.) we mention the following two. A group G is said to be solvable of rank k if it possesses a normal chain of length k such that the factor groups Gj-l/Gj are abelian (commutative). Thus, G may be said to be of type 17 where 17 = {PI, . . .,Pk}, PI = . . . = = P k = P, and P contains the single sentence i.e. the group in question is commutative. We see immediately that P is quasi-elementary and so 2.5.8. applies. Hence 2.5.9. THEOREM. If every finitely generated subgroup of a group G is solvable of rank k then G is solvable of rank k. Now let 17 = {PI, Pz}where P 2 = P is the property of commutativity as in the last section and Pi states, for a given positive integer n, that the group in question does not include more than n different (i.e. unequal) elements. It is easy to check that this property also is quasi-elementary. Hence, by 2.5.8., noting that GlfG2 = Gl/(e) is isomorphic to GI , 2.5.10. THEOREM. If every finitely generated subgroup of a group G contains a normal subgroup of index 6 n then G contains a normal subgroup of index < n.

In view of the important role played in the above theory by the kind of properties called quasi-elementary it is natural to ask what distinguishes a set of sentences which is such that whenever it holds in a group G then it holds also in every subgroup of G. This is a typical problem of Model theory, to which a full answer will be provided in the next chapter (section 3.3). 2.6. Problems 2.6.1. Let S = ( A y } be a set of disjoint non-empty abstract sets. The axiom of choice states that there exists a set which has just one element in common with every set A, in S. Employ the extended completeness theorem 1.5.4. in order to prove the axiom of choice for the case that all Ar are finite. (Hint. Introduce the relation of equality E ( x , y ) and relations

50

ALGEBRAIC THEORIES

[2.6.

R,(x) such that R, (a) holds if and only if a E A,. For an additional relation P (x), ensure by means of sentences in the formal language L that for every R, there exists just one element which satisfies both R1(x) and P(x).) The method does not apply to infinite A,. Why? 2.6.2. Prove the maximal ideal theorem for Boolean algebras by means of

1.5.4. (and, of course, without using the axiom of choice).

2.6.3. A graph is called k-chromatic if k is the smallest positive integer for which there exists a partition of the nodes of G into k sets Gt such that no two elements of any Gt are linked by an edge of G. Prove the theorem

of de Bruijn and Erdos [de Bruijn-Erdos 19511 that every k-chromatic graph contains a finite subgraph which is also k-chromatic.

2.6.4. Let K,,be the set of sentences which are formulated in terms of

the relations of addition, multiplication, and equality, E, S, P,and in terms of individuals 0, 1, 2, . .. (with their usual meaning), and which hold in the system of natural numbers, Mo. Show that there exists a model M of K which is a proper extension of Mo. Such a structure M is called a (strong) non-standard model of Arithmetic (Compare section 9.3. below) (Hint. Define H a s the set of sentences E(a, n) where a is an additional individual, and n = 0, 1,2, . . . .Prove that K u H i s consistent). N

2.6.5. Show that not every element of a model M as defined in 2.7.4. is

the product of a finite number of primes. Why is this compatible with the fact that M is a model of KO?

References. Embedding principles are considered from a mathematical point of view in Lo61955 and A. Robinson 1955. Compare also Neumann 1954. Theorem 2.4.6. is given in A. Robinson, 1951, 2.4.12. in Henkin 1953 and A. Robinson, 1955. For purely mathematical proofs of 2.4.14. see Ostrowski 1919 and Eichler 1939.The analysis of Archimedes’ axiom is in A. Robinson 1951. For more recent work on infinitary languages see Scott-Tarski 1958, Tarski 1958, Engeler 1961. Malcev’s work on normal chains including the main theorem 2.5.8. is in Malcev 1941.

C H A P T E R 111

SOME CONCEPTS AND METHODS OF MODEL THEORY 3.1. Skolem Functions; Relativization. Consider any sentence of L which

is in prenex normal form, e.g.

3.1.1.

(W ( 3 ~ 0'4 ) 0'4( 3 ~ (W ) (30 Q ( x , Y , z, u, v, w,0

where Q does not contain any further quantifiers. Putting

R (x, Y ) = (W (vu) (W (W ( 3 0 Q (x, Y , z, u, v, w, 1) we obtain from 3.1.1.

3.1.2.

(W(3Y) R

(x7

Y)

Let M be any model of 3.1.2. According to the semantic interpretation of 3.1.2. the fact that the sentence in question holds in M signifies that for any element a of M there exists an element b of M such that R(a, b) holds in M. That is to say there exists a function p ( x ) whose domain are the elements of M and which takes values in M such that for every a E M , the sentence R (a, q (a))holds in M. Note that the expression R (a, I (a)) as it stands does not belong to L. It denotes a sentence R(a, b) which belongs to L. We may interpret the remaining existential quantifiers in 3.2.1. in a similar way. Thus, the fact that 3.2.1. holds in M signifies that there (x,), w (x, z, u) x ( x , z, u, w) defined on M and taking exist functions I values in M such that for all elements a, b, c, d of M the sentence

3.1.3.

Q (a, I (4,b, c, w (a, b, c),

d, x (a, by c, 6))

holds in M . It is customary to replace the individuals a, b, c, d in this expression by the original variables, and to state simply that

3.1.4.

Q (x, I (XI, z, u, w ( x , z , 4 , w,x (x, z, u, 4 )

holds in M . 3.1.4. is called the open form (sentence) which corresponds to 3.1.1. and the function symbols q, w, x are known as Skolem functors, or sometimes as Herbrand functors. They were introduced by Skolem and 51

52

SOME CONCEPTS A N D METHODS OF MODEL THEORY

[3.1.

Hilbert, and exploited systematically by Herbrand. In general one obtains the open form of a sentence X in prenex normal form by deleting the prefix of X and by replacing the variables in the matrix which belong to existential quantifiers by distinct symbols denoting functions of all variables which belong to the universal quantifierspreceding the existential quantifier in question. If the sentence X begins with an existential quantifier then the corresponding functor is supposed to denote a function without variables, i.e. a particular element of M. The functions denoted by the functors are defined on all elements of M and take values in M. They are called Skolem functions. In general, we do not find it necessary to introduce separate symbols to denote the Skolem functors on one hand and the Skolem functions on the other. Skolem functions provide a concrete interpretation of the meaning of the existential quantifiers and have been found useful in various connections. Let X be any sentence in a language L of the Lower predicate calculus and let M be a model of X. Any reference in X to a particular element of M , or to the existence of an element of M with a particular property, need not state the fact that the individual in question is an element of M explicitly since it follows from the definition of a model that any individual mentioned is meant to be an individual in M. Now suppose that M is contained as a substructure in a larger structure M'. Then we cannot express the fact that X holds in M in terms of the semantic interpretation of Xin M'. In order to be able to do so, we introduce a one-place relation R ( x ) which is not contained in M (and hence neither in M' nor in X ) such that R (a) holds for an element a of M' if and only if a belongs to M. The resulting structure will be called M i . For any sentence Y which is defined in M , we shall now define a related sentence YR such that YE holds in M i if and only if Y holds in M . We shall in fact define YRquite generally for any wff Y in L. Thus, the passage from Y to YRamounts to a mapping of the wff of L into the set of WEof L. YBis a syntactical transform of Y, an informal term by which we mean that YRis obtained from Y by a rule which is purely syntactical and takes no account of the meaning of the wff. More particularly, in the present case, YRis called the transform of Y by relativization with respect to R,briefly the R - transform of Y and we write YR= p (Y). The mapping is defined following the rules of formation of wff (1.2.1.-1.2.3.). The following expression are written out in fully bracketed form. 3.1.5.

If Xis an atomic formula, then e ( [ X I ) = [XI

3.1.1

SKOLEM FUNCTIONS ;RELATIVIZATION

53

3.1.6.

3.1.7.

P ( [ W ) X ] )= < W " R W ] = P W ) ] p ( [ ( 3 y ) x ~= ) ( 3 y ) [ [ ~ ( yAAX)] )]

where Xis a wff which does not contain y quantified. The two formulae of 3.1.7. are crucial. They express the fact that where in the original wff a universal quantifier applies universally, it now is to apply only to elements that satisfy R,and where in the original wff the existence of a certain element is required such that Xis satisfied, it is now stipulated in addition that the element satisfy the relation R. Now let M and M' be similar structures, M' an extension of M , and X defined in M yi.e. the relations and individuals of X occur in M . Let R be a relation which does not occur in Myand let the structure MA be defined as before. Then 3.1.8. THEOREM.X holds in M if and only if X R holds in MA. The proof is by induction on the order of the sentences X which are defined in M (see section 1.2. for the definition of the order of a wff). If X is of order 1 then it is a bracketed atomic formula and sop ( X ) = XR = X, by 3.1.5. It is evident that in that case X holds in Mif and only if it holds in

MA.

If Xis obtained by the application of a connective to sentences of lower order for which the theorem has already been proved, then the assertion of the theorem is true also for X . To see this, suppose for example that X = [Y A Z ] ,X R = p ( X ) = [ p ( Y )A p ( Z ) ] .If X holds in Mythen both Y and 2 hold in Myhence, by the assumption of our induction p ( Y ) and p (2)hold in MA, hence XR = p ( X ) holds in MA.If X does not hold in M then one of Y and Z , Y say, does not hold in Myhence p ( Y ) does not hold in MA hence X R = p ( X ) does not hold in MA. The procedure is similar for all the remaining connectives. Suppose finally that X is obtained by quantification from sentences of lower order for which the theorem has been proved already. Thus if X = [ (Vy) Z ( y ) ] ,then the theorem is supposed to have been proved for all 2 (a),where a is any element of M. If X holds in M , then Z (a) holds in

54

SOME CONCEPTS AND METHODS

13.2.

OF MODEL THEORY

M for all elements a of M, hence p(Z(a)) holds in MA. It follows that [R (a)] 3 p (Z (a)) holds in M i for all elements a of M i ; for if a is in M then the implicate of the implication holds, and if a is not in M then the implicans of the implication does not hold, showing that the implication as such holds for all a. Hence (Vy)[ [ R ( y ) ]= p ( Z ( y ) ) ] = p ( X ) holds in MA where we take into account that if p (,Z(a))= Z'(a) then p ( Z ( y ) ) = = Z'(y). On the other hand, if X does not hold in M , then Z ( a ) does not hold in M for some element a of M , hence the sentences p ( Z ( a ) ) and [R(a)] 3 p(Z(a)) do not hold in MA for this particular a, hence (Vy)[ [R(y)] 3 p ( Z ( y ) ) ] does not hold in MA. Again, if X = [ (3y) Z ( y ) ] and the theorem has been proved for all Z (a) with a in M , suppose that X holds in M. Then Z (a) holds in M for some a, hence R(a) A p(Z(a)) holds in M i , hence (3y) [R(y)] A A p ( Z ( y ) ) ] holds in MA, as required. Conversely, if ( 3 y )[ [ R ( y ) ]A p ( Z ( y ) ) ] holds in MA then [ [ R ( a ) ) A p ( Z ( a ) ) ] holds in MA for some a, and that a is in M for it satisfies R (a). Hence, by the assumption of our induction Z(a) holds in M , X = [ (3y) Z ( y ) ] holds in M. This completes the proof of 3.1.8. Now let K be any set of sentences which does not contain R and let K' be the set of all sentences p (X)= X R , X E K. Let M' be any model of K'. Then we cannot affirm that set of elements of M' which satisfy R constitute a model of K, because that set may well be empty. For example, if K contains the sentence (Vx) T ( x ) as its only element then K' contains only the sentence (Vx) [R ( x ) 3 T ( x ) ], and this sentence holds in any structure M' in which R and T occur even if R does not hold for a single element of M'. Accordingly, we define the R-transform of the set of sentences K, p (K) = K R as the set of all sentences X R = p (X), X E K together with the sentence (3x) R (x) if no individuals are contained in K, or of all sentences R(a) for the individuals a which occur in K, if there are such individuals. With this definition, let M' be a model of KR. Then the set of elements of M' which satisfy R is not empty, and if we restrict the relations of M' to that set we obtain a substructure M of M' such that the sentences of K are defined in M . We may now conclude from 3.1.8. that M is a model of K.

1

3.2. The Extension of Models. Let K be any set of sentences. K is said to be complete if for any sentence X which is defined in K (i.e. whose relations and individuals are in the vocabulary of K), either K t X or K t X.

-

3.2.1

55

THE EXTENSION OF MODELS

This is one of the fundamental notions of Model theory. It is not to be confused with the notion of completeness of the entire Lower predicate calculus which was introduced in Chapter I. Let M and M‘ be two similar structures and let A be a subset of the set of individuals common to M and M’. A may be chosen so as to be empty even if there are individuals common to M and M’. Then M is said to be elementarily equivalent to M’ with respect to A if any sentence X whose relations are in the vocabulary of M and M’ and whose individualsbelong to A either holds in both M and M‘ or does not hold in either one of these structures (so that X holds in both). This definition may be formulated in a different way if for any structure M y and for any set of elements of M , A (which may be empty) we define S ( M , A ) as the set of all sentences X which hold in M and which do not contain any individuals other than the individuals of A . Then two similar structures, M and M‘ are elementarily equivalent with respect to A if S ( M , A ) = S ( M ’ , A). If A is empty then we say simply that M and M’ are elementarily equivalent. It is not difficult to see that a set of sentences S ( M , A ) must be complete for if X is any sentence defined in S ( M , A ) then either X or X belongs to S ( M , A). If A contains all individuals of M then we shall write also S(M, A) = S(M). Now let M and M’ be two similar structures such that M’ is an extension of M and let A be the set of all individuals which are contained in M. Then we say that M’ is an elementary extension of M ywrite M ee M’ if M and M’ are elementarily equivalent with respect to A. The notion of an elementary extension is due to Tarski and Vaught. A structure M ’may well be an extension of a structure M and at the same time be elementarily equivalent to M y without being an elementary extension of M . Thus, let M‘ be the structure which consists of a set of individuals {ao, al, a2, .. .} and of the single relation Q ( x , y ) such that R(an,am) holds if and only if n < m. Let M be the restriction of M’ to the set (al, u2, as, . . .}. Then M‘ is elementarily equivalent to M (i.e. elementarily equivalent with respect to the empty set), since M’ is isomorphic to M by the correspondence a n t)an+i,n = 0, 1,2, .. On the other hand, M‘ is not elementary extension of M since the sentence (Vx) Q (x, al) holds in M but not in M‘.

-

-

.

3.2.1. THEOREM. Let M’ be an extension of M and let A be the set of individuals of M. In order that M’ be an elementary extension of M it is necessary and sufficient that for every sentence X which begins with an

56

SOME CONCEPTS AND METHODS OF MODEL THEORY

I3.2.

existential qualifier, X = (3y) Z ( y ) , say, and which is defined in M and holds in M', there exists an element a E A such that Z ( a ) holds in M'. PROOF.The condition is necessary. For if M' is an elementary extension of M , then since X holds in M' it must hold also in M . That is to say, for some a E A , Z ( a ) holds in M and hence in M'. The condition is also sufficient. Let X be any sentence which is defined in M and holds in M', then we have to show that X holds also in M . We may suppose for the proof that X is in prenex normal form since any other sentence is equivalent to a sentence in prenex normal form with the same vocabulary. We now prove our assertion by induction with respect to the number of quantifiers in the prefix of X.If X does not contain any quantifiers, then since all the atomic formulae contained in X hold or do not hold equally in M and M', and since the question whether X does or does not hold in a given structure is then decided by the truth-table method of the propositional calculus, it follows that X holds equally in M' and in M . Suppose now that there exists a sentence in prenex normal form which holds in M' but not in M . If so there exists such a sentence for which the number of quantifiers is minimal although as we have seen already, it must be positive. Suppose first that X begins with a universal quantifier, X = (Vy) 2 (y). If X does not hold in M then the sentence Z (a) does not hold in M for some a E A. But in this case Z (a) cannot hold in M' either since the number of quantifiers in Z(a) is less than in X. It follows that (Vy) Z ( y ) does not hold in M' contrary to assumption. Suppose next that X begins with an existential quantifier, X = (3y) Z ( y ) . Since X holds in M' it follows from the assumption of the theorem that for some a E A , Z(a) holds in M'. Again, Z ( u ) contains less quantifiers than X and so holds in M . But if so then (3y) Z ( y ) holds in M , again contrary to assumption. This completes the proof of the theorem. Let M' be an elementary extension of M and let A be the set of individuals of M yas before. Let X be any sentence in prenex normal form which is defined and holds in M y e.g.

3.2.2. X = ( V X ) ( ~ Y(Vz) ) (Vv) ( 3 4 Q (xYY , Z, u, 4 where the matrix Q does not contain any further quantifiers. We introduce the open form of X , i.e. for the above example, the expression

Q ( x , v (42, w (x, z, uY4) so that ~ ( x ) ~, ( xz), are the Skolem functors which belong to X . With these assumptions, we have

3.2.1

57

THE EXTENSION OF MODELS

3.2.3. THEOREM. Consider any definition of the Skolem functions of X (e.g. p (x), ly (x, z ) ) in the set of individuals of M , A. Then these functions can be extended to the entire set of individuals of M' so as to constitute Skolem functions for X within M'.

Indeed, suppose that the prefix of X contains the universal quantifiers . . . (Vxk) in that order together with certain existential quantifiers, which appear interspersed between the (Vxj). Then we have to show that if the variables XI, . . . xf are replaced by individuals which belong to M , and hence the functional values of all Skolem functions which contain no other variables are determined as certain elements of M , then the values of the remaining Skolem functions can still be determined appropriately in M' for arbitrary values of the remaining universally quantified variables. But this is clearly possible, for the substitution of elements of M for XI, ..,xj, and for the Skolem functions in question, turns X into a sentence A" which holds in M and hence in M'. Thus, the remaining functions may be chosen appropriately. For example, if in 3.2.2. we take j = 2 and replace x, z, by elements a, b, of M , then the only Skolem function determined by a and b is q (x). Accordingly, we put X' = ( v ) (3w) Q (a, p(a), b, u, w). Then X' holds in M' and so it must be possible to define ~ ( ab,, u) for arbitrary values of u in M ' . It is clear that if M' is an elementary extension of M and M" is an elementary extension of M ' , then M" is an elementary extension of M , It is also true that if M' is an extension of M and M" is an elementary extension of M' and an elementary extension of M , then M' must be an elementary extension of M. For suppose that X holds in M , but not in M ' , i.e. X holds in M'. If so then both X and X hold in M " which is impossible. Let 0 = ( M y } be a monotonic set of similar structures, that is to say for any two elements M,, M y of @ we have either M p c Mv or MI M,. We may introduce a structure M which is by definition the union of the structures Mv, M = U { M y } in a natural way by taking the set of (Vxl)

.

N

h)

individuals of the My, and by defining that a relation R (XI, . . .,xn) holds between elements al, . ..,apzif M if and only if R(a1, ...,an) holds for any (and hence for all) M , that contains al, . ..,an. V

3.2.4. THEOREM. Let Mybe a set of similar structures such that for any M,, Mv of @ either MBee Myor M y ee M,(so that @ is a monotonic set). Then the union M = U { M y }is an elementary extension of every M y in @. V

58

[3.2.

SOME CONCEPTS AND METHODS OF MODEL THEORY

PROOF.We show first that there exists some structure M' which is an elementary extension of all M y in @. Let S(M,) be the set of all sentences which are defined and hold in Mv, as introduced earlier, S ( M v ) = = S(Mv, Av) where A, is the set of all individuals in Mv. Consider the union S = U ( S ( M v ) } . The S(M,) constitute a monotonic set of V

sentences and so every finite subset S' of S is contained already in a single Mv. Hence S is consistent and possesses a model M'. But for every M y , S(Mv) contains the diagram of Mv. Hence S contains the diagram of Mv and M' is an extension of Mv. Moreover, M' satisfies all sentences of S ( M v )and so it is actually an elementary extension of Mv. Now let M = U {My }.We propose to show that M' is an elementary V

extension of M. It then follows from an observation made earlier in this section that M is an elementary extension of Mv. To prove that M' is an elementary extension of M , we make use of 3.2.1. Thus, let X be a sentence which is defined in M and holds in M', and which begins with an existential quantifier, X = (3y) Z (y). Since the number of individuals in X is finite X must be contained already in some Mv. Hence, for since a 4 Av, Z(a) holds in Mv and hence in M'. But A is contained in M , and so the conditions of 3.2.1. are satisfied and M' is an elementary extension of M . This completes the proof of 3.2.4. 3.2.5. THEOREM. Let M and M' be two similar structures, and let A be the set of individuals common to both. Suppose that M and M are elementarily equivalent with respect to A, S ( M , A) = S ( M ' , A). Then there exists a structure M * which is an elementary extension of both M and M'.

PROOF.Let S ( M ) and S ( M ' ) be the sets of all sentences which hold in M and M' respectively. We shall show that the set S ( M ) U S ( M ' ) = T is consistent. If not then some finite subset T' of T is contradictory. T' must contain elements of both S ( M ) and S ( M ' ) otherwise it would be a subset of one of these sets and hence, would be consistent. Moreover, together with any two sentences, S ( M ) contains also their conjunction, and the same applies to S(M'). Accordingly, we may suppose that T' contains just two sentences, X E S ( M ) and X' E S ( M ' ) such that X A X' is contradictory. Now let al, . . ., Uk be the individuals which occur in X and are not contained in M' and let bl, ..., bj be the individuals which occur in X' and are not contained in M. Thus, we may write X = Y(Ui, ., Uk), x' = Y'(bi, ...,bj),k 2 0,i 2 0. Now if Y(a1, ., U k )

..

..

3.2.1

59

THE EXTENSION OF MODELS

A Y (bl, calcu1us,

. ..,bj) is contradictory, then by the rules of the Lower predicate

3.2.6. (3x1)

...( 3 4 (3yi) ..- @~j) [Y(xi, ... ,xk) A Y'(yi, ...,yj)]

also must be contradictory, where we have introduced xf, yg which did not occur previously. Since bl, . . ., bj do not occur in Y(a, . .,ak) the yr do not occur in Y(x1, . . ., xk) in 3.2.6. and so we may replace 3.2.6. by the equivalent.

.

3.2.7.

[ (3x1) . . . (3xk) Y(xi, . . ., ~ t ) ]A [ ( 3 ~ 1 ). . . (3yz) Y' ( ~ i ,.. ., ~j)]

Now the conjuncts in 3.2.1. are defined in both M and M', and they belong to S ( M , A ) and S ( M ' , A ) respectively. But S ( M , A ) = S ( M ' , A ) and so 3.2.7. holds in both M and M'. This proves that 3.2.7. is consistent. We conclude that T is consistent and possesses a model M*. M * is an elementary extension of M since it satisfies S ( M ) and it is an elementary extension of M' since it satisfies S ( M ' ) . This concludes the proof 3.2.5. We shall now consider the question to what extent we may assign the cardinal number of an extension in which we are interested, arbitrarily. By the cardinal number of a structure, without reference to a relation of equality, we mean the cardinal number of its set of individuals. Given a structure M of cardinal 01, we may find an elementary extension of M of a preassigned cardinal 01' > a in the following trivial fashion. We choose a set of individuals, B, whose elements do not belong to the set of individuals A of M , such that B is of cardinal a' or a' - CY according as a' is infinite or finite. We also select an arbitrary but definite element a of A . The structure M' with set of individuals A U B is then defined as follows. For any relation R (XI, .. ., x,) which is contained in M , R (a:, .. .,a;) shall hold in M ' , a;, . . ., a; E A u B if R(a1, . . . an) holds in M where ac = a; if u; E A and a6 = a if a; E B. The structure M' obtained in this way is an extension of M . We wish to show that it is more particularly, an elementary extension of M , i.e. that all elements of S ( M ) hold in M' or, which is the same, that any sentence which is defined in M and holds in M' holds also in M. Again, it is clearly sufficient to prove the assertion for sentences in prenex normal form since all other sentences are equivalent to sentences of this kind. We map any X' E S ( M )which is in prenex normal form on a sentence X E S ( M ) by replacing any b which occurs in X' by the particular element U E A .In particular, if X' does not contain any elements of B then X' -+X = X ' , i.e. X ' is mapped on itself. Accordingly we shall have proved the theorem if we can show that

60

SOME CONCEPTS AND METHODS OF MODEL THEORY

[3.2.

X holds in M if X' holds in M'. If X' is of order 1 then X holds in M if and only if X' holds in M', by the definition of M'. If X' is free of quantifiers, then it is obtained from certain sentences of order one by the same successive application of connectives by which X is obtained from the corresponding sentences of order one (which are defined in M). It follows that for such sentences also X holds in M precisely when X' holds in M'. For general sentences in prenex normal form we only have to prove that if X' holds in M' then X holds in M and we do this by induction with respect to the number of quantifiers in X'. The assertion is correct if the number of quantifiers is zero. Also if X' = [ (3y) 2' (y)] holds in M' and X' +X = [ (3y) Z ( y ) ] , then Z'(b') holds in M' for some b' E A u B. But Z' (b') +Z ( b ) where b E A and so Z ( b ) holds in M. Hence [ (3y) Z (y)] holds in M , as required. Again, if X' = [ (Vy) Z' (y)] holds in M' and X'+2 = [ (try) 2 (y)] then Z'(b') holds in M' for all b' E A u B and so Z ( b ) holds in M for all b E A. Hence [ (Vy) Z (y)] = X holds in M as required. In spite of the length of the above proof, the result is fairly obvious. There is however a more interesting problem regarding the cardinality of an extension which applies only to systems that include a relation of equality. Thus, let M be a structure which contains a relation of equality. When talking of the cardinal number of M one usually assumes that M is normal. Thus two distinct individuals a, b of M are never equal in the sense of the relation E(x, y), i.e. E(u, 6) holds in M (see section 2.1. above). When M is not normal one implicitly takes the cardinal of the corresponding normal structure, or, which is the same, the cardinal number of the set of equivalence classes of the individuals of M with respect to the relation E(x, y). Let M be a structure with relation of equality such that M is of finite cardinal a, in the sense just defined. In that case, it is not difficult to see that there does not exist any elementary extension of M which is of greater cardinal, indeed there does not exist any normal proper extension of M which is elementarily equivalent with M. For the sentence

-

(3x1)

- - - (3x4 (try)[w, y ) v - v E(xa, y ) ] * *

holds in M but it does not hold in any normal proper extension M' of M whose cardinal, a', is greater than a. The situation is quite different if a is infinite. Suppose first that the number of relations defined in a does not exceed a. In that case the

3.2.1

61

THE EXTENSION OF MODELS

number of sentences in S ( M ) must be of cardinal a also. For by considering all possibilities for sentences which involve 1,2, 3, ..., n, ... atomic symbols, we find that the number of different sentences in S ( M ) cannot exceed a + a 2 + a 3 + . . . + a ~ + . . . = a + a + a + . . . = a . On the other hand, by considering the set of all sentences E (a,b,) where a is an arbitrary but fixed individual of M and b, varies over all the individuals of M which are different from (not equal to) a,we see that the cardinal of S ( M ) cannot be less than a.

-

3.2.8. THEOREM. Suppose that the number of relations in a structure M with equality E ( x , y ) does not exceed the number, a, of different individuals, which is infinite. Then there exists an elementary extension M' of M which is of cardinal a and which contains at least one element that is not equal to any element of M. Moreover, for any cardinal a' > a there exists an elementary extension M' of M which is of cardinal a. To prove the first part of the theorem, we select an individual c which is not contained in M and we then define K as the set of all sentences E(c, b,J where b, varies over all elements of M. Let H = S ( M ) U K. Then we claim that H i s consistent. Indeed, if H where contradictory then a finite subset of H would be contradictory. It follows, that for some finite subset K of K and for some finite S' c S ( M ) the set S' U K' would be contradictory. K' cannot be empty for S(M)alone is certainly consistent. Let

-

K' = {- E(c, bl), . . .,

N

E(c, b k ) } ,

k 2- 1

and let X be the conjunction of all element of S'. Then X also belongs to S ( M ) . If H is contradictory, so is

Y =X A

-

E(c, bi) A

... A

N

E(c, h)

which is the conjunction of the elements of H'. Thus we have for any arbitrary sentence Z of L, e.g. 2 = [A n A], for some relation A of order 0, that Y =I 2 is a theorem. It then follows from the third rule of 1.3.3. that Y =I 2 also is a theorem, where (for a suitable variable z which does not appear in X) N

-

Y' = ( 3 z ) [ X A E(z, bl) A

... A

N

E(z, b k ) ]

But if so then Y' also must be contradictory, and the same then applies to

Y" = X A [ ( 3 ~ ) [ - E(z, b ~A)

... A - E(z, b k ) ] ]

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[3.2.

Y” hoIds in M for this is true of X,and also of the second member of the conjunction, (32) [

N

E(z, bi) A

. .. A

-

E(z, bk)]

for the latter sentence states only that M contains an element different from bl, . . . b g , which is certainly true. Hence Y” is consistent, contrary to the conclusion reached on the assumption that H is contradictory. Hence H i s consistent and possesses a model M ’ . Moreover, a computation made at the end of section 1.5. (see Theorem 1.5.13.) shows that we may suppose that the cardinal of M’ does not exceed the number of sentences in H. But we have seen already that S ( M ) contains exactly a sentences, while it is evident that there are just a elements of K. Hence H contains a sentences, and we may suppose that M’ does not contain more than a different (unequal) individuals. On the other hand, M’ contains at least a different individuals, i.e. the elements b,. It follows that the cardinal of M’ is equal to that of M , i.e. a. At the same time, M’ contains the element c which is different from the elements of M . This proves the first part of the theorem. (From now on, whenever we introduce a quantified variable by virtue of the second or third rule of 1.3.3. it will be taken for granted that the variable so introduced did not occur previously in the sentence under consideration.) In order to prove the second part of the theorem, we choose a set of individual constants C = {c,} of cardinal a‘ such that C has no elements in common with the set of individuals of M , and we introduce the sets K1 = {,- B(c,, by) } where b, varies over a set of different elements of M as before and c, varies over all elements of C, and KZ = { w E(c,, c,)} where cg and c, vary independently over all elements of C. Then the cardinal number of K1 is a’ a = a’ and the cardinal number of K Z is a‘ * a‘ = a’. Consider the set

-

H = S ( M ) U Ki U K z

+ +

The cardinal number of H i s a a‘ a’ = a‘. We propose to show that H is consistent. If not, then (see the first part of the proof) there exist a single element X of S ( M ) and finite subsets K; and Ki of K1 and K2 respectively, such that {X}u K; U Ki is contradictory. K; consists of a finite number of sentences of the form ,- E(c,, by), and K i consists of a finite number of sentences of the form E(c,, c,). We denote the conjunction of all elements of K; u K i by Y(b1, ..., bj, c1, ..., CZ) where we have displayed all the individuals which appear in the sentence. Then N

3.2.1

63

THE EXTENSION OF MODELS

.

.

2 = X A Y(b1, . . , bj, c1, . ., CZ) is contradictory and this entails, similarly as before, that X A [ (321) . (323) Y (bl, ...,bj, z, ... Z I ) ] is contradictory. X holds in M since it is an element of S ( M ) , and ( 3 ~ .~. .) (3zj) Y (bl, . ., bj, ZZ, . ., ZZ) holds in M since it is equivalent to the existence of some b . ., b: in Mwhich satisfy Y (bt, . . .,bjYb t, . .,b *I). But the latter sentence holds in M for any choice of bT, . . ,b*, which are different from each other and different from bl, ... bj. And since the set of individuals in M is infinite, it is certainly possible to find b*,, ... b: of this description. We conclude that H is consistent and possesses a model M’. The cardinal of M’ is 2 a’ since the elements of C are all different (i.e. not equal) in M‘.On the other hand, since a’is the cardinal of H we may suppose, as before, that M’ has been chosen in such a way that its cardinal is not greater than a’.Hence M’ is of cardinal 01’. Moreover, M‘ is an elementary extension of M since it is a model of S ( M ) . This completes the proof of the theorem. In the usual algebraic structures, the number of relations is finite and includes a relation of equality. Thus, if the structure itself is infinite (with respect to E(x, y ) ) the conditions of Theorem 3.2.8. are satisfied. It is nevertheless of considerable interest to investigate the situation for the case that the number of relations exceeds the number of individuals. An examination of the arguments involved in the proof of 3.2.8. shows that if M includes a relation of equality E ( x , y ) and is infinite (with respect to E), and 01’ is any cardinal which is not smaller than the sum of the number of relations and individuals of M (i.e. the number of relations, if it exceeds the number of individuals), then there exists an elementary extension of M which is of cardinal a’. However, we may still ask, whether for the case that the number of relations of M exceeds the number of individuals, there exists a structure M’ which has the same cardinal as M and which is a non-trivial elementary extension of M in the sense that it contains at least one individual which is different from all individuals of M . In certain cases, the answer is negative, as shown by Theorem 3.2.9. below, which is due to Rabin. Let A be a countably infinite set and let a relation of equality E(x, y ) be defined in A such that E coincides with the identity. That is to say, E(a, a) holds for all a E A and E(a, b) does not hold for any two distinct elements a, b of A . Arrange A in an infinite sequence, A = {ao, al, az, . . .} and define that the relation P(az, ak) holds if and only if i < k . Thus, P is a relation of order defining an ordered set of type o.Finally suppose that to every subset B of A there is assigned a one-place relation Q B ( x )

.

T, .

..

.

.

.

64

SOME CONCEPTS AND METHODS OF MODEL THEORY

[3.2.

such that QB(u) holds if and only if a E B. All these relations together define a structure M with set of individuals A. Note that the number of these relations is 2N0. We shall make use of a result due to Sierpinski which states that there exists a collection R of countable subsets of A such that the cardinal of R is 2 N 0 , and such that the intersection of any two elements of R is finite. Thus, let B1, BZ be two elements of R, then there exists a sentence X B ~B~, which states that B1 and Bz have precisely n elements in common where n is a non-negative integer which depends on BI and Bz. A suitable formulation of X B B2 ~ is (3x1). .. ( W W y ) [ E ( x l yY ) A E h , Y ) A . . A E(xnyU) Q l (Y) A Qz ( Y ) ] where Ql and Qz are the one-place relations which correspond to Bi, and B2 respectively. The formulation has to be changed slightly if the intersection of B1 and Bz is empty.

-

3.2.9. THEOREM. Any elementary extension M' of M which contains at least one element that is different from all elements of M, possesses a cardinal which is greater than or equal to 2%O.

PROOF.Let

a be an element of M' which is not equal to any element of M. Then it is not difficult to see that P(at, a) holds for all as E A. Indeed, if P(a, at) held for some at E A, i > 0, then a would have to be equal to some ak E A, k < i, since the fact that there are just i - 1 elements of A which precede at in the ordering of A is expressible as a sentence of S ( M ) . At the same time (Vx)[P(at,x ) V P ( x , at) V E(x, at)] also is a sentence of S (M). This proves that P (at, a)holds in M'. Again, for i = 0, we cannot have E(at, a) and so P(ac, a) holds in M',as before, Now let B be any infinite subset of A and let Q B( x ) be the corresponding one-place relation. Then the following sentence belongs to S ( M ) .

('W(3~) [P( x , Y ) A Q B ( Y ) A

~(Y z) V , E ( Y , z ) ] ] ]. This sentence states that for every x there exists a y greater than x which is contained in By and which is the first element of B greater than x that is contained in B. In other words y is the first element after x which is contained in B. Since this sentence belongs to S ( M ) it holds also in M', [VZ)[QB(Z)

= ~ ( z x, ) v E(z, x) V

and so in particular, the sentence

(34 [P(a, V ) A Q B ( AA [ (VZ) [Q B (z) = p (z, a) v ~ ( za), V P (Y, z) v E (Y, 4 ] ]

3.3.1

65

THE PREFIX PROBLEM

holds in M'. Thus, there exists an individual U B in M' such that [(Z) [ Q B < Z >

[P(a, U B ) A Q B ( ~ B )A =P(z,a)VE(z,u)VP(uB,z)VE(Utr,z)]]]

and this element is unique (in the sense of the relation of equality). Now let B1 and BZ be two different elements of R and let UB = bl for B = El and UB = bz for B = Bz. We claim that bl and bz are different in M', E(b1, bz) holds in M'.Indeed, since X B ~ Qholds in M there exists an u k E A such that

-

(VY) [P( ~ k V, )

N

Q i (7) V

-

Qz

(Y)]

holds in M,and hence in M'.But P ( U k , a) and P(u, bl) and P(u, 62) all hold in M' and so bl and b2 are greater than U k in M' and, accordingly, cannot be equal. Now let A' be the set of all bj corresponding to sets El that belong to C. Then the bj are all different (not equal). Since C is of cardinal 2#0 and the mapping of C on A' is, as we have just shown, one-to-one, it follows that A' is of cardinal 2%O. Hence M' is of cardinal 2%Oat least. This proves the theorem. The above provides an important example of one of the various possibilities which arise if the conditions of Theorem 3.2.8. are not satisfied. 3.3. The Prefix Problem. Consider the class N of all wff X in prenex form in a language L of the Lower predicate calculus. We recall that such a wff X consists of two parts, its sequence of quantifiers or prefix, and the wff which follows upon the prefix and which is called the matrix of X . We now classify the sentences of N according to the nature of their prefix as follows. By a block of existential (universal) quantifiers in the prefix of a wff X of N we mean a sequence of consecutive existential (universal) quantifiers which cannot be extended to left or right (either because the next quantifier is of the other type or because the prefix begins or ends with the block in question). For example, the blocks of quantifiers in the sentence ( V 4 ( 3 4 (Vw)(W (34(34 Q (u,u7 w, x , Y,4 , where Q is the matrix, are (Vu), (h), (Vw) (Vx), (3y)(32). A wff N will be said to belong to the class Bn, n = 0,1,2, . . . if it contains not more than n blocks of quantifiers. Thus, the above-mentioned sentence belongs to En for n 2 4. Moreover, in B n we distinguish two subclasses as follows. An element X E Bn, n = 0,1, 2, .. . will be said to belong to An, if either

66

SOME CONCEPTS AND METHODS OF MODEL THEORY

r3.3.

X contains less than n blocks of quantifiers (i.e. X belongs also to B,-.I) or it contains exactly n blocks of quantifiers of which the first block consists of universal quantifiers. X E Bn will be said to belong to En, again if either X is also in B,-1 or if it contains precisely n blocks beginning with a block of existential quantifiers. Thus B , = A , = E , and, in general B, = A , u En. There are other classifications of the wff of N according to the nature of their prefix, e.g. by taking into account the number of individual quantifiers which appear in each block. The prejix problem deals with the question of discovering set-theoretic (or model-theoretic) properties which characterize the varieties of structures that are given by sentences with prefixes of a certain class. We do not try to delimit the kind of property that we have in mind by any formal definition, but we shall show that, for some classes of sentences at least, the answers that have been provided to the prefix problem are simple and striking. A sentence X will be called persistent under extension if whenever X holds in a structure M it holds also in all extensions of M . X will be called persistent under restriction if whenever X holds in a structure M it holds also in all substructures of M that contain the individuals of X . If X is persistent both under extension and under restriction if will be called invariant. X is persistent without qualification shall mean ‘Xis persistent under extension’. In concrete cases, it is usually more interesting to consider, not all structures, but only those structures, which are models of a certain set of axioms given in advance (e.g., to consider all groups, all fields). Thus, given a set of axioms K, X will be called persistent (under extension) with respect to K if whenever X holds in a model M of K it holds also in all models of K which are extensions of M. X will be called persistent under restriction with respect to K if whenever X holds in a model M of K it holds also in all substructures of M which are models of K and which contain the individuals of X. If X is persistent with respect to K both under extension and under restriction we say that Xis invariant with respect to K. We introduce similar definitions for the case that Xis replaced by a set of sentences, H . Later we shall require similar notions with respect to predicates and it will be appropriate to introduce them at this point. Let Q(x1, . ..,x,) be a predicate which is defined in a structure M , i.e. whose relations and individuals are contained in M . The n-dimensional Cartesian space Mn over M , n 2 1,is defined as the set of all n-element sequences (al, .. .,a,)

3.3.1

67

THJ3 PREFIX PROBLEM

with coordinates ag in Myi = 1, ...,n. These sequences are then called the points of Mn. We may if we wish identify M with MI. The predicate Q is said to hold at a pointP E Mn, P = (ax, . . ., an) if Q(m, ..., 0%) holds in M. Q is said to be persistent (under extension) at a point P E Mn if Q (a1, . . .an)holds in M and in all extensions of M . Q is persistent in M if it is persistent at all points of M n at which it holds. Finally, Q is persistent if it is persistent at all points at which it holds, over all structures in which it is defined. Similarly Q is persistent under restriction at a point P E Mn if Q(u1, . . ., an) holds in M and in all substructures of M which include the individuals of Q (a,. .., an). Q is persistent under restriction in M if it is persistent under restriction at all points of M at which it holds. If this is true for all structures in which Q is defined then Q is said to be persistent under restriction. If Q is persistent both under extension and under restriction, in connection with one of the three kinds of persistence just defined then Q is said to be invariant (at a particular point, in a given structure, or without any qualification). We may relativize these concepts, as before, by restricting our attention to models of a given set of axioms K. In that case, M also is supposed to be a model of K. For example, Q is persistent at a point P E M nover M if Q holds at P in M and in all extensions of M which are models of K. Q is persistent if it is persistent in all models of K. We may regard the definitions given earlier for the different types of persistence and of invariance as special cases of persistence of invariance of a predicate (with respect to a given set of axioms). It is not difficult to see that persistence in any of the above meanings is preserved under conjunction and disjunction. Thus, if two sentences X and Yare persistent under restriction with respect to a given set of axioms K then X A Y and X V Y are persistent under restriction, with respect to K , etc. If a sentence Xis not persistent (under extension) with respect to a set of sentences K , then it is not difficult to see that Xis not persistent under restriction, with respect to K. For the assumption is that there exist two models of K, M I and M2 such that M2 is an extension of M I and such that X holds in M I but not in M2, i.e. X holds in M2 and is defined in M I but does not hold in that structure. Conversely, if a sentence X is not persistent under restriction, with respect to a set of sentences K, then there exist models M1 and M Z of K , M Zan extension of M I and X defined in it such that X holds in M Zbut not in M I . Hence X holds in M I but not in M z . We conclude that X is persistent under extension with respect to a set of sentences K if and only if X is persistent under restriction with N

-

N

N

68

SOME CONCEPTS A N D METHODS OF MODEL THEORY

f3.3.

respect to K. It follows that Xis invariant with respect to K if and only if X is invariant with respect to K. A corresponding result holds for predicates. A set of sentences H is called conjunctive if for any X I and Xz in H there exists an X3 in H such that XI A XZE X3 is a theorem. H is conjuctive relative to a given set of sentencesK if for any XI, XZE Hthere exists an X3 E H such that K k X I A XZE X3. Exactly similar definitions lead to the notion of a disjunctive set, and of a set which is disjunctive relative to a given set. The discussion of the corresponding definitions for general wff requires some care. We shall say that a set of wff H is conjunctive if for any X I and XZin H there exists an X3 E H such that N

(Vy)

... (Vyn) [XiA Xz G X3]

is a theorem, where we have introduced universal quantifiers for all free variables contained in XIA XZE X3. Note, that according to this definition a set H which consists of the predicates Q ( x ) and R ( x ) may well be conjunctive, although the set J = {Q (x), R ( y ) } is not. Thus let A and B be two one place relations and let Q(x) = A(x), R ( x ) = A(x) A B ( x ) . Then (Vx) [Q ( x ) A Q ( x ) E Q ( x ) ] , (Vx) [R( x ) A R ( x ) R ( x ) ] , and ('W [Q ( 4 A R (4 E R ( x ) ] are all theorems, and so H is conjunctive. But neither ( V x ) ( V y ) [ Q ( x ) A R ( y ) Q ( x ) ] nor (Vx) (Vy) [Q ( x ) A R ( y ) R ( y ) ]is a theorem and so J is not conjunctive. - Conjunctivity relative to a given set of sentences and disjunctivity are defined correspondingly. It follows from an earlier remark that the set of all sentences H which is persistent (or, persistent under restriction) relative to a set of sentences K is conjunctive and disjunctive relative to that set. Also, if H is the set of all predicates with free variables XI, ...,X n which are persistent relative to a set K then H i s conjunctive and disjunctive. The same applies if we restrict H to a specified vocabulary. Again, the set BO= A0 = EOdefined at the beginning at this section is conjunctive and disjunctive and so are the sets A n and En. The set B1 is not conjunctive as will be shown later in this section. In order to appreciate that A n and E n are conjunctive, consider for example the class As. Typical elements of class A3 are

Xl

QI

... ( V W g ) (3x1). ..(3Xk) (Vy1) ... (Vyt) ...,wg, XI, ...,~ kJQ, , .. .,yg, ZI, . ..,~ j )

= (VWl)

(WI,

3.3.1

69

THE PREFIX PROBLEM

and

Xz

=WWl)

. . . ( V W k ) (3x1) . . . (3x1)(VZl) . . . (VZ,)

Q Z( ~ 1 ,. . ., WIG, XI, . . .,z,ZI, . . ., ~ mJQ, , . . .,y p )

whereg20, h > O , i > O , j > O , k > O , I > O , r n > O , p > O , a n d where we have displayed the free variables of X I and XZ respectively. Note that in this example, the variables of XIand XZoverlap. To find an appropriate X3, we first replace the variables wq, xq,zqin XZby sq, t,, uq and the variables y , in Xi by uq, respectively. The results are sentences X i and X i such that t. (VZl) . . . (VZj) [Xlz X i ] 1

(VYl) . . . )Vyp)[Xz3 Xi1

Note that in either sentences we have changed only the bound variables so as to avoid an overlap with the free or bound variables of the other sentence. We now define X3 by

. . . (vwg) (vsl) . . . (vspk)(3x1) . . . ( h h ) (ltl) .. . (3tl) Wdvl) . . . - - . (VUm) [QI (w1,. . ., wg, XI,. . .,xh, U1, . . .,UI, z1, .. .,zj) A QZ (~1,. . .,sk, 11, . . ., tt, ~ 1 ,. . .,Urn, .YI, . . .,~ p ) ] . - . (Vy’yp) [XI A XZ X3I Wzl) - - (Vzj)WY~)

X3 = (vwl) ( b t ) (VUl)

Then

is a theorem of the Lower predicate calculus, the interchange in the order of the quantifiers being permitted because of the elimination of any undesirable overlap of variables between X I and XZ.If by chance some free variables appear in both X I and XZthey have to be left unchanged. This example will serve to illustrate the general procedure. It is evident that if we restrict the classes An and En to the sentences contained in them, the resulting sets are still conjunctive and disjunctive. The same applies if we consider only wff or sentences which are defined in terms of a specified vocabulary. A sentence which belongs to the class A0 = Eo will be called quant$erfree. It is not difficult to verify (by means of an argument which has been applied previously to particular free sentences) that a quantifier-free sentence is invariant. A sentence X which belongs to the class A1 will be called a universal sentence,

X = (Vyi) ... (Vyd Q (yi,

... YIC)

where Q is the matrix and (hence) does not contain any further quantifiers. Suppose that X holds in a structure M and is defined in a substructure M‘

70

[3.3.

SOME CONCEPTS A N D METHODS OF MODEL THEORY

of M. That is to say, the individuals which occur in X are contained in M'. We claim that X holds in M'. Indeed, let al, . . .,Uk be any set of individuals which belong to M ' . Then Q(a1, . . ., a g ) holds in M and, being quantifier-free, holds also in M'. It follows that X holds in M'. In other words, Xis persistent under restriction. It is now natural to ask whether the converse is true, i.e. whether any sentence X which is persistent under restriction is by necessity universal. The answer to this is trivially false since, for example, all theorems are persistent under restriction. Clearly, all we can expect is that X be equivalent to a universal sentence, and this will indeed be shown to be true. In order to derive this result, we shall consider immediately a somewhat more general situation. Let K be a set of sentences (which will be kept constant throughout this discussion) and let X and Y be two sentences which are defined in K and hence, in all models of K. We wish to investigate under what conditions it is true that whenever X holds in a model M of K, Y holds in all substructures M' of M which are models of K. It is not difficult to see that this will be the case if there exists a universal sentence Z which is defined in K such that K t X 3 Z and K t Z 3 Y. Indeed, since X and X 3 Z hold in M , 2 holds in Myhence, as a universal sentence, 2 holds in M' and since Z 3 Y also holds in M' this shows that Y holds in M'. We shall now establish a theorem which shows that the converse is also true. 3.3.1. THEOREM. Let the sentences X and Y be defined in K such that whenever X holds in a model M of K, Y holds in all substructures of M which are models of K. Then there exists a universal sentence 2 which is defined in K such that K t X f> Z and K t Z 3 Y, i.e. the sentences X 3 2 and Z 3 Yare deducible from K.

PROOF.Let H be the set of all universal sentences W which are defined in K such that K t X 3 W . It will be seen that H is conjunctive and disjunctive. Consider the set J = K U H u { Y } .Suppose that J i s consistent and let M' be a model of J. It then follows from the assumptions of the theorem that there does not exist any extension M of M' which is a model of K such that X holds in M (otherwise Y would hold in M'). Hence, K u D' t X where D' is the diagram of M'. We conclude as usual that there exists a finite subset D* of D' such that K u D* !- X. Let Q (al, ...,a g ) be the conjunction of all elements of D*, where we have displayed the individuals which are not contained in K(if any). We may always suppose that D* is non-empty, if necessary by introducing an N

N

N

3.3.1

THE PREFIX PROBLEM

-

.

-

71

arbitrary element of D'. Then K t Q(a1, . ., ar) 3 X and so K t X 1 Q (al, . . ., ak:). Since al, .. ., ak: are not contained in K, and hence are not contained in X , we conclude further that

[ (VZI). ..( V Z [~ Q (zi, ...,zk:)]] where the variables z1, . . ., zk did not occur previously in Q. It follows that the sentence (Vz1) ... (Vzk)[- Q(z1, ...,zk:)] = W, say, is an K tX

2

-

.

element of H, and hence holds in M'. But W is equivalent to (321) , . (3zk) Q (ZI, .,zr) and this sentence holds in M' since Q (a,..,a#) is a conjunction of elements of the diagram of M' and hence holds in M'. Clearly W and W cannot hoth hold in M' showing that J is contradictory. It follows that there exists a finite subset H* of H such that K u H* u {- Y} is contradictory. We may suppose that H* is not empty. Indeed, for any relation A (XI, . . .,x,) which belongs to the vocabulary of K, n 2 0, the sentence (Vxl) . (Vx,) [ A(XI, . x), V A (XI, . .,x,)] belongs to H, and we may introduce such sentences into H* at our pleasure. Let 2 be a universal sentence which is equivalent to the conjunction of the elements of H*, then Z E H and K k [Z A Y].This entails K t 2 =) Y, i.e. 2 satisfies the conclusion of 3.3.1., the proof is complete. Suppose in particular that X coincides with Y.Then K t X G 2 and so we obtain the result

..

.

-

..

..

-

.

- -

3.3.2. THEOREM. Let X be a sentence which is defined in a set of sentences K such that whenever X holds in a model M of K it holds also in all substructures of M which are models of K (i.e. such that X is persistent under restriction relative to K). Then there exists a universal sentence Z which is defined in K such that K t- X 2. As they stand, these theorems become empty if Kis empty, since in that case there are no sentences X and Y as described. Nevertheless we may use a simple artifice in order to arrive at 3.3.3. THEOREM. Let Xand Y be two sentences such that whenever Xholds in a structure M and Y is defined in a substructure M' of M then Y holds in M'. Then there exists a universal sentence Z whose relations and individuals are contained in either X or Y such that X t 2 and 2 t Y. In order to reduce 3.3.3. to 3.3.1., we introduce a set of sentences K which consists of a finite number of theorems containing all relations and individuals which occur in X or Y and no others. Thus, if a relation

72

[3.3.

SOME CONCEPTS AND METHODS OF MODEL THEORY

A (x, .. .,x ) is contained in X or Y, n 2 0, then we may include in K the sentence (Vxl) .. . @xN)[ A(XI, ...,xn) V A (XI, . ..,xN).For any individual in X or Y, b say, we may include some A (b, . . ,,b) V A (b . . . b). With this definition of K, the assumptions of Theorem 3.3.1. are satisfied. Indeed, any structure M is a model of K provided only that all the relations and individuals which occur in X or Yare contained in K. Thus, if X holds in a model M of K, Y holds in all substructures of M which are models of K. It follows that for some universal Z which is defined in K, i.e. whose relations or constants are contained in either X or Y, X 3 Z and Z 3 Yare theorems, and this is the assertion. In particular,

-

N

3.3.4. THEOREM. Let X be a sentence which is persistent under restriction. Then there exists a universal sentence 2 whose relations and individuals are contained in X such that Z is equivalent to X . A sentence X which belongs to El is said to be an existential sentence. An existential sentence is persistent. Theorems 3.3.1.-3.3.4. give rise to corresponding results concerning existential sentences of which we mention the first two.

3.3.5. THEOREM. Let the sentences X and Y be defined in the set of sentences K such that whenever X holds in a model M of K, Y holds in all extensions of M which are models of K. Then there exists an existential sentence Z which is defined in K such that K t X 3 Z and K t Z 3 Y. Indeed, the conditions of 3.3.1. are satisfied, for the same K, if we replace X and Y in that theorem by the present Y and Xrespectively. Accordingly, there exists a universal sentence 2' = (Vy,) . . . (Vyk) Q ( y l , . .. yk) which is defined in K such that K t Y 13 Z' and K t 2' u - X I t f o l l o w s t h a t K t - Z ' = ) Y a n d K t X D - - ' . B u t - Z'is equivalentto Z = ( 3 ~ 1.).. ( 3 p ) [ Q ( y l , . . ,,yx)] and this is the required existential sentence. N

-

-

N

3.3.6. THEOREM. Let X be a sentence which is defined in a set of sentences K and persistent relative to it. Then there exists an existential sentence Z which is defined in K such that K t X 2. The proof is by reduction to 3.3.2. Some of the interrelations which are assumed in the above theorems, e.g., if X holds in a structure M then Y holds in all extensions of M in which it is defined, may be regarded as special kinds of consequence relations.

3.3.1

73

THE PREFIX PROBLEM

We may ask whether this consequence relation is subject to a principle of localization or some other principle of this kind. The answer is provided by

3.3.7. THEOREM, Let H be a set of sentences, X a sentence which holds in every extension M' of a model M of H such that Xis defined in M'. Then there exists a finite subset H' of H such that for any extension M' of a model M of H', X holds in M' provided only Xis defined in M'. PROOF.We relativize H with respect to a one-place relation R which is not contained in either K or X (compare section 3.1. above). Let the resulting set be HR. Then X holds in all model of HR in which it is defined. It fdlow that Xis deducible from a finite subset H I of HR. Let H2 be the finite subset of H whose transforms are included in HI. For any individual u such that R(a) E H I we add to HZa sentence of H which contains a. Let the result be H . Then X holds in all extensions M' of models of H' such that Xis defined in M', as required by 3.3.7. Similarly,

3.3.8. THEOREM. Suppose that the sentence X holds in all substructures M' of the models M of a given set of sentences H such that Xis defined in M'. Then there exists a finite subset H' of H such that X holds in all substructures M' of models of H' such that X is defined in M'.

-

-

For the proof, we now relativize X calling the result XR. Then H U XR) is contradictory, otherwise X R would hold in a model M of H which contains at least one element a that satisfies R ( x ) (i.e. such that R(u) holds in M). Hence X would hold in the substructure M' of M which is obtained by restricting the relations of M to the elements that satisfy R.This is contrary to the assumption of the theorem and shows that H u { [ ( 3 x ) R (x)] A X R } is contradictory. It follows that there exists a finite subset H' of H such that H' 1 [ ( 3 x ) R (x)] =I XR. H' satisfies the conditions of the conclusion of 3.3.8. 3.3.7. and 3.3.8. can again be relativized with respect to a given set of sentences K. For example, corresponding to 3.3.8. we have, assuming for simplicity that Xis defined in K,

{ [ (3x) R (x)] A

-

3.3.9. THEOREM. Suppose that the sentence X holds in all models of a given set of sentences K which are substructures of models of K u H. Then there exists a finite subset H of H such that X holds in all models of K which are substructures of models of K U H'.

74

SOME CONCEPTS AND METHODS OF MODEL THEORY

r3.3.

It will be sufficient to give the proof in outline. We note that K U H U KR u {- X R ) is contradictory, similarly as before and we deduce that K u H u KR u {- X R } is contradictory for some finite subset H of H. This is the required set. We are now in a position to prove the counterparts of Theorems 3.3.2. and 3.3.6. for sets of sentences.

u

3.3.10. THEOREM.Let H be a set of sentences which are defined in a given set of sentences K such that H i s persistent under restriction relative to K. That is to say whenever a model M of K is also a model of H, then all substructures of M which are models of K are also models of H.Then there exists a set J of sentences which are defined in K and belong to A1 such that J is equivalent to H with respect to K, i.e. K u H I- J and K u JkH.

PROOF.Let J be the set of all universal sentences Y which are defined in K such that K U H t- Y. Then K U H t J. Let X E H then we have to show that K u J t X. Since X belongs to H, it holds in any substructure M'of a model of K u Hand such that M'is itself a model of K. It follows, by 3.3.9., that there exists a finite subset H' of H such that X holds in all models of K which are substructures of models of K u H'. Let Z be the conjunction of all elements of H .Then by 3.3.1. there exists a universal sentence W which is defined in K such that K t Z 3 W,K t W 3 X . But Z is a conjunction of elements of H, and so K U H t W, W belongs to J. Hence K t W 3 X entails K U J t X, which was the result to be proved. There is a corresponding result for persistence under extension. In that case, J is a set of existential sentences. Several of the above results can be adapted to predicates instead of sentences. For example 3.3.11. THEOREM. Let Q(x1 . . ., xn), n 2 1, be a predicate which is defined in a set of sentences K and persistent relative to it. Thus Q is persistent at all points in all spaces Mnover models M of K. Then there exists an existential predicate P(x1, . . ., xn) (i.e. a predicate which belongs to El) such that P(x1, .. ., xn) is defined in K and K t ('4x1)

. . . ( V X ~ )[Q (xi, . . .,x,)

P (XI, . . .,x.)].

PROOF.Excepting trivial cases, we may suppose that K contains at least one relation of positive order A(x1, . . ., xk). Now let al, . . .,an be a set

3.4.1

75

OBSTRUCTIONS TO ARITHMETICAL EXTENSION

of distinct individuals which do not appear in K and add to K a single theorem Ywhich contains al, . . .,an.Such a theorem is Y = [A (a,...,al) V A(a1, .. . a)]A [A(az, . ., U Z ) V A ( a z , . . U Z ) ] A .. . A [ A (an, . . ., an) V A (an . . an)]. Let K' = K u { Y } then the sentence X = Q (al, . . .,an)is defined in K' and is persistent relative to K, and hence, relative to K'. It follows by 3.3.6. that there exists an existentialsentence Z which is defined in K', 2 = P(a1, ...,an)say - where we have displayed only the individuals which are not included in K - such that K I- Q (al, . . .,an) P ( a , . . . an), or N

N

.

K I- Y =I [ Q ( a l ,

.

.

N

.. ., an)

P(a1,

. . ., an)].

But Y is a theorem and so

K I- Q(a1, . ..,a,)

Again, al,

G P(a1,

...,a n ) .

. ..,an do not appear in K and so finally

K I - ( x l ) ... ( x n ) [ Q ( x l ,..., x n ) - P ( x l ,

. e m ,

-

xn)]

This proves 3.3.11. It will be seen that the reduction of 3.3.11. to 3.3.6. involves only technicalities. This concludes our discussion of sentences or predicates that belong to the classes A1 and El. It is equally possible to find a satisfactory answer to the problem as regards sentences of the class Az. This will be done in the next section. 3.4. Obstructions to Elementary Extension. Let M be any structure and let M' be an extension of M. We say that M' obstructs M if there does not exist any extension of M' which is an elementary extension of M .

3.4.1. THEOREM.In order that a structure M be obstructed by one of its extensions, M', it is necessary and sufficient that there exist a sentence X E A1 such that X holds in M but not in M'.

PROOF.Let H = S ( M ) u D(M') where we recall that S ( M ) is the set

of all sentences which are defined and hold in M and D ( M ' ) is the diagram of M'. If H is consistent then it possesses a model, M * M* is an extension of M' since it is a model of D(M'), and it is an arithmetical extension of M since it satisfies the sentences of S ( M ) . But no such structure exists, by the assumption of the theorem and so H must be contradictory. Thus there exists a finite subset J of D ( M ) such that S ( M ) u J is contradictory. Let Y(a1, . . .,an) be the conjunction of all

-

76

[3.4.

SOME CONCEPTS AND METHODS OF MODEL THEORY

elements of J, where we have displayed the individuals which occur in Y and which do not belong to M, n 2 0.Then S ( M ) F Y(m, . ,an) and, since the al, ..., a, do not belong to M, S ( M ) F ( " X I )... (Vx,) [ Y(x1, . . .,x.)]. The sentence X = (Vxl) .. . (Vx,) [ Y(x1, . . . x,)] is defined and holds in M since it is deducible from S ( M ) and so it actually belongs to S ( M ) . On the other hand, since Y ( m , . . .,an) is a conjunction of elements of the diagram of M', it holds in M' and so therefore does the sentence (3x1) . . . ( 3 ~Y ~( x 1) , . . ., x,). But this sentence is equivalent to X and so the condition of the theorem is necessary. To prove sufficiency, let X be a sentence as described in 3.4.1., e.g. X = (Vxl ). . . (Vx,) Y ( X I , . . ., x,) where Y does not contain any further quantifiers. Then X and hence, the sentence 2 = (3x1) . . (3xn)[- Y(x1, .. ., x,)] holds in M', by assumption. But 2 is an existential sentence and so it holds in all extensions of M'. On the other hand 2 is defined but does not hold in M since X holds in that structure. It follows that there does not exist any extension of M' which is an elementary extension of M . This proves 3.4.1. Let it4 be a structure and let K be a set of sentences. K may include relations that are not contained in M. When saying that a model M' of K' is an extension of M , we shall ignore such relations. In this sense, we shall say that K' obstructs M if every model M' of K' which is an extension of M, obstructs M .

..

N

-

N

-

-

.

3.4.2. THEOREM. In order that the set of sentences K' obstruct the structure M , it is necessary and sufficient that there exist a sentence X B AI, such that Xis defined and holds in M and Xis deducible from K'. N

PROOF.Suppose that X is a sentence of the kind described in 3.4.2., and let M be an extension of M and a model of K'.Then Xis defined N

in M' since it is defined in M and it holds in M' since it is deducible from K'. On the other hand, X holds in M. Hence, no extension of M' can be an elementary extension of M, by 3.4.1. Thus the condition is sufficient. It is also necessary, for suppose that it is not satisfied and let H be the set of all sentences of class A1 which are defined and hold in M. Then K' U H must be consistent. For if K' is inconsistent, then the condition of 3.4.2. is satisfied, trivially. While if K is consistent but K' U H i s inconsistent then Y = [ X I A .. . A X,] is deducible from K for certain elements X I , . . .,X, of H, n 2 1. But A1 is conjunctive and so for some X that belongs to H, Z = X is deducible from K'. This shows that the condition of the theorem is satisfied, contrary to N

N

3.4.1

77

OBSTRUCTIONS TO ARITHMETICAL EXTENSION

assumption. It follows that K'

u

H is consistent and possesses a model

M'. M' is an extension of M since it ( M ' ) is a model of H and H contains

the diagram of M. Suppose that M' obstructs M. Then, by theorem 3.4.1., there exists a sentence X P A1 which holds in M but not in M'. Since X holds in M it belongs to H and so, must hold also in M'. This yields a contradiction and proves that the condition of the theorem is also necessary. A set of sentences K will be said to obstruct the set of sentences K if there exists a model M of K such that K' obstructs M.

3.4.3. THEOREM. In order that a set of sentences K' obstruct a set of

-

sentences K it is necessary and sufficient that there exist a sentence Y EA2, such that Y is defined in K' and deducible from it while Y is consistent with K (i.e. such that K U {- Y } is consistent).

PROOF.Suppose that the condition of the theorem is satisfied, and let M be a model of K u {- Y > .Since Y belongs to A2 it must be of the

form

Y

= (VXl)

... (VXk)($l)

. . (3ym) z(Xl, . . . *

y

k 2 0,m 2 0

X k , y1,

-

. . ., y m ) ,

where 2 does not contain any quantifiers. Y holds in M y so there exist individuals al, . . .,ak in M such that the sentence

x = (VYl) ... (Vym)[-

Z(a1,

.- .,

m

y

y1,

...,r m ) ]

holds in M. But Y is deducible from K' and so the same applies to the ~ (a, ) . . ., ak, yl, . . . , y m ) and hence to sentence (3yl) . . . ( 3 ~ Z (Vyl) . . . (Vym)[ Z(al, . . ,,ak, y l , . . .,ym)] which is X . This shows that the conditions of 3.4.2. are satisfied, K obstructs M , the condition of the theorem 3.4.3. is sufficient. In order to see that the condition of the theorem is also necessary, suppose that K' obstructs the model M of the set of sentences K. Then there exists a sentence X P AI,

-

N

x= VYl) ... (Vym) Z(a1, - -

*,

ak,yl,

.

*

.,ym)

where Z is free of quantifiers such that X holds in M and Xis deducible from K . In this connection, we have displayed the individuals al, . .., U K which are contained in X but not in K', if any. Applying the second rule of 1.3.3., we find that the sentence N

Y = (VXl)

. .. ( V X k ) (3y1) ... (3Ym) [

-

Z(X1,

*

. .,Xk, Yl, ... rm)] Y

78

13.4.

SOME CONCEPTS A N D METHODS OF MODEL THEORY

is deducible from K'. On the other hand, the sentence (3x1)

.. .

( 3 X k ) (Vy1)

..

*

(Vym) z (x1,

. . .,Xk, y1, . .,y m ) *

-

holds in M (since X holds in M). This sentence is equivalent to Y which therefore holds in M. It follows that Y is consistent with K, since the elements of that set all hold in M. This concludes the proof of 3.4.3. Now let X be any sentence of class A2, and let {Mr} be a monotonic set of models of X. Then it is not difficult to see that X holds in U { M y } = M. V Indeed, suppose that X is given by N

. . .Wxn)(3y1) . . .(3ym) Z ( X 1 , . . .,Xn,y1, . . * ,y m ) where Z does not contain any further quantifiers. The corresponding open form sentence is 3.4.4. X = Wx1)

3.4.5.

Z(x1,

. . .,x n , ~ l ( x 1 ., - ., xn), - .., qm(x1, - .,,xn))

and we have to show that there are functions of n variables which realize the functors pg(XI, ...,xn) such that

z (a1, . ..,an, p (a1, . .,an), . pm (a1, - . . an) holds for all al, ...,an in M. Now for every sequence (a,...,a,) * *Y

Y

in M we can find an Mv which contains all elements of the sequence, although the choice of Mv is not general unique. We select the functional values of the p~in M for xg = at, i = 1, . . . n as any possible set of values of the p in M,,. Such values exist since M,, satisfies X. This proves our assertion. A sentence X i s called o-persistent if whenever X holds in all elements of an increasing sequence of structures ( M n } , n = 1,2,3, . . ., then X holds also in (Ma}. (Mn) is increasing if Mn C Mk for n < k. We haveproved n

u

3.4.6. THEOREM. Any sentence X

E

A Zis o-persistent.

It has been shown by EoS and Suszko that the converse of 3.4.6. is also true (Theorem 3.4.9. below). We consider a somewhat more general situation, as follows. A sentence X is said to be a-persistent with respect to a given set of sentences KOif the following condition is satisfied. Let (Mn},n = 1,2, . .. be any increasing sequence of structures, and let M = U {Mn}. Then if MI, M2,

n

... are models of KOand of X and M is a model of KO,M is also

a model of X.

3.4.1

79

OBSTRUCTIONS TO ARITHMETICAL EXTENSION

We introduce a similar definition for a set of sentences H in place of the single sentence X. 3.4.7. THEOREM. If a set of sentences H is a-persistent with respect to a set of sentences KOthen there exists a set of H' of sentences of class Az such that H i s equivalent to H' with respect to KO.

PROOF. Let H' be the set of sentences of class A2 which are defined in, and deducible from, H U KO.We propose to show that H is deducible from H' u KO.This will prove the theorem. Suppose on the contrary that there exists an X E H which is not deducible from H u KO.Then the set K = H' u KO U { X } is consistent, and possesses a model, M I . Suppose that H u KOdoes not obstruct K. Then there exists an extension M2 of M I which is a model of H u KO such that some extension M3 of M2 is an elementary extension of M I . M3 is a model of K so there exist extensions M4 of M3 and M5 of M4 such that M4 is a model of H U KOand M5 is an elementary extension of M3. Continuing in this way, we obtain an increasing sequence of structures {Mn} such that M 2 k is a model of H U KOand M2k+l is an elementary extension of MZk-1, k = 1,2, ... Then

-

kf =

(Mn)

n

-

=

u k

(kf2k)

=

u

{M2k+l)

k

is an elementary extension of each M2k+l (as proved in section 3.2.) and, accordingly satisfies X and KO. But M is also a model of H since H is a-persistent with respect to KOand the MZk satisfy H and KO,while M satisfies KO. Hence, M satisfies X . This is impossible and shows that H u KOobstructs K. Hence, by 3.4.4. there exists a sentence Y of class A2 which is defined in, and deducible from, H u KO,while Y is consistent with K. But Y EH ,by the definition of H', and H' c K. Accordingly, Y belongs to K, a contradiction. It follows that H' U KO t X for all X E H, as required. This completes the proof of 3.4.7. Now suppose that the set H in 3.4.7. contains a single sentence, X. Then the theorem shows that there exists a set of sentences, H' c A2 such that { X } u KO !- H' and H U KO !- X . Hence, there exists a finite subset H*of H' such that { X } U KO t H * and H * u KO t X. LetX* be a sentence of class A Z which is equivalent to the conjunction of the elements of H*. Such a sentence exists since A Z is conjunctive. Then { X } u KO t X * and { X * } U KO t X and so KO t X z X * . Hence

-

3.4.8. THEOREM. If a sentence X is a-persistent with respect to a set

80

SOME CONCEPTS A N D METHODS OF MODEL THEORY

p.5.

of sentences& then there exists a sentence X * E AZsuch that KO k X E X*. In particular, if KOis empty, we obtain 3.4.9. THEOREM. If a sentence X is a-persistent then there exists a

sentence X *

E

A2 such that Xis equivalent to X*.

3.5. Convex Systems. We shall now consider an important class of sets of axioms which are characterized by the set-theoretic properties of the varieties of structures which belong to them. We suppose throughout this section that a relation of equality is included in the sets of axioms and structures under consideration. Moreover, unless the matter is raised specifically we shall assume that the relation of inclusion between two structures is always normal (compare section 2.1.). That is to say, if M c M' and a and b are two individuals of M' such that E(a, b) holds in M' and a belongs to M then b also belongs to M . A fundamental property common to all the usual algebraic concepts is, roughly speaking, that the intersection of two models, or indeed of any number of models, is again a model. Accordingly, we introduce the following definition. A set of sentences K which includes a relation of equality E is called convex if for any two models M I and M2 of K that are included in a model M of K the intersection M I n MZ is again a model of K provided it is not empty. By the intersection of M I and M2 regarded as a structure we mean the restriction of M to the intersection of M I and M Z(provided that set is not empty). It is not difficult to check that the sets introduced earlier for groups, rings, and fields are all convex. It is important to bear in mind the condition that M is a model of K and a normal extension of M I and Ma. For example, if we take K = KG, a set of axions for the notion of a group, and drop either of these conditions then the intersection M I n M Zmay well be non-empty without constituting a group. A natural example of a set which is not convex is provided by the concept of a densely ordered set with different first and last elements. 3.5.1. THEOREM. Let { M y }be a monotonic set of models of a convex set of axioms K. Then A4 = U { M y }is a model of K. V

PROOF.Let D be the diagram of M. Then the diagrams D,of the structures M y are subsets of D. Consider the set of sentences K u D. This set is consistent. For if D' is any finite subset of D then there exists an M , such

3.5.1

81

CONVEX SYSTJlMS

that D' c D,. But M y satisfies K, and so K U Dv is consistent, and the same applies a fortiori to K U D' which is a subset of K U Dy.It follows that K u D is consistent and possesses a model M*. M * contains A4 as a substructure which is not necessarily normal. However, this situation can be corrected without difficulty by replacing any individual a of M * which is outside M but equal to an element b of M simply by that b. Now let R ( x ) be a one-place relation that is not contained in M. For any a E M * , R (a) is defined to hold precisely when a is in M. We call the structure which is obtained from M * in this way, MR.We propose to show that M is a model of K. Let D* be the diagram of M * and let K R be the relativized transform of K with respect to R (see section 3.1. above). Consider the set

H = D* U K U K R U {R(uA), R(bJ, [ W'dx) (VY) [JV, Y ) A R (4 R ( Y ) ]I N

= I

where ax varies over the elements of M and bp varies over the elements of M * - M . We claim that H i s consistent. Otherwise, there exist elements al, . ., at in M and bl, .., b , in M* - M such that the set

.

.

...,

H ' = D' U K U KR U {R(ai), R(uz), R (bl), * * * Y R (bm),( V X ) WY) [ W Y Y ) A R (4 = R ( Y ) ] I N

is contradictory. Now, there exists an M y which contains all individuals al, .., at. Consider the structure M' which is obtained from M* by introducing the relation R into M* in such a way that R(a) holds in M' if and only if a is in M . Then R(al), . . ., R(az), holds in M' while R (bl), . . ., R (bm)does not hold in it. Moreover, M * is a model of D* and K, and is a normal extension of My. It follows that H' holds in M', be a model of H. Then W is a H is consistent, H is consistent. Let model of Kand an extension of M*. By a suitable choice of the individuals is a normal extension of M*. On of A?, we may ensure, as before, that the other hand, W is a normal extension of the structure W Rwhich is the restriction of the relations of i@ to the set of elements for which R ( x ) holds in A?. If we disregard the relation R in i @we ~ obtain a model &TO of K which is an extension of M . Hence the intersection M * n must be a model of K and an extension of M. But this intersection actually coincides with M since all elements of M* which are not in M satisfy R ( x ) in i@ and therefore cannot belong to W R and to Hence M * n i@o = M is a model of K, proving 3.5.1. The theorem shows that a convex set of axioms is o-persistent. This

.

a,,

-

no.

82

13.5.

SOME CONCEPTS A N D METHODS OF MODEL THEORY

implies, by 3.4.9. that it is equivalent to a set of sentences of class As. However, the converse is not generally true. Thus, if we add to the axioms for a commutative field KCFan axiom which states that the polynomial x3 - 2 possesses a root, then the resulting set is a-persistent but not convex. 3.5.2. THEOREM. Let { M y }be a set of models of a convex set of axioms K such that all M , are contained in a joint extension M which is a model of K. Suppose that { M y }= M' is not empty. Then M' is a model of K. V

The intersection (I{ M y }regarded as a structure is the restriction of M to

n {My} regarded as a set. V

V

PROOF,Let D be the diagram of M and, for any My, let D, c D be the diagram of Mv. Introduce a new one-place relation R ( x ) and consider the following set of sentences H = D U K U KR U { R @A),

-

R ( b d ,(VX)(VY) [E(x,Y ) A R (4 3 R W] }

where aA varies over all elements of M' and bp varies over all elements of M - M'. If H is not consistent then there exist elements al, . . ., a2 in M' and bl, . . .,bm in M - M' such that the following set, H is contradictory

H =D U K But M' =

.

-

KR U (R(uI), . ., R(uz), R(bl), (W (VY) v) A R (XI = R ( Y ) ] -

U

[m,

n {My},so there exists a particular

.

...,

-

R(bm)

belonging to that set

such that M f iincludes al, . . .,al but excludes bl, . .,bm. It follows that if we introduce a relation R into M such that R (a) holds if and only if a belongs to M f ithen the resulting structure is a model of H'. We conclude that H is consistent and possesses a model M *. Now consider the intersection M A n M*, where MAis an arbitrary element of the set {My} and M*, is obtained by restricting M* to the set of elements that satisfy R ( x ) and by neglecting the relation R in the resulting structure. Then M * may be supposed to be a normal extension of M and hence of MA, as before, and M * is a normal extension of M*,, since it satisfies V(x)(Vy) [ E ( x ,y ) A R ( x ) 1 R ( y ) ] . Hence M A n M*, is a model of K. We claim that this structure coincides with M'. Indeed, by the definition of H, M c M A n M*,. On the other hand, if a E M A does not belong to M' = { M y }then R (a)holds in M * and so a does not belong to M*,. V

V

-

3.5.1

CONVEX SYSTEMS

83

Hence M’ = M A n M*, is a model of K, as required. This completes the proof of 3.5.2. A set K is called strongly convex if it is convex and if the intersection of any set of models of K which are contained jointly in a model of K,is not empty. As a simple example of a set K which is convex but not strongly convex we consider the set K which consists of the usual axioms of equivalence for the relation E ( x , y) (2.1 .l.above). Clearly any substructure of a model of K satisfies K, but the intersection of two such structures may well be empty. On the other hand if K contains at least one individual a and is convex then it must be strongly convex, since all models of K must contain a. Let K be strongly convex. Any model M Oof K which is such that no proper substructure M‘ of MO(M’ # Mo) is a model of K will be called a core model of K. Let M be any model of K. Then M contains a unique core model Mo, the intersection of all substructures of M which are models of K. MOis called the core of M. Let K = KCF,a set of axioms for the notion of a commutative field, then the core models of K are precisely the prime fields of the various characteristics p = 0,2, 3, 5,. , . Let K be convex, M a model of K and S a non-empty set of individuals of M . Then the intersection of all substructures of M which contain S and are models of K is again a model of K. It will be said to be generated by S. Varying this situation we consider a substructure M’ of M which is a model of K and a set S of individuals of M which do not belong to M‘. Then the intersection M’(S) of all substructures of M which are models of K and contain both M’ and S is said to have been obtained by the adjunction of S to M. If S contains a single element only, S = {a}, then we write M’ (a) instead of M ( S ) and we say that M’ (a) is a simple extension of M’. In algebraic field theory and in its extensions and related theories one tries to gain an insight into all possible extensions of a model of the set of axioms under consideration by first studying the simple extensions of the given model. All other models are then obtained by the successive adjunction of single elements. The same procedure can be applied to arbitrary convex systems as is shown by the following theorem. 3.5.8. THEOREM. Let M be a model of a convex set of axioms K and let M’

be a substructure of M which is a model of K , M’ c M. Then there exists a well-ordered sequence { M y }of models of K such that MO= M’, if v is not a limit number then M vis a simple extension of MY-1; if v is a limit

84

SOME CONCEPTS AND METHODS OF MODEL THEORY

[3.6.

limit number then, Mi is defined as the union of all M p for p < v, and MV is a simple extension of M i ; finally, M is the union of all M7. In order to show that there exists a sequence {Mv) as described, we select a choice function f ( M * ) which assigns to every substructure M * of M which is different from M and which is a model of K, an element of M which does not belong to M * . The transfinite sequence (My} is now defined by transfinite induction. We first define MO = M'. Suppose that M y has been defined for all ordinals v < p. If p is not a limit number and M A is a proper substructure of M (i.e. MA # M ) where 3, is the predecessor of p then we put M p = MA MA)). If M A = M , we put MP = M . If p is a limit number and { M y }= M : is a proper

n

V
substructure of M , we put Mh = M r ( f ( M * J ) . If M* = M then we put Mh = M . By this definition all structures Mh and M : (where MZ is defined only for limit ordinals p) are models of K. From some ordinal rl onward all Mh are equal to M where the cardinal of Iz cannot exceed the cardinal of M . In other words the procedure terminates as soon as MA= M . Thus, all extensions of M' which are models of K can be obtained by the successive adjunction of single elements. The notion of convexity will be applied later, in Chapter VI, in order to elucidate certain standard concepts of Algebra.

3.6. Model-consistency. Let K be a consistent set of sentences. The sentence X will be said to be model-consistent with K if for every model of K, the set K U D ( M ) U { X } is consistent, where D ( M ) is the diagram of M, as usual. Thus, X is model-consistent with K if and only if for every model M of K there exists a structure M' which is an extension of M and a model of K such that X holds in M'. Similarly,the set of sentences H is model-consistent with K if K U D ( M ) u H is consistent for all models M of K.

3.6.1. THEOREM. Let K be a consistent set of sentences which belong to the class A2 and let X be a sentence of that class, where Z is an existential predicate. In order that X be model-consistent with K i t is necessary and sufficient that for every model M of K and for every set al, .. ., a,, of elements of M , there exists an extension M' of M which is a model of K such that the sentence Z(a1, .. ., an) holds in M'.

3.6.1

85

MODEL-CONSISTENCY

PROOF.The condition is necessary. For if the model M of K contains elements al, . . ., a n and if X is model-consistent with K then there exists an extension M' of M which is a model of K such that X holds in M'. But X

3

Z(a1,

.. .,a,)

is a theorem and so Z(a1, . . .,a,) also holds in M'. To show that the condition is also sufficient, suppose that it is satisfied, for given K and X , and consider any model M of K. Let S be the set of all sentences Z(a1, . . ., an) where al, . . .,an is any array of n individuals of M , and let D be the diagram of M . We claim that K U D U S' is consistent for any finite subset S' of S. Indeed, let S' = (21,. . ., Zm} where 2, = ~ ( a i.,..,a:), j = 1,

..., m

and where the a{ are individuals of M. Then K U D U { Z l } is consistent according to the condition of the theorem. Let M(1) be a model of K u D u { Z l } and let D(l) be the diagram of M ( 9 . Then the condition of the theorem ensures that K U D(1) U ( 2 2 ) is consistent and possesses a model M(2) with diagram D(Z). M(2) satisfies also Z1, since 2 1 is existential. Similarly, the model K u D(2) U ( 2 3 ) is consistent, and possesses a model M ( 3 )which satisfies Z l , Z z , Z3. Continuing in this way, we finally obtain a model M(mt of K which is an extension of M and which satisfies (21, . ..,Z,} = S'. Hence K U D U S' is consistent. Since this is true for any finite subset S' of S, K u D U S is consistent. Let M I be a model of K u D u S, with diagram D1, and let SIbe the set of all sentences Z(m, . . ., a,) with al, . .., a n in M I . Then K U Di u S1 is consistent, similarly as before, and possesses a model M Zwith diagram Dz.Let S2 be the set of all sentences Z (a,. ..,a,) with ar, . . ,a n in Mz,then K u D Z u S2 is consistent and possesses a model M3. Continuing in this way, we obtain a sequence of models of K,

.

.

T = { M I ,Mz,. ., Mk...}

such that Mk c Ma for k

< I.

Let M* =

u k

(kfk}.

Then M * is an

extension of M which satisfies the sentences of K since these sentences are of class Az. Now let al, ..., a, be any set of individuals in M * . Then a, . . ., an E Mk for some positive integer k. By our construction, the sentence Z (al, ...,an) holds in Mk+l and hence in MZfor all I > k, and SO in M*. This shows that X = (Vyi) . . . ( V y n ) Z ( y l , . . . y,) holds in M * which is an extension of M and a model of K. The proof of 3.6.1. is now complete.

86

[3.7.

SOME CONCEPTS A N D METHODS OF MODEL THEORY

The same method of proof leads to the following theorem which is a slight generalization of 3.6.1. 3.6.2. THEOREM. Let K be a consistent set of sentences of class A2 and let H be a set of sentences of the same class, H = {X?} when

xv = (VYl) .. . (Vyn) zv (Yl, - . .,Y n J ,

n 2

0

9

Zvexistential. In order that H be model-consistent with K, it is necessary and sufficient that for every model M of K and every Zv,and for every set of elements al, .. .,an, of M ythere exists an extension M‘ of M which is a model of K such that Zv(al, . . ., any)holds in M‘.

The only noticeable difference in the situation is that we now have to consider finite sets of sentences S’ whose elements are obtained by substituting individuals for the variables in different Zv.Apart from that, the proof proceeds exactly as before. 3.6.2. may be used in place of Zorn’s lemma, or the well-ordering theorem in order to show that every field can be embedded in an algebraically closed field. Thus let K = K C Fbe a set of axioms for the notion of a commutative field as in 2.2, so that the sentences of K belong to class Az. Let H = { Y,}, n = 2,3,, . . be a sequence of sentenceswhich state that every monic polynomial of degree n possesses a root. All these sentences also may be supposed to belong to class A2. The condition of 3.6.2. requires that every particular monic polynomial with coefficients in a given field M possesses a root in some extension of M. Supposing that this has been established, by the familiar construction of KroneckerSteinitz, we may conclude by means of 3.6.2. that every commutative field can be embedded in an algebraically closed commutative field. This argument is not sufficient to prove that every field possesses an algebraically closed algebraic extension. 3.7. Problems 3.7.1. Prove the following theorem. In order that any two models of a set of axioms K that have only individuals of K in common, possess a joint extension it is necessary and sufficient that the set of sentences which are defined in K and are consistent with K be conjunctive. The sentence Xis said to be consistent with K if K u { X } is consistent. 3.7.2. Prove that the concept of a ring cannot be formulated in terms of sentences of class B1 (i.e. existential and universal sentences) alone.

3.7.1

PROBLEMS

87

3.7.3. Prove the following theorem. In order that a sentence X be such that whenever X holds in a structure M then it holds also in the homomorphic images of all extensions of M it is necessary and sufficient that X be equivalent to an existential sentence whose matrix is a conjunction of atomic wff. 3.7.4. Give another example of a consistent set of axioms which is a-persistent but not convex. 3.7.5. Prove the following theorem. In order that a sentence X be modelconsistent with a set of sentences K i t is necessary and sufficient that for any existential sentence Y which is defined in, and consistent with, K the sentence X A Y also be consistent with K. References. For some uses of Skolem functors see Herbrand 1928 and Hilbert-Bernays 1934/1939. For the operation of relativization see Herbrand 1928 and Tarski-Mostowski-Robinson 1953. The notion of an elementary extension is introduced in Tarski-Vaught 1957. Another version of 3.2.5. is proved by the present method in A. Robinson 1956b (with the weaker conclusion that there exists a joint extension M* which is elementarily equivalent to M and W ) .The present version was given by Keisler For the remainder of section 3.2., compare A. Robinson 1951 and Rabin 1959. The theorem of Sierpinski's which is used in the proof of 3.2.9. is given in Sierpinski 1928. The notion of persistence is introduced in A. Robinson 1951. Theorem 3.2.9. is proved in Los 1955a and A. Robinson 1956c. It is a generalization of Theorem 3.2.4. which is given in Tarski 1954/1955. The argument in section 3.3. follows A. Robinson 1962. Theorem 3.4.9. is proved in LoS-Suszko 1957 and in Chang 1959. For the method and other results of section 3.4. compare A. Robinson 1959a. Additional work on the prefk problem is summarized in Lyndon 1959.3.5.1. is proved in A. Robinson 1951 on the basis of a somewhat stronger definition of convexity. Improvements are due to Chang 1959 and to Rabin, who suggested 3.5.2. Rabin has recently obtained a complete syntactical characterization of convex sets.

C H A P T E R IV

COMPLETENESS

4.1. A Test for Completeness. We recall (compare section 3.2.) that the

set of sentences K is said to be complete if for every sentence X which is defined in K (i.e. whose relations and individuals all occur in the sentences of K ) either X or X is deducible from K, in symbols, either R t X or K t X. There are two trivial cases when K is complete, one when K is contradictory, the other when K is empty. The following basic fact about completeness is due to Lindenbaum.

-

N

4.1.1. THEOREM. Let K be a non-empty and consistent set of sentences.

Then there exists an extension K' of K which contains no additional relations or individuals, such that K' is consistent and complete.

PROOF. Given K, non-empty and consistent as assumed in the theorem, let H be the set of all sentences which are defined in K,and let J = {Ky} be the set of all consistent subsets of H, which include K. Then J includes K. J is partially ordered by inclusion, and every totally ordered (i.e. monotonic) subset J' of J has an upper bound, which is the union of the elements of J', KOsay. Indeed, since J' is monotonic, every finite subset of KOis contained in some element of J', Hence, if KOwere contradictory

then some element of J' would be contradictory, and this is contrary to assumption. It now follows from Zorn's lemma that J contains at least one maximal element, K* say. K* is consistent and contains K . We shall show that it is also complete. Indeed, let X be any element of H, such that X i s not deducible from K*. Then we claim that X is deducible from K*. Consider the set K* u ( X } = K I . Since K* is maximal it follows that either Xis included in K* or Ki is contradictory. Since X is not deducible from K* it can certainly not be contained in that set. Hence K* u { X } is contradictory. It follows that there exists a finite subset of K * , e.g. the set { Y1, . . ., Ym}, such that { Y l , . . ., Ym, X } is contradictory and hence, such that X are theorems. [YI A . .. A Ym A X ] and Y i A , . . . A Ym 3 The last sentence shows that Xis deducible from K*, as asserted. Note

-

N

-

N

88

4.1.1

89

A TEST FOR COMPLETENESS

that if m = 0 to begin with, then we may always introduce arbitrary elements of Kin order to ensure that the implicans in Y1 A .. . A Ym 3 3 X i s not empty. Thus, the proof of 4.1.9. is complete. Except in the trivial case when K is itself complete, there are usually numerous complete extensions of K which are not equivalent to one another. Thus, let K be non-empty and consistent, and let X be a sentence of H such that neither Xnor Xis deducible from K . It follows that both KI = K U {- X > and Kz = K U { X } are consistent. Indeed, by an argument which has been used before and which will be used again in the sequel, if K I = K U {X} is contradictory then XI, . . . A X , A X is contradictory, for some XI, . . ., X n E K and so X I A . . ., AX, 3 Xis a theorem, Xis deducible from K . (In the trivial case that K is empty X is a theorem and is regarded as deducible from the empty set, by convention). K Z is consistent for an exactly similar reason. Hence, by 4.1.1 ., K1 and Kz, respectively, possess complete extensions K; and Ki. Thus K; and Ki are two distinct complete extensions of K. The importance of the notion of completeness for the mathematical applications is largely due to the fact that if we have established that two non-isomorphic structures are models of the same complete set of sentences K then every set of sentences which is defined in K and which holds in one of the structures holds also in the other one. Accordingly, it is of great interest to establish that certain familiar notions of Algebra are given by complete sets of axioms. A simple but effective test for completeness is as follows N

-

N

4.1.2. THEOREM (VAUGHT’S TEST).Let K be a consistent set of axioms which includes a relation of equality E ( x , y), and let a be the (finite or

infinite) cardinal of K. Suppose that all models of K are infinite and that for some infinite cardinal a’ 2 a all models of K are isomorphic (in the sense defined for sets with a relation of equality). Then K is complete. - If K includes any individuals then it is taken for granted that the models of K are isomorphic by mappings under which the individuals that are contained in K correspond to themselves. PROOF.Suppose that the conditions of 4.1.2. are satisfied and let X be a sentence which is defined in K. Then we have to show that either X or X is deducible from K. Suppose on the contrary that neither X nor Xis deducible from K then, by the usual argument, the sets K I = K U {- X} and KZ = K U {X}are both consistent. Their cardinal number is a 1, and since a’ is infinite we have a’ 2 a 1. It then follows from

--

+

+

90

comLJ3TENESs

[4.1.

Theorem 1.5.13. that there exist models M Iand M Zof KI and K2 respectively such that both M I and M Zare of cardinal a’.But if so then Mi and Ma are isomorphic, by one of the assumptions of 4.1.2. and hence satisfy either Xor X, simultaneously. This is contrary to the construction of M I or of M2 and proves the theorem. As a first example of a complete set of axioms which arises naturally in Mathematics, we consider a set of axioms K for the concept of a densely ordered set without first or last element. K may be expressed by a finite number of sentences formulated in terms of a relation of equality B(x, y) and a relation of order Q ( x , y ) (x smaller than or equal to y) and without individuals. Then all models of K are infinite and all countable models of K are isomorphic by a well-known result of Cantor’s. Hence, by Vaught’s text, K is complete. We formulate this result, which is essentially due to Langford as

-

4.1.3. THEOREM. The concept of a densely ordered set without first or last element is complete.

Corresponding results hold for the concepts of densely ordered infinite sets with first and last elements, and of densely ordered sets with first but without last element or with last but without first element. Consider now the set of sentences Rc:)(section 2.2.) for the notion of an algebraically closed field of specified characteristicp (p = 0,2, 3, .. .). It follows from a theorem of Steinitz’ that any two algebraically closed fields of the same characteristic and of cardinal a’ > No are isomorphic. Hence, by Vaught’s test 4.1.4. THEOREM. The concept of an algebraically closed field of specified

characteristic is complete.

This result shows that if a theorem X which can be formulated in the Lower predicate calculus in terms of equality, addition and multiplication, has been proved in any manner whatever for the field of complex numbers then it holds also for all other algebraically closed fields of characteristic 0. This is a transfer theorem of the kind discussed previously in section 2.4. It lends a limited but unambiguous meaning to the heuristic argument known as Lefschetz’ principle. However, it is possible that Lefschetz’ principle has some heuristic value in cases which go beyond the confines of the Lower predicate calculus. Combining 4.1.4. with the methods of section 2.4. we obtain the following result

4.2.1

91

MODEL-COMPLETENESS

4.1.5. THEOREM. Let X be a sentence formulated in the Lower predicate

calculus in terms of the relations of equality, addition, and multiplication and with individuals 0 and 1 (or, without individuals - the difference is irrelevant). Then if X holds in the field of complex numbers it holds also in all algebraically closed fields of characteristic p > po where PO is a constant which depends on X . 4.2. Model-completeness. In spite of the remarkable efficacityof Vaught's test, there are many important examples of complete sets of axioms whose completeness can be established only by different means. We shall now present a method which proves successful in at least some of these cases. We first introduce a modified form of completeness which will be called model-completeness. Let K be a non-empty and consistent set of sentences. Whenever we mention that a structure M is a model of K we shall suppose that all relations of M are contained in K. K will be called model-complete if for every model M of K with diagram D, the set K u D is complete. An equivalent definition is as follows. K is model-complete if for any model M of K, any extension M' which is a model of K is an elementary extension of M . Indeed, if K is model-complete by the first definition, and M and M' are models of K such that M' is an extension of M , let D be the diagram of M. Then M' satisfies both K and D and K u D is complete. Now let X be any sentence which is defined and holds in M. Then X is defmed in K u D and hence, is deducible from K U D. It follows that X holds also in M ' , M' is an elementary extension of M. Conversely, suppose that K is model-complete according to the second definition, and let D be the diagram of a model M of K. Then we have to show that K U D is complete. If K U D is not complete then there exists a sentence X which is defined in K U D and that neither X nor X is deducible from K U D.One of these sentences must hold in M and we may suppose without any essential limitation of the generality of the argument that X holds in M. If X is not deducible from M then K u D U { X } is consistent and possesses a model M'. But M' is an extension of M and a model of K. It follows that M' must be an elementary extension of M , and hence that X holds in M . This contradiction shows that K u D is complete, the two definitions are indeed equivalent. The notions of completeness and of model-completeness overlap but neither one of them includes the other. In fact, let K be the set of sentences formulated in terms of the relation of order Q ( x , y ) (i.e. x < y ) and

-

-

92

r4.2.

COMPLETENESS

without individuals which hold in any ordinary sequence M = {ul, a2, as, . . .} for instance in the sequence of positive integers. Note that the relation of equality may in the present case be replaced by the defined predicate Q (x, y ) A Q (y, x). The diagram D of M is the set of all sentences Q (a$,a,) for i < j , Q (a, uj) for i >j , i, j = 1, 2, .. . The sentence (Vx) Q (al, x) is defined in K u D and holds in M . However, this sentence does not hold in the structure M' which is given by the infinite sequence {UO, al, a2, . . .}, a0 an additional individual, although M' is isomorphic to M and hence a model of K, while at the same time it is an extension of M. On the other hand, we shall show presently that the concept of an algebraically closed field formulated in terms of the usual relations E, S, P is model-complete. The concept is certainly not complete for as long as the characteristic of the field is not specified, the sentence N

can be neither proved nor refuted (i.e. neither X nor X is deducible). A necessary and sufficient condition for model-completeness will now be derived. A well-formed formula (wff) will be called primitive if it is in prenex normal form of class El (existential) and if its matrix consists of formulae of order one - i.e. bracketed atomic formulae, in the full notation - and of the negation of such formulae. Thus, if we take a nuniber of elements of a diagram in conjunction, and we replace some of the individuals in the conjunction by variables and prefix the corresponding existential quantifiers to the result then we obtain a primitive sentence. N

4.2.1. THEOREM. (MODEL-COMPLETENESS TEST). Let K be a non-empty and consistent set of sentences. In order that K be model - complete it is necessary and sufficient that for any two models, M and M', of K, such that M c M' any primitive sentence which is defined in M may hold in M' only if it holds in M. PROOF.The condition of the theorem is necessary. Let M and M' be models of a non-empty K, M' an extension of My such that for some primitive sentence X which is defined in M y X holds in M' but X holds in M . Then the second definition of model-completeness given above shows that K cannot be model-complete. The condition is also sufficient. Let S =- {MyX } be the set of all ordered pairs such that M is a model of K with diagram D and X is a sentence defined in M (and hence in K u D ) such that neither X nor N

-

4.2.1

93

MODEL-COMPLETENESS

Xis deducible from K U D. If K is not model-complete then S is not empty, Moreover, in that case, S contains pairs in which the second element is in prenex normal form. For if ( M , X ) is any element of Xand X' is equivalent to X and in prenex normal form and contains the same relations and individuals as X (and such an X' exists) then ( M , X ' ) also belongs to S. Moreover, in any element of S the second number of the pair must contain at least one quantifier. For if Xis free of quantifiers and holds in M then it holds also in all extensions of M and hence, is deducible from D alone, while if X holds in M then X i s deducible from D. Thus, ( M ,X ) cannot belong to S. Finally, if S is not empty then it contains a pair whose second member is in prenex normal form and begins with an existential quantifier. For if ( M , X ) belongs to S then (M, X) also belongs to S, and if X begins with a universal quantifier then the usual reduction of X to prenex normal form (e.g. from [ (Vx) 2 ( x ) ] to (3x) [ Z ( x ) ] ) leads to a sentence which begins with an existential quantifier. Thus, let S' = ( M , X) be the set of ordered pairs such that M is a model of K with diagram D and Xis a sentence in prenex normal form beginning with an existential quantifier and defined in M such that neither X nor X is deducible from K.Then we have just shown that S' cannot be empty if K is not model-complete. Supposing that K is not modelcomplete, let (M, XO) be an element of S' for which the number of quantifiers in XO is a minimum. Thus, XO= (32) Q (2)say, where the number of quantifiers in Q is one less than in XO.By assumption, the set K U D u { X O }is consistent, and possesses a model, M'. M' is therefore a model of K and an extension of M such that for some a in M',Q(a) holds in that structure. Since Q(a) contains less quantifiers than XO, the pair ( W ,Q ( u ) ) does not belong to S'. On the other hand Q ( a ) is defined in M' and so either Q(u) or Q(u) must be deducible from K U D' where D' is the diagram of M'. But Q (a)cannot be deducible from K U D' since Q (u) holds in M' which is a model of K U D'. Hence Q(a) is deducible from K U D'. Also, Q (a) 2 [ (32) Q (41,i.e. Q (a) =) XO,is a theorem of the Lower predicate calculus and so XOis deducible from K u D'. It follows that there exists a sentence Y which is a conjunction of elements of D' and hence is free of quantifiers such that Y 2 XO is deducible from K. Now let Y = Y(a1, . ..,am),where we have displayed the individuals which are contained in Y but which do not belong to M , m 2 0. Since XO is defined in M and since all individuals of K occur in M , the individuals ul, . .., a m do not appear in K. Moreover, m must actually be positive, otherwise Y would N

N

-

N

N

-

N

94

[4.2.

COMPLETENESS

be a conjunction of elements of D , and hence deducible from D . But Y 3 XOis deducible from K, and so XOwould be deducible from K u D, contrary to assumption. Now K t Y 3 XO implies that Z h Y 3 XO is a theorem of the Lower predicate calculus for some coniunction 2 of a finite number of elements of K. Hence Y(a1, . . . a,) 3 [ Z 3 XO]is a theorem and hence, by one of the rules of deduction of the predicate calculus, [ (3Yl) * * * (3Ym) Y(Y1, . . .,y m ) ] = [z = XO] is a theorem for suitable variables yg (i.e. yt which do not occur in Y(a1, . . .,am)).Putting XI = (3y1) . . . (gym) Y ( y l , . . . y,) we then see that 2 3 [XI3 XO]is a theorem, and hence K k [XI 3 XO].We claim that ( M , X I ) belongs to S’. Indeed, XI is defined in M , and in prenex normal form, and begins with an existential quantifier. At the same time XI is not deducible from K u D since K k 1x1 =I XO]and so XO would be deducible from K as well. On the other hand X I is not deducible from K U D since XI holds in M’ which is a model of K U D. It follows that ( M , XI) belongs to S’. But XIis primitive and so, if the conditions of the theorem are satisfied, XI can hold in M’ only if it holds in M. Since XI is existential it would then hold also in all extensions of M and hence, be deducible from D . This contradicts the above conclusion that XI is not deducible from K u D . We infer that if the conditions of 4.2.1. are satisfied then S’ and S are empty, K is model-complete. This proves the theorem. It will be shown in the sequel that the notion of model-completeness possesses some importance of its own. However, in many cases it can also be used to establish ordinary completeness, in the following way. Let K be a non-empty and consistent set of sentences. A model MOof K will be said to be a prime model of K if every model M of K possesses a substructure which is isomorphic to Mo. For example if K is a set of axioms for the notion of a densely ordered set, then a prime model for K is constituted by any countable densely ordered set without first or last elements. Indeed, it is well known (and can easily be checked) that every densely ordered infinite set contains a densely ordered countable subset without first or last element. As this example shows, a given model of K may contain many prime models. In fact, in the present example, the ordered sets of type w also constitute prime models for K so that two prime models of a given K need not even be isomorphic. On the other hand, suppose that K is a set of axioms for the notion of a commutative N

4.2.1

95

MODEL-COMPLETENESS

field of specified characteristicp (p = 0,2,3, . ..). In this case, all prime models are isomorphic to the prime field of characteristicp , i.e. the field of rational number’s for p = 0, or the residue ring modulo p, forp # 0. Moreover, every model of K now includes only a single prime model. 4.2.2. THEOREM. Let M Obe a prime model of a set of sentences K and let DObe the diagram of Mo. Then any sentence X which is defined in K and deducible from K u DOis deducible also from K alone.

PROOF.By assumption there exist elements Y1, . .., Ym of D (where m may always be supposed positive, for ease of notation), such that K k Yi A .. . A Ym 3 X , or K k Y 2 X where Y = Y1 A . . A Ym. Let Y = Y(a1, . . ., am), n > 0 where the displayed individuals are those that occur in Y but not in K. Then by an argument used previously, more particularly in the proof of 4.2.1.,

.

Now the existential sentence Z = (3yl) .. . (3yn) Y ( y l , .. .,yn) holds in Mo and so it holds also in any structure which is isomorphic to MOand in any extension of such a structure. But according to our assumption, all models of K are extensions of structures which are isomorphic to M and so 2 holds in all models of K , It follows that X also holds in all models of K, and hence that K k X. This proves the theorem. 4.2.3. THEOREM. (PRIME MODELTEST).Let K be a model-complete set

of sentences which possesses a prime model. Then K is complete.

PROOF.Let M Obe a prime model of K , and let X be any sentence which is defined in K. Then we have to prove that either K t X or K t X. Since X is defined in Mo, either X or X holds in MOand we may suppose, without limiting the generality of our considerations, that X holds in Mo. Since K is model-complete, it follows that X is deducible from K u DO where DOis the diagram of Mo. And since MOis a prime model of K, 4.2.2. applies and we may conclude that X is deducible from K alone. N

N

This proves 4.2.3. Another test by means of which completeness can be inferred from model-completeness depends on the existence of joint extensions. It is sometimes useful, though less frequently than 4.2.3. 4.2.4. THEOREM. Suppose that any two models MI and Mz, of the model-

complete set of sentences K which have only the individuals of K in

96

[4.2.

coMPLETENF.ss

common can be embedded in a joint extension, M , which is a model of K. Then K is complete.

-

PROOF.Suppose that K is not complete. Then for some sentence X which is defined in K neither K t X nor K t X . It follows that the sets K u { X} and K U { X } are consistent and possess models M I and M Z respectively. Clearly, we may choose M I and M2 in such a way that they have only the individuals of K in common. If K satisfies the condition of the theorem then there exists a model M of K which is an extension of both M I and M2.But K is model-complete and so the fact that X holds in M I must entail that it holds also in M. By the same argument, the fact that X holds in M2 entails that X holds also in M. This is a contradiction and proves that K is complete. As a first application of Theorems 4.2.1. and 4.2.3. we consider again the case that K is a set of axioms for the concept of a densely ordered set without first or last element formulated in terms of the relations of equality E(x, y ) and order, Q (x, y), (i.e. x Q y). We wish to show that K is model-complete. Let then M and M’ be two densely ordered sets without first or last element such that M’ is an extension of M. Let X be any primitive sentence formulated in terms of the relations E and Q, and in terms of a number of individuals of M , such that X holds in M’. Then we have to show that X holds already in M. Let N

N

4.2.5.

x = (3y1) . . . (3yn)z(y1, . . .,y,)

where Z is the matrix and does not contain any quantifiers. Then the assumption is that the predicate Z ( y 1 , . . ., y,) is satisfied by a set of individuals bl, . . ,,b which belong to M‘ (i.e. by a point (bl, . . ., bn) in the space M’n) and we have to show that this predicate holds also for some set (c1, . , .,c,) of individuals of M . Interpreting 4.2.5. in ordinary mathematical language, we see that it affirms the existence of a solution to a finite system of equations and inequalities of the form

a=B,

4.2.6.

a>B

a#P,

- the

second and fourth expression being the negations of the preceding ones, respectively. In this notation, and /3 denote either elements of M o r y t , i = l , 2 ,,.., n. The assumption is that the system in question possesses a solution yc = br, i = 1,2, . ., n in M’ and we have to show that it possesses a (Y

.

4.2.1

97

MODEL-COMPLETENESS

solution already in M. We first reduce the list 4.2.6. by the following considerations. If the system contains a relation of the second kind mentioned in 4.2.6., a # p, then this may be replaced either by a < p or by p < OT according to the relation which holds in M' foryt = bt, i = 1, . . .,n. If a relation of the third kind mentioned in 4.2.6, occurs in the system, i.e. a < p, then we may replace this by a < p or a = p, in accordance with the relation which holds in M' for yc = bl, i = 1, ..., n. Finally we may eliminate all relations of the form a = jl successively by replacing the symbol which is written on the right hand side of this equation B in all the remaining expressions of the system by a. We see therefore that in order to prove model-completeness in the present case, we may confine ourselves to a finite system of inequalities 4.2.7.

aj

< pj

j = 1,

.. .,m

where q,pj denote either elements of M , at, or unknowns, yr, and where it is known that the inequalities are satisfied simultaneously by yt = bc in M'. If in the set of a6 and bt there are still pairs which are equal to one another (in the sense of the relation of equality of M and W )then we may again eliminate one of the elements of any such pair. Doing this successively,we finally arrive at a situation in which all at are different from one another, and the bt are different from the ar and different from one another. Suppose that the at which occur in the resulting system, which will again be assumed to be given by 4.2.7., are ranged in their specified order, so a1 < a2 < . . . < ak, k 2 0. These elements divide M' into k 1 intervals I;, I;; . . .,I;, which are given by y < al, a1 < y < az, ..., a& < y, in that order. If k = 0, then I: contains all elements of M'. The solutions yg = bt are in the interior of the intervals I;, since no br is equal to an u5. Suppose then that an interval I ; as introduced above contains a set of elements bt of M ' , e.g. b({)< b([)< . . . < b({,)where aj-l< b({)if 1; possesses a left end point, and b(i,)< aj if Zj possesses a right end-point. Now let (ct3), . . ., be any set of Ij elements in the interior of the interval Ij which is the restriction of I; to M , i.e. the set of individuals of Ij which belong to M , such that cy) < c $ ) . . . < c\{). Map the set S' = (at, b(g}, i = 1, .. ., k,j = 0, . ..,k, m = 1, . . . l j on the set S = {a, cE}with the same range for i, k, m,by the correspondence ug H a t , b c <+ CEand note that the set of b z coincides with (bl, . . .,bn). Then the correspondence S t)S' in order-preserving. It follows that we obtain a solution of 4.2.7. by replacing the bt everywhere by the corresponding c1. But the individuals ct all belong io M and so we have proved

+

~$3)

98

i4.2.

COMPLETENESS

4.2.8. THEOREM. The concept of a densely ordered set without first or last element is model-complete.

Observing that any countable densely ordered set without first or last element is a prime model for the concept in question, we conclude by means of 4.2.3. that the concept of a densely ordered set without first or last element is complete. This is 4.1.3. which was proved previously by a different method. The following result is of a somewhat deeper character. Let K be a set of axioms for the concept of a discretely ordered set with first but without last element, formulated in terms of the relations E ( x , y) and Q (x,y) (i.e. x 6 y, as before). K comprises the axioms of equivalence for E together with the axiom of substitutivity with respect to Q (see 2.1. above). In addition, K includes the following sentences 4.2-9.

(W 09)(W [Q(xyr) A Q 0, 4 Q (x, 41 (‘W VY)[Q (x, Y ) A Q (Y, E(x, Y ) ] (W WY)[Q (xyY ) V Q (Y, -41 ( 3 4 (VY) [Q (Y, x) 3 E (x, A] (V-4 (39 (Vz) [Q (x,V )A E (x,Y ) A [Q (x, 4 A Q (2,v) 3 E (x, 4 V =)

-

J W , Y ) ] ]* Of these the first two sentences require no comment. The fourth affirms that there is a first element while the last establishes the existence of an immediate successor to every element. 4.2.10. THEOREM. The set Kis complete.

We wish to prove this theorem via model-completeness but we observe immediately that the present K is not model-complete. Indeed, the sentence (Vy) [Q (y, a) =) E(a, y ) ] which affirms that the individual a is the first element of the set, may well hold in a model M of K but not in an extension M ’ of M which is a model of K. In order to obtain such an M’ from a given M y we only have to add to M a single predecessor to its first element, a. Accordingly, we supplement K as follows. We introduce an individual a and a two-place predicate S ( x , y) - “ y is the successor of x” - and we define K’as the set of axioms which is obtained from K by replacing the last two sentences of 4.2.9. by 4.2.11.

(W [Q ( Y , 4 2 E(a, Y ) ] ( W V Y ) W 4 [ [ S ( x ,Y ) A S ( x y2) =. Q (x, Y ) A E(x, Y ) A E ( Y , ~ ) ] A [S ( x A A Q ( x 4 A Q ( ~ 39E ( X J ) V E ( z , ~ )] ] (W (WJ’ (x,Y ) N

4.2.1

99

MODEL-COMPLETENESS

It will be seen that K I- K.We propose to show that K' is model-complete. Indeed, let M be a model of K' and let M' be an extension of M which is also a model of K'. The model-completeness test, 4.2.1., now amounts to the following. Let there be given a system of relations of the type 4.2.12.

a = P,

a

#

P,

a

< P, a > P,

S(a,P),

N

S(a,P)

where a,P denote either elements of M or unknowns, yg, i = 1,2, .. .,a. Suppose that the system has a solution yg = bg in M', then we have to show that the system possesses a solution also in M. We first consider any expression of the form S(a, p), which by assumption is satisfied in M'. This may happen either if p < a or if P = a or, finally, if there exists an element b such that a < b and b < p, and precisely one of these S(a,P) situations is realized in each particular case. We then omit and add fi < to the system in the first case and p = a in the second case. In the third case, we introduce a new unknown y' and we add 01 < y' and y' < P to the given system. Applying this method to a11 expressions of the form S(a,B) in the given system, we obtain another system which consists of expressions of the first five types mentioned in 4.2.12. We may now apply the methods of reduction used in the proof of 4.2.8. in order to eliminate all but the expressions of the form N

-

(Y

-

Suppose that the individuals of M which appear in the system are al, ...,U k , in that order, and that the unknowns are yr, i = 1, ..., m and that the system is satisfied by yr = bt, i = 1, . . . m in that order, where the bg are elements of M' and where we may again suppose that all the elements of the array al, . . ., U H , bl, . . . bm are different. 1 intervals Iz,I ; respectively The ut again divide M and M' into k I = 0, . . . k,which are given by the conditions y < al, a1 < y < as, .. ., u k < y. Then the bt are divided up into k 1 sets, b({)< b",' < .. . < b t ) , each in the interior of the interval I; and it is sufficient to show that we may replace the b($)which are elements of M' by certain C'Z) which are elements of My in such a way that the order is preserved and, moreover, such that if S(q-I, b({))or S(b$', b$) or S (b({,),uj) holds in M' then the corresponding relation S(q-1, c({)), S(c(:', S(cyi, a,) holds in M. Now we note that if an element c in M is nthin order after an element b, where n is a positive integer, then no additional elements can be introduced into the interval from b to c on passing from M to M'. Accordingly, we distinguish between I, which contain only a finite number of individuals

+ +

4+,),

100

[4.2.

coMPLETEm

and infinite Zj. In the former case all the Z$ are elements of M from the outset. In the latter case those by) which are nth successors of ag-1, n any positive integer, are also by necessity in M and the same applies to the b y which are nth predecessors of a5. The remaining by) decompose into a finite number of segments, where a segment is a set of consecutive elements of M'. However, since our system contains expressions of the form S(a, /3) but none of the form S(a, p), it is sufficient to map all these by) on a set of consecutive elements in I5 provided these elements are neither nth successors of 03-1 nor nth predecessors of aj, There must exist such a set in M since the number of elements in the interval between c~5-1 and aj is infinite, by assumption. This completes the argument, except that we observe that slight verbal modifications are required in order to deal with the first and last intervals. Accordingly, we have shown N

4.2.13. THEOREM. The set K' is model-complete.

Moreover, K' possesses a prime-model M O and this prime-model is unique, except for isomorphism. As M Owe may in fact take any ordered set of ordinal o,which begins with a. Hence 4.2.14. THEOREM.K' is complete.

It remains to be shown that the completeness of K' entails the completeness of K. Let then X be any sentence which is defined in K. Then Xis defined also in K' and since K' is complete either X or Xis deducible from K'. We may suppose that Xis deducible from K' and we shall show that in that case Xis deducible from K. Denote the axioms of 4.2.1 1. by 21,22,Zs respectively, and denote the conjunction of the remaining sentences by W. Since these sentences belong to K, we have K k W. Then N

or, equivalently,

ZiAZzhZ3hW2X 21 3 [ Z z A Z s h W 2

XI,

are theorems.

Since u does not appear in the square brackets we may replace Z1 by 2; = ( w v Y ) [ e c x , Y ) = E ( x , y ) ]

sothatZr2 [ Z z h 2 3 h W 3 X ] a n d h e n c e Z 2 h Z s 3 [ Z r h W ~ X ] are theorems. Now suppose that there exists a model M of Kin which X does not hold. We supplement the relations of M by introducing S ( x , y)

4.2.1

101

MODEL-COMPLETENESS

and by defining that S(b, c) holds between elements b and c of M if and only if holds in M . Let M’ be the supplemented model, then 2 2 and 2 3 hold in M‘. Also, 2: and W hold in M‘ since these sentences hold in M. It follows that M’ is a model of K‘, and so X holds in M’. This involves a contradiction and shows that X is deducible also from K. Thus, K is complete. The above example shows that we may increase the range of applicability of the model-completeness test by the introduction of auxiliary relations and individuals. Further applications of the notion of modelcompleteness will be found in A. Robinson 1956. Here we shall establish model-completeness only for two of the most important cases, in addition to those considered above. Let K = R’CF(see section 2.2.) be the specified set of axioms for the concept of an algebraically closed field. We propose to show that K is model-complete. Among the various proofs available for the purpose, we select one that requires relatively few algebraic results. Thus, let M be an algebraically closed field of arbitrary characteristic and let M‘ be another model of K which is a proper extension of M . Suppose that the primitive sentence Xis defined in M and holds in M ‘ . Then we have to show that X holds also in M . In the present case, X states that there exists a solution to a finite system T of equations and inequalities of the following types, 4.2.15.

a=j?

a#j?

a+j?=r

a f Q # ~ aP=y

aj?#y

In these relations, which are here written in ordinary algebraic notation, the Greek letter denote either individuals ai, i, = 1 . . .,k,which belong to M or “unknowns” yt, i = 1, .. .,m. The assumption is that the system T is satisfied by elements yt = bt of M‘. Consider the algebraic closure M * of the field M(b1, . . .,bm). Then the degree of transcendence of M * over M is at most m, and the given system is satisfied in M*. If the bl, . . ., bm all belong to M then M* = M and we have finished. If not there exists a chain of 1 1 algebraically closed fields

+

M=MocMlc

... c M l = M * ,

I<m

such that each field is of degree of transcendence one over its predecessor, if any. Now if T has a solution already in MO then we have finished. If not, then there exist two fields Mt c Mt+l such that T has a solution in Mi+l but not in Ma, i.e. such that X holds in Mi+1 but not in Mc. Since

102

r4.2.

COMPLETENESS

X is defined in Mi, we may replace M and M' by MZand Mt+i. Thus, we have shown that it is sufficient to prove that X holds in M , given that it holds in M', for the particular case that the degree of transcendence of M' over M is precisely one. Since every algebraically closed field M a which is a proper extension of M contains a subfield which is isomorphic to M' under a correspondence under which the elements of M remain fixed, it follows that X holds in M". Thus, if c is an individual which does not appear in M then X must be deducible from the set of sentences K' which is the union of K and of the diagram D of M and of a set of sentences H which is defined as follows. For any polynomial p ( x ) with coefficients in M select a predicate Q p ( x ) which is defined in M and which states that p ( x ) is equal to 0. Let H be the set of all sentences Qp(c) for p ( x ) not identically 0. H states that c is transcendental over M. Thus, every model of K is of degree of transcendence one at least over M and hence, satisfies X . It follows that Xis deducible from K' and hence from K U D together with a finite number of sentences of the form w Q p ( c ) ,e.g. Qpl(c), .. . Q p j(c). These sentences may be replaced by the single sentence Qpo( c ) where PO ( x ) = p l ( x ) . .. p j (x). Thus, we have N

N

-

N

N

K U D k

N

Q p o ( ~ )3

X

and so, by an argument used several times already, K U D k [ (3x1"

4.2.16.

Q ~ o < x ) ] ]2

X

Now K and D hold in M. So does the sentence ( 3 x ) [ Qpo( x ) ] since it merely affirms the existence of an element which is different from all roots of Q p o ( x ) ,i.e. from the elements of a certain finite set. But any algebraically closed field is infinite and so there exists an individual d such that Qpo ( d ) hold in M. This shows that ( 3 x ) [ Qpo( x ) ] holds in M. Referring to 4.2.16., we conclude that X also holds in M. This proves an assertion. We have established N

N

-

4.2.17. THEOREM. The concept of an algebraically closed field is model-

complete.

Since K'CF is not complete this provides the example, promised above, of a set of sentences which is model-complete but not complete. However, if we add to R'CFan axiom (for p # 0) or set of axioms (for p = 0) which fixes the characteristic of the field, then the resulting set R';) possesses a prime model, Ma, i.e. the algebraic closure of the prime field

4.2.1

103

MODEL-COMPLETENESS

of the given characteristic. It follows that the sets Kc;), which are still model-complete, are also complete as proved previously by a different method (Theorem 4.1.4., above). As a last application of the model completeness test, we shall consider the case where K = ROF,a set of axioms for the concept of a real-closed ordered field, formulated in terms of the relations of equality, addition, multiplication, and order, E, S, P, Q. We propose to show that ROF is model-complete. We begin by proving the following auxiliary result. 4.2.18. THEOREM. Let M be a real-closed field, and let M(b) be a simple

transcendental ordered extension of M. Then the order of M ( b ) is determined uniquely by the set of relations

4.2.19.

b

< av

a,, < b

or

whichever holds in M(b), where as varies over all elements of M .

PROOF.It is sufficient to show that 4.2.19. determines whether an arbitrary element of M(b), other than 0, is positive or negative. Let c E M(b), so that t=a

4.2.20.

+ a, b + a2b2 + ... + bk k 2 0 , m 2 0 a; + a;b + aib2 + ... + bm’ a0

where a, ao, a,, a2, ...,ao,a,, a2, ... belong to M . If t belongs to M and is different from 0 then t is positive according as it is or is not the square of an element of M. In the general case, both the numerator and the denominator of 4.2.20. can be decomposed into linear and quadratic factors, I

4.2.21.

,

,

t = a Z7(b - bj) Z7( (b - bj)2

+ ci)/Z7(b- bj) Z7( (b - b5)2 + ci)

with distinct ranges for j in the different product symbols, where b5, c5 belong to M. The quadratic factors in 4.2.21. are positive, while the sign of the linear factors is determined by 4.2.19., and the sign of a is determinate since a belongs to M . All these together determine the sign oft. This proves 4.2.18. In order to prove that ROFis model-complete we consider a system T of relations of types 4.2.22.

a=

P.

B,

f Y

a#

P,

a

< 8, a > P,

a

+ P = y, + P # y, a@ = y, a

where the Greek letters denote either elements at of a model M of ROP,

104

[4.2.

COMPLETENESS

. .. k, or unknowns yg, i = 1, . . ., m. We have to show that if T has a solution yr = bs in some extension M' of M which is a model of ROFthen is possesses a solution already in M . The fact that T possesses a solution is expressed by a primitive statement defined in M'

i = 1,

X = W )

... (3ym)QCul, ..., y m L

whichis therefore supposed to hold in M' and of which we wish to show that it holds also in M . An argument similar to one used in the proof of 4.2.17. shows that we may suppose the degree of transcendence ofM' over M to be exactly one. Let b be an arbitrary but fixed element of M' which is not contained in M , then M' is the real closure of M(b). Let H be the set of all sentences E(b, av) where av varies over the elements of M , together with all sentences N

Q (by av) and Q (an b) which hold in M' and put K = ROFu D u H, where D is the diagram

4.2.23.

of M. Let M" be any model of K' and let M* be the real closure of M ( b ) in M". Then M* is isomorphic to M', by 4.2.18. and so X holds in M * and hence, in M". This shows that X is deducible from K' and hence, from ROFU D together with a finite subset of H, H' = { Yl, . . ., Yh}, say. Let Y = Y ( b ) = Yi A . . . A Yh where we have displayed the individual b which occurs in Y, and where we note that all other individuals which occur in Yare contained in M . To avoid trivial difficulties, we may see to it that H is not empty. We then have and hence, as usual,

ROFU D ROFU D

t- Y ( b ) 3 X

t- [ @ y ) y ( y ) ] 2

x

Thus, in order to show that X holds in M we only have to establish that the sentence 2 = (3y) Y ( y ) holds in M. 2 is a primitive sentence, so that we are again faced with the existence of a solution to a particular system of relations, T'.However, T'has the special property that it contains only a single unknown, y, and that it contains only relations of one of the types 4.2.24.

a
.#B

where the Greek letters stand either for y or for elements of M . We know that T' is satisfied in M' by y = b, where b is different from all elements of M , and so we may replace 4.2.24. by relations of the type 01 < p only. The resultant system is of the form

4.3.1

RELATIVE MODEL-COMPLETENESS

4.2.26.

ug<x, X
105

i=l, ...,j i = j + 1, ... 1

where one or the other of these two sets may be empty and where we may suppose that the at are all different and that we have ranged them in each set according their given order in M. However, we know that the equations of 4.2.26. are compatible since they possess a solution in M' and so uj < uj+l if both sets are non-empty. In that case, the system is solved in M by x = (uj uj+1)/2. Again, if the second set is empty, then x = uj 1 is a solution, while if the first set is empty, we havej = 0, uj+l = U I , and we may choose x = ul - 1. This shows that ROFis modelcomplete. We have proved

+

+

4.2.27. THEOREM. The concept of a real-closed ordered field is modelcomplete.

Now ROFpossesses a (unique) prime model, i.e. the field of algebraic real numbers. Hence, by 4.2.3. 4.2.28. THEOREM. (TARSKI).The concept of a real-closed ordered field is

complete.

This important result was first proved by Tarski by an entirely different method. Relative Model-Completeness. Throughout this section we shall suppose that the sets of sentences Kand K* are non-empty and consistent. We recall that a set K* is said to be model-consistent relative to a set K if every model M of K can be embedded in a model M * of K*.(That is to say M * is a model of the diagram D of M). Thus, K* is model-consistent relative to K if for every model M of K, with diagram D,the set K* U D is consistent. K* will be said to be model-complete relative to K if for every model M of K, with diagram D,the set K* u D is complete. 4.3.

4.3.1. THEOREM. Let K* be an extension of the set of sentences K such

that K* does not contain any relations or individuals that are not also contained in K, and such that K* is model-consistent relative to K. In order that K* be model-complete relative to K it is necessary and sufficient that for every model M of K, and for every primitive sentence X defined in M, X either holds in all extensions of M which are models of K* or in none. The proof is a refinement of the proof of 4.2.1. The conclusion of the

106

COMPLETENESS

r4.3.

-

theorem is equivalentto the requirement that either Xor Xbe deducible from K* U D where D is the diagram of M. Since for any K* which is model-complete relative to K this condition must be satisfied for all X (and not only for primitive X) the condition of the theorem is clearly necessary. The condition is also sufficient. For suppose that K* is not modelcomplete relative to K. Then there exists a model M of K with diagram D and a sentence X which is defined in M such that neither X nor X is deducible from K* u D. We may suppose that X is in prenex normal form and that we have chosen a model of K and a corresponding sentence X such that the number of quantifiers in the prefix is a minimum. This minimum must be positive, for if X does not include any quantifiers then either X or X is deducible from D alone. Moreover, we may suppose that X begins with an existential quantifier, for if X as chosen begins with a universal quantifier, then we may pass to the negation of X and transform it to prenex normal form in the usual way. Thus, X = (32) Q (z) where Q ( z ) is a wff in prenex normal form which may contain further quantifiers. Xis defined in a model M of K but neither Xnor Xis deducible from K* U D,where D is the diagram of M. It follows that there exists a model M * of K* which is an extension of M such that X holds in K*, Q ( a ) holds in M * for some individual a. Let D* be the diagram of M * . Since M * is a model of K and Q(a) holds in M * it follows from the minimal property of X that Q (a)is deducible from K* u D*.Hence also, K* U D* t X since Q ( a ) =I X is a theorem. We propose to show that K* U D I- X. The set K* U D* contains in addition to the set K* U D only certain sentences of order one (bracketed atomic formulae) or the negations of such sentences. Let D' be the set of these sentences D' = D* - D.D' is not empty, so (in any case) there exists a sentence Y(ai, .. .,am)which is a conjunction of a finite number of elements of D' such that K* U D I- Y(a1, ...,a m ) 3 X.

-

-

In Y, we have displayed the individuals which do not belong to M , and hence do not occur in K* or D or X. It follows that

K* U D I- [ (321) . .. (32,) Y ( z ~ ., . .,zm)]

2

X

NOW (321) . . . (32,) Y (zi, . . .,z,) is a primitive sentence which is defined in M and which holds in M * . Hence, by the condition of 4.3.1., this sentence is deducible from K* u D.It follows that Xalso is deducible

4.3.J

RELATIVE

MODEL-COMPLETENESS

107

from K* U D.This contradicts our original assumptions on Xand proves 4.3.1.

In certain cases it is easier to derive relative model-completeness from ordinary model-completeness than to apply the above test directly. This can be done by means of the following theorem. 4.3.2. THEOREM. Let K* be an extension of a set of sentences K such that

K* does not contain any additional relations or individuals and such that K* is model-consistent relative to K. Suppose that K* is model-complete in the ordinary sense and that for every model M of K, with diagram D, the set K* U D possesses a prime model. Then K* is model-complete relative to K. PROOF.Let M be any model of K, with diagram D. Then we have to show that K* U D is complete. Let M*, be any prime model of K* u D with diagram D*, and let X b e any sentence which is defined in K* u D,and hence, in D alone. Since K* is model-complete, either X or X is deducible from K* U D*,. It will be sufficient to consider the former possibility. Then X is deducible from K* U D together with a finite set S of sentences which belong to D*, - D. If S is empty then we have finished. If S is not empty let Y = Y(a1, . . ,an) be the conjunction of its elements, where we have displayed the individuals which do not belong to M , n 2 1. Thus

-

.

and hence

K* U D I-

-

Y(Ul,

...,an)

3

X

-

K* U D [ (3y1) -.(3yn) y ( y l , . , Y n ) ] x In order to show that K* u D t Xis therefore sufficient to establish that K* U D I- 2 where 2 = [ (3y1) . . . (3yn) Y ( y l , . . .,y n ) ] , in other words, we have to verify that Z holds in all models of K* U D.Let M1

be any model of that set then by the definition of a prime model M I contains a substructure MT which is isomorphic to M*,, such that the individuals of D correspond to themselves under the isomorphism. It follows that 2 holds in MT and, being an existential sentence, that it holds in M I which is an extension of M:. This completes the proof of the theorem. Let us now consider some examples of relative model-completeness. Clearly, ordinary model-completeness is a special case of relative modelcompleteness, corresponding to K = K * . Now let K = KAF be a set of axioms for the concept of a commutative field as described in section 2.2.

108

COrnLETENESs

14.3.

above, and let K* be the set RhF for the notion of an algebraically closed field. Then it is easy to check that K* is an extension of K with no additional individuals or relations and that it is model-consistent relative to K, i.e. that every field can be embedded in an algebraically closed field. Moreover, by 4.2.17., K* is model- complete, and if M is any model of K, with diagram D,then K* U D possesses a prime model, namely the algebraic closure of M. This shows that the conditions of 4.3.2. apply. We conclude 4.3.3. THEOREM. The concept of an algebraically closed field is modelcomplete relative to the concept of a (commutative) field.

Now let K* = R($ be a set of axioms for the concept of an algebraically closed field of specified characteristic p as above, p = 0,2,3, . . . Then K* is still model-complete (since it is an extension of RbF with no additional relations or individuals) and it is model-consistent relative to set K = KiF for the notion of a field of the same characteristic. The remaining conditions of 4.3.2. also apply, and so 4.3.4. THEOREM. The concept of an algebraically closed field of specified

characteristic is model-complete relative to the concept of a commutative field of the same characteristic. Next, let K* be a set of axioms for the concept of a real closed ordered field, K* = ROF, then K* is model-complete by 4.2.27. Let K be a set of axioms for the concept of an ordered field, K = KOF.Since every ordered field can be embedded in a real-closed ordered field, more particularly in its real closure, the conditions of 4.3.2. apply, and we obtain the theorem 4.3.5. THEOREM. The concept of a real-closed ordered field is model-

complete relative to the concept of an ordered field.

Now let K and K* be sets of axioms for the notions of a formally real and of a real-closed field formulated in terms of the relations E, S, P,and in terms of the individuals 0 and 1 (compare section 2.2.). It is then casy to ensure that K* is an extension of K. Then K and K* have the same vocabulary and K* is model-consistent with K. Taking into account 4.2.27., it is not difficult to show that K* is model-complete. Nevertheless, we are going to show that K* is not model-complete relative to K. Indeed, let M be the field obtained by adjoining a square root 6 of 2 to the rational numbers. Then M is a model of K. Consider the sentence

4.3.1

109

RELATIVE MODEL-COMPLETENESS

x = (39fJ (Y, Y , 5) We claim that neither X nor Xis deducible from K* U D where D is the diagram of M. Indeed, there exist real-closed extensions of K , M * , and M : say, such that E, is positive in MT, and negative in M : . Then X holds in M:, and X holds in M*,. But M : and M*, both are models of K* u D,and this proves our assertion. It is quite possible that a set K* is model-complete relative to two nonequivalent K. Thus, if K* is a set of axioms for an algebraically closed field as in 4.3.3. then K* is model-complete not only relative to the concept of a field but also relative to the concept of a field of specified characteristic, in addition to being model-complete relative to itself. On the other hand, we shall now show that if a set K possesses an extension K* with the same vocabulary such that K* is model-consistent and modelcomplete relative to K, then K* is essentially unique. In this case, we are justified in calling such a set the model-completion of K. N

N

4.3.6. THEOREM. Let K,KT, KZ be three non-empty and consistent sets of sentences, K c KT, K c K*,, such that the individuals and relations

of Kf and K*,are contained in K and such that both KT and K*, are model-consistent and model-complete relative to K. Then KT and K*,are equivalent, i s . they are deducible from one another and hence, possess the same set of models.

PROOF.Let X E K : then we have to prove K*, t X . Suppose on the contrary that X E K: but that K*, U {-J X ) is consistent. In that case, let M be a model of K i U {- X } with diagram D.Since K*,is modelcomplete relative to K it follows that X is deducible from K*, u D

-

while at the same time Xis deducible from K : U D, trivially. In order to prove the theorem, it is therefore sufficient to show that for every model M of K with diagram D,and for every sentence X which is definable in M, either X is deducible both from K : u D and from K*, U D or X is deducible from each of these sets, separately. Suppose on the contrary that there exists a model M of Kand a sentence X defined in K such that Xis deducible from K: U D and Xis deducible from K*, U D where D is the diagram of M . Familiar arguments then show that X may be supposed to be in prenex normal form, and that it must contain at least one quantifier. We select an M and a sentence X satisfying these conditions such that the number of quantifiers in X is as small as possible. Moreover, we may suppose that X begins with an N

-J

110

COMPLETENESS

14.4.

existential quantifier, otherwise we consider the sentence X transformed into prenex normal form, and we interchange the sets KT and K:. Let X = (3y) Z (y), where Z may contain further quantifiers. Let MT be a model of KT U D, with diagram DT,then X holds in MT since it is deducible from K : U D. Thus M*, contains an individual a such that Z ( a ) holds in MT. But MT is a model of K and so KT u DT is complete, and K : U D t Z (a). Moreover, 2 (a)contains one quantifier less than X and so Z ( a ) must be deducible also from K : u DT. This entails that Xis deducible from K*, U DT. On the other hand, K : u D t k Xby assumption and so K : U D t X. This implies that K : u DT is contradictory. But that is impossible since K*, is model-consistent relative to KI and MT is a model of K. We conclude that we cannot have K : U D t Xand K*, U D t Xfor any model M of K. This completes the proof of 4.3.6. The theorem shows that for a given set of axioms K there can be essentially no more than one extension of K, K*, which satisfies the conditions of 4.3.6. In particular, the variety of all algebraically closed fields of specified characteristic occupies a unique position relative to the (more extensive) variety of all fields of the same characteristic. A similar position is occupied by the variety of all real-closed ordered fields relative to the variety of all ordered fields. We shall see later (section 5.4.) that a corresponding concept can be formulated relative to the notion of a differential field. On the other hand, there exist K for which a K* of the kind described does not exist at all (see Problem 5.6.4., below).

-

N

-

4.4. Problems 4.4.1. Establish the completeness of the concept of an infinite abelian

group in which every non-zero element is of given prime order p.

4.4.2. Let K be a non-empty and consistent set of sentences. Suppose

that for any two models of K, M and M’ such that M c M‘ any primitive sentence which is defined in M may hold in M’ only if it holds in M. Show that under these conditions any existential predicate Q ( X I ,.. .,xn) which is defined in K is equivalent with respect to K to some universal predicate R(x1, . . .,x) (i.e. K t (XI) . . . (xn) [Q(xl, . . ., xn) zz R(x1, . . ., 41). Hence, give an alternative proof for the model-completeness test 4.2.1. 4.4.3. An abelian group G is completely divisible (briefly, divisible)

4.4.1

PROBLEMS

111

if for every a E G and for every positive integer n there exists a b E G such that nb = a. Prove that the concept of a completely divisible ordered abelian group which contains at least two different elements is complete. 4.4.4. Prove the model-completenessof the concept of a discretely ordered set without first and last element formulated in terms of E and Q and in terms of a relation S(x, y ) for successiveelements. Show that this concept

is model-complete relative to a set K which includes the axiom for ordered sets (compare 4.2.9.) together with and together with the axiom of substitutivity for S(x, y).

References. Concerning Lindenbaum’s theorem, see Tarski 1935/1936 Vaught’s test is given in Vaught 1954. 4.1.3. is proved by a different method in Langford 1927. For characteristic 0, 4.1.4. is a consequence of Tarski’s theorem 4.2.28., for arbitrary characteristic it was first proved in A. Robinson 1951. The theorem of Steinitz’ which was used in the proof of 4.1.4. is in Steinitz 1910. The theory of model-completenessis developed in detail in A. Robinson 1956b. 4.2.10. is stated without proof in Tarski 1935/1936 and is proved by the present method in Robinson-Zakon 1960. For Tarski’s theorem, 4.2.28., see Tarski-McKinsey 1948/1951. The theory of relative model-completeness is developed in A. Robinson 1957.

CHAPTER V

DEFINABILITY

5.1. The Consistency Lemma. In the present chapter we shall be concerned

with connections between sentences, or sets of sentences, with different vocabularies. It will be convenient to use the following notation. Given a set of sentences K, we denote by Wt ( K ) the set of individuals which are contained in the sentences of K1 and by Wr( K ) the set of relations which are contained in the sentences of K. We write W ( K ) for the entire vocabulary of K, so that W ( K ) = Wg ( K ) U Wr (K). If K contains only a single sentence, K = { X } then we write Wr ( X ) in place of W g ( { X } ) , and so on. 5.1.1. THEOREM. Let K be a non-empty complete set of sentences, and let

KO,K1 be two consistent sets of sentences such that

KO 3 K, Ki 3 K, Wt(K0) C K, Wr(K1)C K Then the set KO U Ki is consistent. PROOF.Suppose, contrary to the assertion of 5.1.1. that there exist K, KO,KI,which satisfy the conditions of the theorem but not its conclusion. Then there exist sentences X I , . . ., Xj E KO - K, and YI, ., YmE K, such that the sentence

..

XiA

... A X j A Yi ... A Y m

is contradictory, i.e. such that

... A Y m 3 [XiA ... A X j ] Neither the set { YI,. . ., Ym}nor { X I , .. .,X j } can be empty, since the 5.1.2.

YIA

N

sets KOand K1 are, separately, consistent. Now let Yi A ... A Y m = Y(ai,

...,0%)

where we have displayed the individuals which occur in this conjunction but which do not belong to the vocabulary of K and, hence, do not belong to the vocabulary of XI . . . X I . Then, from 5.1.2. 112

5.1.]

113

T€!E coNsIsTENcy LEMMA

5.1.3.

t

2 -~ [ XiA

... AXj]

where Z = ( 3 ~ 1 ). . . (3zn)Y (z1, . . ., z,), n 2 0. Also, the sentence Y(a1, . . .,am) 3 2 is a theorem and so K1 t 2. Since W ( Z ) c K and K is complete, either K t 2 or K t Z . Now if the latter applied, then we should have also K1 t Z since K c KI . But we have just shown that K1 t Z , and we know that K1 is consistent. Hence, K t Z is impossible, and we must have K t 2. Also, KO 3 Kand so we certainly have KO t Z . Hence, by 5.1.3., KO t [XIA . . . A X j ] . But this is impossible since Xg E K, i = 1, .. . , j . Thus, we have arrived at a contradiction, proving

-

-

N

-

5.1.1. 5.1.4. THEOREM. Let K be a non-empty and complete set of sentences and let X be a sentence which is consistent with K (i.e. K u { X } is consistent), such that Wc(X) c W t ( K ) .Let M be any model of K. Let K1 be the set of all sentences which are defined and hold in M. Then X is consistent with Ki.

PROOF.Put KO= KU {X}.Then the sets K, KO,K1, satisfy the conditions of theorem 5.1.1. This leads to the conclusion that KO U Ki = KI u { X } is consistent - as required. Note that if Wr(X)c Wr (K)then the theorem is trivial. 5.1.5. THEOREM. Let M I be any structure, and let MT be a structure which is obtained from Mi by ignoring some (but not all) of the relations of M I . Let M*, be any elementary extension of M:. Then there exists an elementary extension of Mi which is also an elementary extension of M z as far as the relations which occur in M*, are concerned.

PROOF.Let K1 be the set of all sentences which are defined and hold in M I , and let K*, be the set of all sentences which are defined and hold in M;. Any model of H = K1 U K*, is an elementary extension of both M I and M*,, so in order to prove 5.15, we only have to show that H is consistent. If H is contradictory, then there exists a finite subset {XI, .. .,X k } of K*,,k 2 1, such that Ki u {XI, ...,Xk) is contradictory. Let XI A . . . A Xk = Y(a1, . . .,a,) where we have displayed the individuals in that conjunction which are not contained in M I . Then K1 t Y(a1, . . ., an) and, since the al, . . . an are not contained in K I , K1 I- (XI) . . . (xn) [- Y(x1, . . ., x.)] Now the sentence

-

2 = (Xl)

...

(&a)

[

-

Y(X1,

... 4 1 y

114

IS.1.

DEFINABILITY

is defined in A41 and contains only relations which occur in M ; and hence in MT. It follows that Z is defined in MT, and holds in that structure (since 2 holds in M I which is a model of K1). Thus, 2 holds in M t . On the other hand, Y(a1, . . .,an) holds in M*, and the same therefore is true of the sentence 2' = (3x1) .. . (3xn) Y (x,1 . . .,xn). But Z' is equivalent to the negation of 2, and so we have arrived at a contradiction. This shows that His consistent and proves the theorem. 5.1.6. THEOREM (THECONSISTENCY LEMMA). Let K be a non-empty and complete set of sentences and let H I , HZbe two consistent sets of sentences which include K, H I n HZ 2 K. Suppose that W(H1) n W ( H z )= W ( K ) . Then H I U HZis consistent.

We begin the proof with the following reduction. If HI u HZis contradictory then there exist finite subsets { X I , . .., X h } , { Y l , ..., Yk) of HI and HZrespectively such that X A Y is contradictory, where X = XI A ... A Xh, Y = Y1 A ... A Yk. Note that h 2 1, k 2 1. X and Yare, separately, consistent with K, since K u { X I is deducible from HI and K U { Y > is deducible from Hz.Thus, in order to prove 5.1.6., it suffices to show that X A Y must be consistent. Accordingly, it remains for us to prove the following theorem. 5.1.7. THEOREM. Let K be a complete set of sentences and let X I and XZ be two sentences such that W(X1) n W(X2) = W ( K ) . Suppose that X I and X2 are, separately, consistent with K. Then the conjunction X I A XZ is consistent.

For the proof, we may suppose that Wt(X1)c K and Wt(X2) c K. Indeed, suppose that we have already proved the theorem with this restriction. In the general case, we put

XI = Yl(a1, - .., am), X2 = Yz(b1,

*

., bn)

where we have displayed the individuals which are not contained in K. The sets ai, . . ., am and bl, . . ., bn are disjoint. Put z 1=

[ (3x1) ... (3xm) Y l ( X 1 , ...,xm)]

zz = [ ($9) ..- (3yn) Y2 (Yl, ...,yn)]

(where the variables xf, ya did not occur previously). Then Z1 and Z2 are, separately, consistent with K. Hence, according to the special case of 5.1.7. which is supposed to have been proved already, 2 1 A 2 2 is consistent. Now if X I A X Zwere contradictory, then by a familiar argument, the

5.1.1

115

THE CONSISTENCY LEMMA

sentence Z which is obtained from X I A XZ by r;placing al, ..., am, X I , ..., X m , y l , ...,yn, with subsequent existential quantification with respect to these variables -

bi, ..., bn by

Z = (3x1)

.. . ( 3 x m ) (3~1). . . (3yn) [ YI( X I , . .,x m ) A Y2 ( ~ 1 ,.. .,y n ) ]

would be contradictory as well. But t. 2 2 1 A ZZ and 21A 22 has already been proved consistent. This shows that we may suppose Wr(Xi) c W W ) , Wi(X2) c W ( K ) . Now let M I be a model of K u (XI] and let M be the structure which is obtained from A41 by ignoring the relations which do not occur in K. Let K: be the set of all sentences which are defined and hold in MT. Since X2 is consistent with K and MT is a model of K, the conditions of Theorem 5.1.4., are satisfied and the set KT U ( X Z }is consistent. Let M2 be a model of this set and let M*,be the structure which is obtained from M2 by ignoring the relations which do not occur in K. Then M2 is an elementary extension of MI, since it is a model of KT. Thus, the structures MI, M:, M*, satisfy the conditions of 5.1.5. It follows that there exists an elementary extension M3 of M I which is also an elementary extension of M*, as far as the relations which are contained in M*, are concerned. Thus, if M ; is obtained from M3 by ignoring the relations which are not contained in M*, then M*, is an elementary extension of M*,. It follows, again by 5.1.5. that there exists a structure M4 which is an elementary extension of M2, such that MZ is an elementary extension of M : where M t is obtained from M4 by ignoring the relations which do not occur in M:. Continuing in this way, we obtain a sequence of structures MI, M2 M3, . . ., Mn, . . . and a corresponding sequence MT, M*,, M : , . . ., M:, ... such that the following conditions are satisfied. The structures MI,M3, M5, ... contain precisely the same relations, and MI ee M3 ee M5 ee . . . (i.e. it43 is an elementary extension of MI, M5 is an elementary extension of M3, etc.). Similarly, the structures M2, M4, M6 contain the same relations and M2 ee M4 ee M6 ee . .. Finally, M : is obtained from Ma by taking into account only the relations of K, which are contained jointly in the structures with odd and with even subscripts and MT ee M*,ee M : ee . . . Now let M* = U { M : } , M' = {M2k+l} and M" = U { M z k } .

T

n

u k

k

Then the three structures M * , M', M" coincide so far as their sets of individuals are concerned and, moreover, M* is obtained from M' and from M" by ignoring the relations which are not contained in K. We may

116

[5.1.

DEFINABILJTY

now define a structure M, by supplementing M* simultaneously with the relations of M' and of M" as given in these structures. Since the relations of M' and M" do not overlap except for the relations of M*, the definition is unique. But MI satisfies XI and so Ms, M5, . . . and finally M' satisfy XI. Similarly M Z satisfies XZand so Mz, M4, . . . and finally M" satisfy Xz.It follows that M satisfies both XI and XZ.This shows that M satisfies X I A XZand proves the theorem. Clearly, the assumption that K is complete is essential for the proof. Otherwise, if K is not empty, we may take any X defined in K such that neither K t X nor K t- X. Then both K U {-J X} and K u {A') are consistent, but X A X certainly is not consistent. The following theorem which is essentially equivalent to the consistency lemma differs from it by its purely syntactical character.

-

N

5.1.8. THEOREM. (CRAIG'SLEMMA). Let X 2 Y be a theorem of the n Wr(Y) is not empty Lower predicate calculus, t X 2 Y. If Wy(X)

there exists a sentence ZOsuch that W(Z0) c W ( X ) n W ( Y )and such that t X =) ZOand t ZO3 Y. If Wr(X) n Wr(Y) is empty and X is consistent then t Y.

PROOF.Suppose first that W y ( X ) n W,(Y) is not empty. Let S be the (non-empty) set of sentences 2 with W ( 2 ) c W ( X ) flW ( Y ) .We may suppose that X is consistent for if X is contradictory then an arbitrary contradictory element of S, e.g. a sentence 20of the form 2 A Z with Z e S, satisfies the conditions of the theorem. For consistent X, let S' be the set of elements Z of S such that t X 2 2. We have to show that 1 Z 3 Y for some Z E S'. If no such Z exists then S' u {- Y )is consistent. For otherwise Y would be deducible from a finite subset S" of S', and hence from the conjunction of the elements of S", which is itself in S'. Supposing that S' u { Y} is consistent, let M be a model of that set and let K be the set of sentences of S that hold in M. Then K is complete. Clearly, Y is consistent with K since Y holds in M . We claim that X also is consistent with K. Indeed, if this were not the case then for some finite subset K' of K, K = {XI, .. ., Xn},n 2 1, the set K' u {A'} would be contradictory and so X 2 N [XIA . . . A Xn], or X 3 X' where X' = X I A .. . A Xnis an element of K'. But if so then X' E S' by the definition of S' and hence X' E K. This is impossible since K is consistent and so X is consistent with K. Hence by the special case of the consistency lemma, 5.1.7., X A Y is consistent. But this contradicts the assumption t X 2 Y. It follows that for some ZOE S' t

-

-

-

N

N

N

-

N

5.2.1

117

BETH’S THEOREM

k ZO1 Y,while I-X 3 20by the definition of S‘. This proves the first part of the theorem. Suppose next that Wr(X) n W r ( Y ) is empty but that Y is not a theorem. Then there exists a model M of Y such that the individual constants of Y appear as individuals in M. Moreover, we may suppose that the cardinal of M is precisely No, where the cardinal of M is defined without reference to any relation of equality. We may also suppose that the individuals of X which do not appear in Y have been included in M y and, we may define the relations of X in M in such a way that M becomes a model of X also. Thus X A Y is consistent, contrary to assumption. This completes the proof of the theorem. Conversely, it is not difficult to deduce 5.1.7. and hence, 5.1.6. from 5.1.8. Suppose that XI and XZare, separately, consistent with the set of sentences K. We have to show that XI A XZis consistent. Suppose on the contrary that XI A XZ is contradictory, then t- X I 3 XZ.Since XZ is consistent with K, X2 is not a theorem and so, by 5.1.8., the intersection Wr(X1) n Wr(Xz)is not empty. Accordingly, for some Z with W ( Z ) c W(X1) n W(X2), we have !-XI 3 2 and t- Z 1 XZ.Now, by the assumption of 5.1.7., W ( K ) 3 W(X1) n W(X2) and so 2 is defined in K.It follows that either 2 or 2 belongs to K. But if 2 belongs to K then K t XZ since t- Z 3 ZZ,and so K u {XZ}cannot be consistent, while if Z belongs to Kthen K t- XI,since t- Z 2 XZ, and so K u {XI} cannot be consistent. Thus, either assumption leads to a contradiction, and we may conclude that X I A XZ is consistent. This completes the proof.

-

-

N

-- -

--

-

-

N

5.2. Beth’s Theorem. In this section we shall use the notation X(F, GI,..., Gm) in order to indicate a sentence of the Lower predicate

calculus which is formulated in terms of a relation F(x1, ..., xk), k 2 0 and in terms of other relations GI, .. .,Gm, m 2 1 , In addition, X may contain a number of individuals, which are not displayed in the above notation. Let Y = (Vxl) . . . (Vxk) [F(xl, .. ., xk) G F’(x, .. ., x k ) ] where F’ is an additional k-place relation (and where the quantifiers are omitted in case k = 0), and let Z

..., Gm) A X ( F ’ , GI,..., Gm)

= X(F, GI,

where the second element of the conjunction is obtained from the first by replacing F everywhere by F’. We shall say that X defines F implicitly in

118

t5.2.

DEFINABILITY

terms of G I , . .., Gm if Z 3 Y is a theorem, t- Z 3 Y.We shall say that X defines G I , . . ., Gm explicitly in terms of G I , . . ., G m if there exists a , is defined in terms of G I , . . ., Gm and in terms wff Q ( X I , .. .,X X )which of the individuals of X ( F , GI, . . .,Gnr),such that

- - - ( V X k ) [ F - .. Xk) = Q (Xl, ... X k ) ] is deducible from X(F, GI, ...,Gm). Beth‘s result on definitions may (VX1)

(XI,

y

y

now be formulated as follows.

5.2.1. THEOREM (BETH).If a relation F(x1, . . .,xk) is defined implicitly in terms of relations G I , . , Gm by a sentence X(F, G I , . . ., Gm) then F can also be defined explicitly in terms of G I , .. .,Gm and in terms of the individuals of X .

..

PROOF.Let al, ...,a& be k individuals which are not contained in X . Then it follows from the assumptions of the theorem that

5.2.2. t- X ( F , G I ,

Hence

. .., Gm) A X(F’, G I , .. ., Gm) 2 = F’(a1, . . .)U X ) ]

I- X ( F , G I , 3

..., Gm) A F ( u I , . . .,

[X(F’, G I ,

. .., Gm)

3

[F(uI,

..., ak) E

ak) 3

F’ ( ~ 1 ,. .

.,~ k ) ]

By Craig’s lemma 5.1.8., there exists a sentence W = Q(a1, ...,ak), formulated in terms of the vocabulary of X, excepting F, such that 5.2.3.

t- X ( F , G I ,

. . ., Gm) A F(a1, . ..,a&)

Q (01,

2

...,U X )

and 5.2.4.

. ..,ax) [ X(F’, G I , .. ., Gm)

t. Q ( U I ,

3

2

F’ ( ~ 1 ,. . ., U X ) ]

Modifying 5.2.3. slightly, we obtain 5.2.5.

t X(F, G I ,

...,Gm)

2

[F(ai,

..., a&)

2

Q ( ~ 1 ,...,u.)]

.

Similarly, substituting F for F‘ in 5.2.4. and using another simple transformation of the propositional calculus, 5.2.6.

t- X ( F yG I , .

. ., Gm) 3 [Q(czI,. .., a&) 2 I”(a1, .

.,U X ) ] -

Combining 5.2.5. and 5.2.6. and employing one of the rules of inference

.

5.2.7. t- X ( F , G I , . . ,Gm) 3 ( X I )

. .. (x&)[ F ( x I ,. . .,XX) E Q ( X I , . . .,X X ) ]

and this is the conclusion of Beth‘s theorem.

5.3.1

119

RELATIVE D E F I " S

It is understood that the proof requires some trivial mod,fications for

k = 0. If k 2 1 and at least one G6 is of positive order GI = G ( X I ,. ..,

xz), say, then we can always ensure that all at appear in W, if necessary by adding the sentence [G(al,. .., al) G(a1, ., a l ) ] A . .. A [G (a,, . . .,a,) G (am,. . .,a,)] to the given Win conjunction. If all Gt are of order 0, then W is by necessity a sentence which does not include any variables.

..

5.3. Relative Dehitions. The following section has some points of contact with 5.2. but it deals with a different situation, which arises in Algebra on several important occasions. Let K and K* be two sets of sentences. A sentence X' which is defined in K will be said to be invariant relative to K* over K if for every model M of K,Xis either satisfied in all models of K* which are extensions of M or in none. 5.3.1. THEOREM. Let K and K* be two non-empty and consistent sets of sentences such that all relations and individuals of K* are contained in K and such that K* is model-consistent with K. Let X * be a sentence which is defined in K* (and hence in K ) and is invariant relative to K* over K. Then there exists a sentence X which is defined in K such that X holds in any model M of K if and only if X* holds in all models of K* which are extensions of M.

PROOF.Given K, K*, and X*, as described in the theorem, let KR be the set that is obtained by relativizing K with respect to a relation R (x) which is not contained in K and hence is not contained in K* or X *

either. We recall (compare section 3.1.) that KR contains, in addition to the R-transforms of the individual sentences of K, the sentences R (a) for all individuals a which occur in K, and the sentence (3x) R (x) if no such individual exists. We shall show presently that we can find a sentence X which is defined in K such that

5.3.2.

K R U K* I - X * _ X R .

Accepting this result, it is not difficult to derive 5.3.1. Indeed, supposing that such an X exists, let M and M * be models of K and K* respectively such that M * is an extension of M. Introduce the relation R ( x ) into M * by defining that R(u) holds in M * precisely for all elements a of M y and call the resulting structure M:. Then M: is a model of K R as well as

120

15.3.

DEFINABILITY

of K*, and so X * X R holds in M i , by 5.3.2. Suppose that X * holds in M * and hence, in M*. Since X * X R holds in M * , X R also holds in the same structure, and this in turn entails that X holds in M. On the other hand, if X * holds in M * , then the same method shows that X holds in M. Thus, in order to prove 5.3.1., it is sufficient to find a sentence X such that X R satisfies 5.3.2. Let S be the set of sentences X which are defined in K such that X * 2 X R is deducible from KR U K*. S is not empty for it includes all theorems (of the Lower predicate calculus) which are defined in K. Moreover, S is conjunctive for if X * =, X R and X * =, YR are deducible from KR U K*, then X * 2 [ X Rh YR] is deducible from that set as well. But [XRh YR] coincides with [ X h Y]R, and so X h Y also belongs to S. Let T be the set of all sentences X R for X E S, and put

-

-

H = K R U K* U TU { - X * } .

If this set is consistent then it possesses a model, M:. The individuals of M: which satisfy R ( x ) constitute a model M R of KR, and if we ignore the relation R(x) in M R then we obtain a model M of K. Let D and D R be the diagram of M and M R respectively, so that DR is obtained from D by adding the sentence R(a) for all individuals a of M. The set H = D R U KR U K * is consistent since M i is a model of this set. We propose to prove that H' t X * . Indeed, if H U { X * } = D R U KR u K* u { X * } is consistent then it possesses a model N:. The individuals of N: which satisfy R (x)constitute a model N R of KR which is at the same time an extension of M R . Thus, N: and M: are models of K* and extensions o f M , but N: satisfies X * while M: satisfies X*. It follows that the structures N * and M * which are obtained from N i and M: by ignoring R ( x ) also satisfy X * and X * , respectively, although they are both extensions of M. This is contrary to the assumptions of 5.3.1. and proves that H k X * . Thus, there exists a conjunction Y of a finite number of elements of DR, Y = YI h . . . h Yk such that the sentences Y 1 X * and, hence, X* 3 Yare deducible from KR u K*. Every YZis either of the form R(a) or a formula of order one (a bracketed atomic formula) defined in M or a negation of such a formula. Accordingly, Y may be written as a disjunction, or as an implication of the form N

N

N

-

-

-

-

R(a1) h

... h R ( u ~1) Z

where Z is a disjunction of negations of elements of D. By adding, if

5.3.1

? 21

RELATIVE DEFINITIONS

necessary, suitable elements to the original { Y1, . .., YR),we may always ensure that Z is not empty, and that it contains as individuals precisely U I , . . . , an, Z = Z(a1, . . ., an).Then KR U K* I-X* 3 [ R ( a l ) A ... A R ( u ~3) Z ]

.

The conjunction R(a1) A . . . A R(an)may be shortened by deleting all terms R ( q ) such that uj is contained in K, if any, for these terms are included in KR. Suppose then that aj is in K for j = m 1, ...,n, then 5.3.3.

KR U K* I- X *

3

[R(ai)A

... A R(u,)

+

2

Z]

.

However, if m = 0, then this sentence is to be read simply as X * 3 Z . Now the individuals al, ...,am are not included in K R or K* or X * and so 5.3.3. entails that the sentence

X* 2 [ (Vxl) .. . (Vxm) [ R( X I ) A . . . AR (x,)

--

Z(x1,

3

.. .,x,,

+ 1,

an)]]

is deducible from KR U K*. It follows that the sentence X* deducible from KR U K*, where 3 *. *

am

WR = (vxl)[R(x1) = [ (vx2) [ R(x2) [R(x,) = Z(X1, .* .,X m , U m + l , * .,an)]

.

3

W R is

...] .

But, as anticipated, by our notation, WEis the relativized transform of W = ( V X ~ .)

.. ( V X ~ )2 ( X I , ..., Xm, am+l, - .- an)

and so the fact that KR U K* I- X* 3 WRimplies that W E Sand W RE T. We now recall that M i is a model of H = KR U K* U T U {N X*}. This implies that M i satisfies WR. It follows that M i satisfies also the sentence R(a1) A . . . A R(am) 3 Z(a1, . . .,am, u ~ + I.,. ., an) .

..

But R(a1) A . A R(anb)holds in M i since that structure is a model of KR, and so Z(a1, . .,an) also holds in M:. Also, Z(a1, ..., am) is equivalent to the negation of a conjunction Z’ of elements of the diagram D of M , and so Z’ holds in M and M:. Thus Z(a1, . . .,an) holds in M i , yielding a contradiction which proves that the set His contradictory. Since T is conjunctive, it follows that there exists an element XR of T such that KR U K* I-XR 2 X * . But at the same time KR U K* k X * 3 XR

.

N

122

15.3.

DEFINABILITY

by the defining property of T and so

This completes the proof of Theorem 5.3.1. We note that the sentence X

whose existence is affirmed by 5.3.1. is essentially unique, i.e. if both X = XI and X = X2 satisfy the conclusion of the theorem, then XI X2 is deducible from K. There is a corresponding result for predicates. Let K and K* be two sets of sentences. A predicate Q* (xi, .. .,xn), n 2 1,which is defined in K will be said to be invariant relative to K* over K if for any given model M of Kand for any set of elements al, . . .,an of Myeither Q * (a,. . . an) holds in all extensions of M which are models of K* or Q * (al, . ..,a n ) holds in all extensions of M which are models of K*.

-

5.3.4. THEOREM. Let K and K* be two non-empty and consistent sets of sentences such that the relations and individuals of K* are contained in K and such that K* is modeI-consistent with K. Let Q * (XI, .. .,X n ) , n 2 1 , be a predicate which is defined in K* and is invariant relative to K* over K. Then there exists a predicate Q(x1, ..,Xn) which is defined in K, such that if M is any model of K, and (a,. .. an) is any set of elements of M , the sentence Q (al, .. ., a%)holds in Mif and only if Q* (al, .,an) holds in all models of K* which are extensions of M.

.

..

..

PROOF.Let bl, . ,bn be a set of individuals which are not contained in K We extend K and K* without extending their sets of relations by adding otherwise arbitrary sentences which involve bl, . .., bn and no other individuals and which are theorems of the Lower predicate calculus. Let the resulting sets be called KOand K*,. Consider the sentence X * = Q* (bl, . ..,bn). X* is defined in K*, and in KO.We are going to verify that X* is invariant relative to K*, over KO.Indeed, if MOis any model of KOand M*,, M t are models of K*,,then bl, .. ., bn are contained in all three of these structures. Since Q*( X I , . . ., xn) is invariant relative to K* over K, and since Mo is a model of K, and M : and M*, are models of K*, it follows that X * holds either in both M: and M*,or it holds neither in M: nor in M*,. Moreover, the relations and individuals of K*, are contained in KOand if MOis a model of KO, and hence of K then it possesses an extension M*,which is a model of K*. But bl, . . .,bn belong to MO and hence to M*,, and K*, contains, in addition to K* only a number of theorems of the Lower predicate

5.3.1

123

RELATWE D-ONS

calculus. Hence M*, is a model of K*,, K*, is model-consistent with KO. This shows that the conditions of Theorem 5.3.1. are satisfied, and we may conclude that there exists a sentence X which is defined KOsuch that X holds in any model MOof KOif and only if X* holds in all models of K*, which are extensions of Mo. We may suppose that X contains the individuals bl, . . .,bn for if this is not the case from the outset then we may achieve the desired situation by adding to the original X a theorem (of the Lower predicate calculus) which contains bl, ..., bn. Accordingly, we may write X = Q (bi, . . ., bn). We maintain that Q (xi, . . .,X n ) satisfies the conditions laid down in the conclusion of 5.3.4. To see this, let M be any model of K and let al, . ..,an be any set of elements of M. Let M* be an extension of M which is a model of K*. It is evident that the definition of Q (XI, . . .,xn) is independent of the choice of bl, . ..,bn subject only to the condition that these individuals do not occur in K, and we may therefore suppose that they were selected from outside M*. We extend M by adding the constants bl, ...,b, to its set of individuals and by defining that any relation A (XI, .,x k ) shall hold for a set involving some of the br precisely if A ( X I , . . .,x k ) holds in M for the set which is obtained by replacing the bt by the at with equal subscript. The enlarged structure will be called Mo. Enlarging M * in a similar way by means of the same set bl, . . ., bn we obtain a structure M*, which is an extension of Mo. MOand M*, are models of KOand K*,, respectively. Suppose now that Q* (al, .. .,an)holds in M , then X* = Q*(bl, .. .,bn) holds in M*, by the construction of M z , and so X = Q (bl, ...,bn) holds in Mo by 5.3.1. Hence, Q (al, ...,an) holds in M, by the construction of Mo. Similarly, if Q*(ai, . . ., an) holds in M* then Q(a1, . . .,an) holds in M. This completes the proof of 5.3.4. The sentence Xwhose existence is established by 5.3.1. ispersistent with respect to K (see section 3.3.) To see this consider any two models M and M' of K such that M' is an extension of M and X holds in M. Suppose that X does not hold in M', so that X holds in that structure. Let M * be any extension of M' which is a model of K*. Such an M * exists since K* is model-consistent with K. Then X* holds in M*, by one of the defining properties of X.On the other hand, M* is also an extension of M which satisfies X. Hence, M * satisfies X*.This involves a contradiction and shows that X holds also in M', X is persistent with respect to K. A similar argument shows that X also is persistent with respect to K, i.e. X is invariant with respect to K. Hence, taking into account Theorem 3.3.2., 3.3.3., we obtain

..

-

-

N

N

N

12.4

15.3.

DEFIN ABILITY

5.3.5. COROLLARY. The sentence X in 5.3.1. may be selected as an existential sentence, or as a universal sentence.

Similarly, the predicate Q (XI, . . ., xn) which appears in the conclusion of 5.3.4. is invariant with respect to K. Hence, 5.3.6. COROLLARY. The predicate Q (XI, . . .,x,) in 5.3.4. may be selected as an existential predicate, or as a universal predicate. For practical purposes, it is useful to link the above results with the notion of relative model-completeness. 5.3.7. THEOREM. Let K and K* be two non-empty and consistent sets of

sentences such that the relations and individuals of K* are contained in K and such that K* is model-consistent with K and model-complete relative to K. Let X * be any sentence which is defined in K*. Then there exists a sentence X which is defined in K such that if M is any model of K and M * is a model of K* which is an extension of M , X holds in M if and only if X * holds in M*. Moreover, X may be chosen as an existential sentence, or as a universal sentence. Similarly, to any predicate Q* (XI, . . .,x,) which is defined in K*, n 2 1 there exists a predicate Q (XI, . .,x,) which is defined in K such that if u1, . . .,an are elements of any model M of K and M * is a model of K* which is an extension of M,then Q(a1, . .,an) holds in M if and only if Q* (a,. ., a,) holds in M*. Q is invariant with respect to K and hence, may be chosen as an existential predicate or as a universal predicate.

.

.

.

PROOF.To prove the first part of the theorem we have to show, in view of 5.3.1. that any sentence X * which is defined in K* is invariant relative to K* over K. Let M be any model of K with diagram D. Then K* U D is complete, since K* is model-complete relative to K. It follows that either X* is deducible from K* u D and hence, holds in all models of K* which are extensions of M , or X * is deducible from K* u D and hence, holds in all models of K* which are extensions of M. This proves the assertion. The truth of the second part of the theorem is established in a similar way. To give an example of a situation in which the conditions of 5.3.7. are satisfied, we consider the case that K is a set of axioms for the concept of an ordered field K = KOFas defined in section 2.2. and K* is a set of axioms for the concept of a real closed ordered field K* = ROF.Then the assumption of 5.3.7. are satisfied with respect to K and K * . Now let p (u) be any monic polynomial of pre-assigned degree m > 0 with coeffiN

5.4.1

125

APPLICATION TO THE NWLUTELLENSATZ

cients in an ordered field M,and let I be a closed interval in M.If k is any specified natural number, k 2 0, then the property (of p ( u ) ) to possess exactly k distinct roots in I may be regarded as a predicate of the coefficients xo, . . .,Xm-1 of p (u) - the leading coefficient being 1, by assumption - and of the two end-points of I, y and 2. However, in Algebra as opposed to Arithmetic, one is in general interested not in the number of roots which belong to the given M, but in the “total” number of roots in I, more precisely, in the number of roots in I which belong to the real closure of M.Thus, if M is the field of rational numbers, one wishes to determine the number of real roots in Z or, which is the same, the number of real algebraic roots. Accordingly, the predicate Q* (XO, .. .,Xm-1, y, z) which states that p ( u ) = xo XI u . . . xm-l um-1 u m possesses precisely k roots in the closed interval y < u < z, refers to M*. Theorem 5.3.7. now tells us that there exists a predicate Q(x0, . . ., xm-l, y, z) which holds in M,i.e. in the original field, precisely when p possesses k roots in the interval determined by y and z in any real-closed extension of M.The reader will recognize that we have thus disposed of a problem which, in standard Algebra, is settled by means of Sturm’s method. Similarly, the problem whether or not a system of equations in a given field possesses a solution in the algebraic closure, or in any algebraically closed extension of that field, comes within the scope of 5.3.7. In standard Algebra, this problem is dealt with by means of the construction of resultants. In the next section, we shall give an application of 5.3.7. which shows that our methods provide a ready means for establishing the existence of numerical bounds in certain algebraic problems.

+

+

5.4. Application to the Nullstellensatz. Hilbert’s theorem on the zeros of a system of polynomials, briefly the Nullstellensatz, is commonly formulated as follows.

5.4.1. “Let f ( x 1 , . . .,x,), f1 (XI, . . .,xn), . . ., fr (XI, .. .,x,) be polynomials in the polynomial ring M [XI, . .,xn] over a given field M such that f vanishes for all joint zeros of the polynomials fi, ...,&. Then there exists a positive integer p such that fe belongs to the ideal (fl, . . ,

.

.

fr)”.

We observe that the above statement is correct if by “all joint zeros” we mean all joint zeros in all commutative fields which are extensions of M” or “all joint zeros in all algebraically closed extensions of M ” or even “all joint zeros in the algebraic closure of M”. The statement is not

126

15.4.

DEFINABILITY

correct, generally speaking, if by “all joint zeros” we mean “all joint zeros in My’, and one of the earlier interpretations is usually taken for granted, implicitly. The second and third interpretation are equivalent by virtue of the model-completeness of the notion of an algebraically closed field. More precisely, let Q* be the formula 5.4.2.

[fi (XI, .. ., x,)

=0 A

(VXl) .- . (vxn) . .. A & (xi, . . .,x,)

=O 3

XI,

. . .,x,)

= 0]

when written out in detail within the Lower predicate calculus in terms of the relations of equality, addition, and multiplication, E, S, and P. We may regard Q* as a predicate of the coefficients of the polynomials f., . . ., & andf’taken in an arbitrary but definite order, yl, . . .,y k , say. Thus, Q* = Q*(yl, . . .,y k ) may be formulated in terms of E, S, and P, and without individuals. It follows that if we take K = KCF,a set of axioms for the notion of a commutative field and K* = RCF, a set of axioms for the notion of an algebraically closed field, then the conditions of Theorem 5.3.7. are satisfied. Accordingly, there exists a predicate Q (yl, .. .,y k ) formulated in terms of E, S, and P,and without individuals such that for any field M and elements al, .. ., Uk of M , the predicate Q ( a 1 , . . ., a k ) holds in M if and only iff vanishes for all joint zeros of fly , ., fry in the algebraic closure of M (or, in any other algebraically closed extension of M ) . It is of interest to note that the definition of Q is independent of the characteristic of the field under consideration. The conclusion of 5.4.1. asserts, in more detail, that there exists a positive integer p, and polynomials gl (XI, . . .,x,), . .,gr (XI, ..., x,) such that

.

.

5.4.3. g1 (Xl,

.. . X,)fl y

(Xl,

.. .

y

x,)

- ..,x,))Q=

+ . . . + gr

(f(x1,

(Xl,

...,xn)& (Xl, . .. x,) . y

Observe that in this context, it is taken for granted that the coefficients of 81, ...,g r belong to the field M. As it stands, the conclusion of 5.4.1. cannot be formulated within the Lower predicate calculus. However, given any pair of integers p and y, it is not difficult to verify that the assertion “There exist polynomials gl, . . ., g r of degrees not exceeding ,u such that the identity 5.4.3. is satisfied” can be formulated within the Lower predicate calculus as a predicate of the coefficients y l , . . .,yk of the polynomials fly . ..,A and f. For

5.4.1

127

APPLICATION TO THE NULLSTELLENSATZ

given p and p we denote a suitable predicate of this kind by Q e p ( y l , . . ., contain any individuals. Let al, . . .,uk be an arbitrary set of individuals, and let H be the set of sentences Q,,(al, . . ., a k ) , p, p = 1,2,3, . . . Let H’ = KCF u H u U (Q(a1, . . ., a)}. If H is consistent then there exists a field M , and polynomials f;,...,&,A with coefficients al, ...,Uk in M such that f vanishes for all joint zeros of f1, ...,f@ in all algebraically closed extensions of M , yet 5.4.3. does not hold for any e and gt(x1, ..., xn). Thus, the conclusion of the Nullstellensatz does not hold, which is impossible. Hence, H is contradictory and, accordingly, there exist positive integers p1, . . ., pz, p1, . . ., pl such that the sentence yk). Qea does not

-

.

5.4.4. Q(a1, . -,a k )

2 Qpl

.q(

a ~.,. .,uk) V

. . . V Q e p l (a;, - . .,ak).

is deducible from KCF.Now the identity f e =cgifr

implies

Gift

f@+A =

for all positive integers I, where Gt =f A gt, and so Q,. (ul, . .., a k ) 3 Qer (a;, .. .,a k ) is deducible from KCBfor all p’ < p. On the other hand,

Qsut (al, . . ., a k ) 2 Qer (a, . . .,a t ) evidently is deducible from KCF for all p‘ < p. It follows that if we set

PO = max pt,

then, by 5.4.4. and hence 5.4-5. KCF 1(

1< i < l

KCF Q (a17 - -

+

PO = max pi 1
ak) 3

QeoPo (a17

. - . ( ~[Q 4( Y L - - - ,~d

~ 1 )

3

* *

QeoPo

-

arc)

. . - ~ d- ]

(~1,

9

The direct interpretation of 5.4.5. now yields the following result. 5.4.6. THEOREM. For given bounds on the degrees of-fi, . . .&fin the formulation of the Nullstellensatz, there exist bounds po, po on the exponent off and on the degrees of the polynomialsgr which occur in the conclusion of the theorem (see 5.4.3.).

These bounds are independent of the particular choice of the field M and of its characteristic. Notice that in the above arguments we have accepted Hilbert’s theorem as a standard result of Algebra. We shall see later that the result as such also can be given a metamathematical interpretation. On the other hand,

128

DEFINABILITY

15.5.

the existence of the bounds po and can also be established by conventional algebraic methods, although the procedure is laborious. The method given here is effective in various similar cases, and another application of it will be discussed in section 8.5. in connection with the theory of definite functions.The idea of the method is related to the treatment of infinite disjunctions given in section 2.4.above.

5.5. Model-Completion. We showed in section 4.3. (Theorem 4.3.6.)that to a given set K there can be, up to equivalence, not more than one extension K* which has the same vocabulary as K and which is model-complete and model-consistent relative to K. Such a set K* was called the modelcompletion of K. We mentioned that not every consistent set of axioms possesses a model-completion. However, there exists an important class of sets K for which the existence of a model-completion can be checked by the application of a concrete test. In order to appreciate the scope of this result, which is given below, it is useful to recall (see Theorem 3.4.7.) that every consistent set of axioms Kis equivalent to a set of sentences K' of class A2 (sentences in prenex normal form in which no existential quantifier precedes a universal quantifier) if and only if K is a-persistent, i.e. if and only if the union of any monotonic increasing sequence of models of K is again a model of K. Moreover, if K is a-persistent then K' may be formulated within the vocabulary of K. 5.5.1. THEOREM. Suppose that the set of sentences K is non-empty, consistent, and a-persistent. In order that K possess a model-completion K* it is necessary and sufficient that for every primitive wff Q* (XI,. . .,x,) which is defined in K, there exists an existential wff Q (XI, .. .,xn) which is also defined in K such that for any model M of K and for any set of n individuals which are contained in M, a ~ .,. ., an, say, Q(a1, . . ., a,) holds in M if and only if Q* (al, . . ,a,) holds in some extension of M which is a model of K.

.

PROOF.The condition is necessary. Indeed, suppose that K possesses a model-completion K* and let Q * (XI, . . .,x,) be any primitive wff which is defined in K. Thus, Q* is an existential wff whose matrix consists of formulae of order one (bracketed atomic formulae) and of negations of such formulae. Let Q ( x 1 , . . ., x,) be the wff which exists, for the given Q*, according to the conclusions of 5.3.1.and 5.3.4. By 5.3.5.and 5.3.6., we may suppose that Q is existential. We propose to show that Q satisfies also the remaining condition laid down in the conclusion of 5.5.1. Thus,

5.5.1

129

MODEL-COMPLETION

let M be any model of K, containing elements al, ..., an, and suppose that Q* (al, ...,an) holds in an extension M’ of M which is a model of K*. Let M * be any extension of M’ which is a model of K*. Such a structure M * exists, by the definition of a model-completion. Then Q* (al, ...,an) holds also in M* since Q* is existential. It follows that Q(uI, . . ., an) holds in M. Suppose on the other hand that Q*(a,..., an) does not hold in any extension of M which is a model of K. Let M * be any extension of M which is a model of K*. Then Q* (al, .. .,an) cannot hold in M* and so Q(a1, . . ., an) cannot hold in M. Thus, Q satisfies the required conditions in relation to Q*, proving that the condition of the theorem is necessary. The condition is also sufficient. Suppose that it is satisfied for a set K which is non-empty, consistent and o-persistent. Then we construct the model-completion K* of K in the following way. Let J* = {Q:(XI, . . .,xk,,)}, k, 0 be the set of all primitive wff which are defined in K, and let J = (Q,,(XI, . . .,xk,,) } be a set of corresponding existential wff Q such as exist according to 5.3.1., 5.3.4-5.3.6. Any Qr E J may be written in the form.

...,

5.5.2. Qi(~1,

- .- ( 3 ~ k )R (XI, - .,,x*,,

XB,) = (3~1)

where R is free of quantifiers. For every Xv by

Qr

E

~ 1 *,

..,~ k )

J*,we define the sentence

Let H be the set of all sentences X,..Then we claim that K* = K U H is the required model-completion of K. Clearly, K* is an extension of K, with the same vocabulary. In order to show that K* is model-consistent relative to K, let M be any model of K. We have to establish the existence of an extension M * of M which is a model of K*.Put m = 2mo where

mo = max (1 K

I, I M I,

KO)

where K I and I M I are the cardinals of the set of sentences K and of the set of individuals in M respectively. Then the cardinal of J cannot exceed m. Let SObe the set of all sentences of the form Q*,(al, . . ., a,) where all .. .,a k , E M and Q,(al, . . .,m,,)holds in M. Let be the set of all models of K which are extensions of M and whose cardinal does not exceed m. Define A as the set of all ordered pairs arc= (crc,M,))

I

130

DEFlNABILITY

c,

15.5.

such that a, is a subset of SOand M , is in and such that every element X =Q : (al, ...,alcy) which belongs to a, holds in M p The set A is not empty since it contains the ordered pair (0, M) where 0 is the empty set. We introduce a relation < between (some) of the elements of A by defining that for any elements of A, a1 = (a, MA),a, = (a,, M,) a A < a, if art c a# and at the same time MA c M,. Moreover, if a~ = a, then we require M A = M , as well, i.e. in that case QA = a,. Thus we do not define < between any two pairs aa, aBsuch that UA = a, and M A c M , but MA# M,. We claim that a2 < a, d e h e s a partial ordering in A. Indeed, it is clear that a1 4 C ~ Aand if cxA < a, and a, < a~ then a,t = a,. Moreover, if CXA< aB and a, < % where a, = (av,M,) then 01 c av,MA< My. Suppose now that CA = av,then a~ = a, = a? and so M A= M , = M,, i.e. aa = a,. This shows that the relation is indeed transitive. Thus < defines a partial ordering in A. Now let A’ = {or,} be a totally ordered non-empty subset of A. Then we claim that 3 = (e, i@)= (U {a,}, U {M,}) is an upper bound and P

P

even a lowest upper bound for A’. Indeed, if X as above is a sentence which belongs to ti = U {a,} then it is contained in some particular a,. P

Accordingly, X holds in M y and, being an existential sentence, it holds in every extension of M,, e.g. in Moreover, &f is an extension of M and a model of K since K is a-persistent. Thus 11 E A. Now it is clear that for every a, = (oh,M,) E A’, a, c 5 and M c A?. However, in order to show that a, < rZ we still have to make use of the fact that a, = 5 entails M f i= i@. Now if a, = 5 then it follows that if a, < a, where ay is any element of A’, a, = (a,My), then a , = uv and so, by the definition of a totally ordered set, M , = M,.Hence = U (Mt) = M,, i.e. a, = or. Moreover 1 < m. 7 Accordingly, Zorn’s lemma is applicable and A contains at least one maximal set a1 = (01, MI). We propose to show that a1 = SO. Suppose on the contrary that Q*,(al, . . ., does not belong to 01 although it belongs to SO.Then QV(m,.. ., a,)holds in M , by the definition of SO,and since Q, is existential this sentence holds also in MI. It follows that there exists an extension M‘ of MI which is a model of K such that Qv (a,.. .,a k > holds in M’. A familiar argument shows that we may suppose I M’ 1 < m so that M ’ E Moreover, the sentences of a1 hold in M‘ since they are all existential. It follows that a’ = (a1 u {Q*, (al, . .., my)},M’) is an element of A and a1 < a’,a1 # a’.But this contradicts the assumption that aro is maximal and proves that 01 = SO.

a.

1.

5.5.1

131

MODEL-COMPLETION

Next we apply the same procedure to M I in place of M and obtain an extension MZof MI which is a model of K and that for all Qv(ul, ...,a d which hold in M I , the corresponding Qr(a1, ,..,ak,,) holds in Mz. Continuing in this way, we obtain a sequence of models of K whose cardinal does not exceed m,

M C MI Let M *

=

C

M2

C

... c Mn

C

... ,

U {Mn}. Then M* is a model of K since K is o-persistent. 12

We claim that moreover all sentences of H are satisfied in M*. Indeed, let X, E H, then we have to show that X , holds in M * or, which is the same, that

[Qv (Q . - ak,,) 3 Q f ( ~ 1 ., . .,ax$] holds in M * for all al, .. ., ak, E M*. Now Qv(m, . . ., u k v ) = ($1) . . . (3yz) Rv(a1, ...,a,,y1, ...,yz), where Rv is free of quantifiers so that if this sentence holds in M * then there exist individuals bl, .. .,bz such 5.5.4.

5

...,ax?,bl, ...,bz) holds in M*. But M* = u {M,} n so there exists some particular Ms such that al, ..., ax,, bl, ...,br belong to Ma and so the sentences 2, and, hence, Qv(al, .. ., uk,) hold

that 2, = RP’(a1,

in MS. But if so then Q*,(al, . . .,ax$ holds in M s + l , by construction. Again using the fact that Qi is existential, we conclude that Qf (a,, . .,ax,) holds also in M*. It follows that 5.5.4. holds in M*, as asserted. Since K* = K u H we have therefore shown that M * is a model of K*, and hence, that K* is model-consistent relative to K. In order to show that K* is also model-completerelative to K,we make use of a test for relative model-completeness proved earlier (4.3.1.). Applying this test to the present case, we have to show that if M is a model of K and Xis a primitive sentence defined in K then X holds either in all extensions M * of M which are models of K* or in none. Let then M be a model of K and let X be a primitive sentence which is defined in M. We may write X = Q* (a,...,ak) displaying the individuals that appear in M but not in K. By assumption there exists a wff Q ( X I , ...,x k ) which is defined in K such that the sentence 5.5.5.

(Vxi)

. .. ( V ~ k ) [ Q ( x .i ,. ., xx)

2

Q* ( X I ,

. . ., xk)]

belongs to K* and such that Q(u1, . . ., uk) holds in M if and only if Q* (al, ..., ak) holds in some extension of M which is a model of K*. Suppose that Q* (ui,. . .,a)holds in such an extension. Then Q (ul, . . .,ax)

132

p.5.

DEFINABILITY

holds in M and so Q*(al, .. ., a t ) holds in all extensions of M which are models of K * . This proves that K* is model-complete relative to K, as required. We observe that K* as constructed is also model-complete. For if M* is any model of K* with diagram N* then we only have to show that K* u N * is complete and this follows from the fact that K* is modelcomplete relative to K while M* is a model of K. The above construction of K* from K is canonical in the following sense. Whenever a set K possesses a model-completion K* at all then (ignoring the difference between equivalent sets of sentences) K* can be obtained from Kin the manner described. For some of the most important cases, e.g. for commutative fields and for ordered fields the corresponding model-completions were formulated in a simpler fashion. However, we shall now consider an example in which K* is obtained in a natural way by means of the above procedure. Let K = KDFbe a set of axioms for the notion of a diflerentialfield of characteristic 0. A differential field is a commutative field in which an additional operation is defined, which will be called A (x), such that for all a, b in the field

+

5.5.6.

d ( a b) = d a d (ab) = d (a)b

+ db + ad (b)

The reader is referred to Ritt 1950 for some simple notions and results on differential fields which will be used here. It is not difficult to formulate KDFwithin the Lower predicate calculus in terms of the individuals 0 and 1 and in terms of the relations E,S, P together with an additional relation A (x, y ) (read y = A (x)). For example, the two axioms of 5.5.6. can be expressed by

WY)(V.4 w>Wu) (Vw)[ A(x, r)

A A (z, 4 A s (x, 2, u,) A S(Y, u, w) = A ( % 4 3 (VX) (VY) (W (W (Vt)Vu)Wu) (W[ A (XY Y ) A A ( r , s) A P(x, r, t ) A P(y,r, u) A P(x, s, u) A S(u, u, w) 3 A ( t , w)] . A diferential polynomial p {XI, . . ., Xn} of the variables XI, . . .,X n with coefficients in a differential field M is by definition an ordinary polynomial with coefficients in M of the variables x(:), i = 1, . . .,n, k = 0, 1,2, . . . (where the x ( t ) , k 2 1 are interpreted as the derivatives of the x$') = XI). We evaluate p {XI, . . .,xn} at a point (al, . . ., an) with coordinates in an extension M' of M in the usual way, i.e. by substituting aa, a; . . . a(:), . . . for x(t),x(:), . . .,x(f),. . . . 5.5.7.

(VX)

5.5.1

133

MODEL-COMPLETION

Now let n 2 1, m 2 0, r 2 0, s 2 0, d 2 0 be a specified set of integers and let

--

--

Pl (Yl, * -,m>, -,p m (Yl, .Yn}, 41(YlY...,m},...,qri(Y1, ..., Y n )

5.5.8.

be a set of general differential polynomials of n variables and of order s and degree d with coefficients which are indeterminate and different for different polynomials in the set. The degree d of a differential polynomial is the joint degree of the y(F). The polynomial is general if all monomials up to the given order and degree actually appear in it. Denote the indeterminate coefficients which are involved in 5.5.8. in some arbitrary but definite order by z1, . . .,zk. For any set a = (al, . . .,uk) we denote the polynomials obtained by substituting uj for zj, j = 1, . ,, k, in a polynomial p~or qg by pi [a1 {ul, . . .,yn}, qi [a1 {YI, . . .,vn} . It has been shown by Seidenberg that there exists a set of differential polynomials of z1, . . .,zk: with integer coefficients

.

5.5.9. Prl

. . .,zk}, . .. Ptz, {ZI, ...,zk},

{ZI,

Qi {zi,

...,zt), i = 1, . . ., A

such that for every differential field M y and for every set of elements of M yul, ...,ak, the set of polynomial equations and inequations (i.e. negations of equations) 5.5.10. PI [a]

{rl,. ..,yn} = 0, . .., p m [a] {YI, .

41[aI {Yl,

-

*

-,Yn} # 0,

-

- 9

qr [a1 { Y l ,

9

* * *Y

~ n = } 0

Yn} f 0

possesses a solution in some differential field M' which is an extension of M if and only if for some i, 1 < i > s, the system 5.5.11. Pgl(a1,

. . .,L Z ~ = } 0, . . .,&i,

{ai, .

..,a k } = 0, Qt {UI, . ..,a k } # 0

holds in M. It is not difficult to deduce that the conclusion still applies if the coefficients of the pg, qf are not the indeterminates zi, . . ., zk themselves but certain polynomials of these indeterminates with integer coefficients. Now let Q* (z1, .. .,zk), k 2 0, be any primitive wff which is defined in K. Q*(z1,

..., Z~C)= (3~1)... @ ~ n ) R*

(Yl,

.-.,yn, z1, - - - ,Z~C)

where R* does not contain any further quantifiers. R* can be represented as a system of equations and inequations whose coefficients are certain (very simple) polynomials of z1, . . .,zk. Hence, by the result just quoted, there exists an existential wff of z1, . . ., Zk, Q(z1, . . ., zk), such that

134

DEFINABILITY

[5.5.

.

Q(a1, , ,,at) holds in any model M of K that contains al, . . ., at, if and only if Q* (al,. . .,at) holds in some extension M' of M that is a model of K. Q is obtained directly by formalizing 5.5.11. within the vocabulary of K. It is now easy to check that the conditions of 5.5.1. are satisfied, and we conclude that K possesses a model-completion K* = RDFwhich may be constructed as described above. The models of K* will be called di~eremtiallyclosedjelds. We noted already that the set K* obtained in this way is not only model-complete relative to K but also model-complete. Moreover, it is not difficult to see that K* is complete. For any model of K* contains the field of rational numbers, Mo, and Mo becomes a model of K if we define A (a) = 0 identically for all a E Mo. Let DObe the diagram of the structure MO modified by the inclusion of the relation A ( x , y), which corresponds to y = A (x). Then K* u Do is complete, by relative modelcompleteness, and so all sentences X which are defined in K* hold either in all models of K* or in none. Hence, 5.5.12. THEOREM. The concept of a differentially closed field is modelcomplete and complete.

We emphasize that the set of axioms K* does not single out any unique extension of a given differential field. On the contrary in general a given model of K will have many extensions which are models of K. We may ask whether to any given differential field M there exists a differentially closed field M * which is an extension of M and which is the smallest structure of this kind just as the algebraic closure of a given field is its smallest algebraically closed extension. However, it does not appear possible to define the notion of a closure (as distinct from a closed extension) for differential fields. The existence of differentially closed fields provides a quick proof of the following theorem. 5.5.13. THEOREM. Let M I and M Z be two extensions of a differential

field MOwhich have no elements in common other than the elements of Mo.Then M I and M Z can be embedded in ajoint extension M3. The somewhat elaborate wording of the theorem is due to the fact that we wish to exclude certain trivial anomalies. Unless we require explicitly that M I and M Zhave only the elements of Mo in common it might happen that two individuals, a and by are inchded in both M I and Mz and that a b = Oin M I but a b = 1 in Mz.

+

+

5.5.1

135

MODEL-COMPLETION

PROOFOF 5.5.13. Let M t and M*, be two differentially closed extensions of M I and M Z respectively, which have no elements in common other than the elements of M Oand let DO,DT,D*,, be the diagram of Mo,M:, M i , formulated as usual in terms of the individuals of the corresponding structures so that DO c DT and DO= 0:. We claim that K* U DT u D*, is consistent. Indeed, if this were not true, there would exist conjunctions of elements of DT and D*, to be denoted by Y I and Y2 respectively, such that K* u { Y1 A Y z } is contradictory. Write Y1 = Ql (al, . . .,am), Y2 = Q ~ ( a 1., . ., an), displaying the individuals of M*, and M*, which do not belong to Mo.Then

...

H = K* U DO U ((3x1) ( h n ) Ql(x1, (3x1) . (3xn) Q Z ( X I., .,xn)

..

.

..., xm),

is contradictory. But the sentence Z1 = (3x1) . . . (bm) QI (XI, . .., xm) is defined in K* u DO and holds in M*, and K* u DOis complete. It follows that K* u DO t Z1. Similarly K* U DO t Z Z ,where Z Z = (3x1) ... (h) Q2 (XI, . ., xn). Since K* u DO is consistent, it follows that H = K* U Do U {Zl, Z Z } also is consistent, a contradiction. Thus K* u D: u D*,is consistent. Any model M3 of K* U D: u D*,satisfies the conclusion of 5.5.13. It can be shown that 5.5.13. is not true for differential fields of characteristic p > 0. On the other hand, it will be seen that the method used in the proof of 5.5.13. actually applies to all cases which come within the framework of the general theory of this section. It follows that this theory cannot be valid for differential fields of characteristic > 0. As another application, we shall now prove a theorem on the specialization of parameters in differential fields.

.

.

5.5.14. THEOREM. Let MO be a differential field and let Q ( y i , . .,yn) n 2 1, be a predicate which is defined in Mo.Let M* be a differentially closed extension of MO and let al, . ..,an be a set of elements of M * such that for any non-vanishing differential polynomial q(y1, .. .,y n } with coefficients in Mo, q{m, . . ., an} is different from zero. (The elements al, . . ., an are then said to be differentially independent over M ) . Suppose that Q (al, . . .,an) holds in M*. Then there exists a non-vanishing differential polynomial 4{y1, . .. ,y n } with coefficients in Mo such that for any set bl, ...,bn in any differentially closed extension M of MOthe predicate Q (bl, , . .,bn)holds in M , provided q {bl, . . ,bn} # 0.

.

PROOF.Let M I be defined by adjoining n differential indeterminates

136

[5.5.

DEFIN A BILITY

c1, . . ., cn to MOand let M2 be obtained by adjoining U I , . . ., an to Mo in M*. Then M2 c M * and M2 is differentially isomorphic to M I by an isomorphism which leaves the elements of MO fixed while mapping uc on cr, i = 1, ...,n. Let M : be any differentially closed extension of M I . Then there exists a differentially closed extension M z of MZ which is isomorphic to MT by an extension of the specified isomorphism between M1 and Mz. Now the sentence Q(u1, . . ., un), which is defined in M2 holds in M*, which is a model of K*, and K* is model-complete relative to K. It follows that Q (UI, ...,an) holds also in M : and, in view of the isomorphism between M : and M ; , that Q(c1, . ..,cn) holds in M:. Again making use of the model-completeness of K* relative to K, we conclude that Q (c1, . . ., c,) holds in all differentially closed extensions of M1. Now let d1, , . .,dn be any set of differentiallyindependent elements in a differentially closed extension M*, of Mo. Then Q(d1, .. ., dn) holds in M : for if M*, satisfied the sentence Q(d1, .. ., dn) then a repetition of the above argument would lead to the conclusion that Q (c1, . ..,cn) holds in all differentially closed extensions of M I , and this is impossible. For any non-vanishing differential polynomial p ( y l , . . .,v n } with coefficients in MOlet Q p { y l , . . .,yn> be a predicate which is defined in M and which affirms that p (yl, ...,yn} = 0. Let el, ...,en be a set of individuals which are not contained in M and let R be the set of sentences Qp(e1, . . .,en) where Q p varies over all the predicates that were defined just now. Let DObe the diagram of MOand put H = K* u DO u R. Then Q (el, . . ., en) must be deducible from H. Indeed, the models of H are the differentially closed extensions of M O in which the elements el, ..., en are differentially independent over MO and we have shown that Q(e1, ..., en) holds in all such structures. It follows that Q(e1, ..., en) is deducible from K u DOtogether with some finite subset R' of R. Let R' = {- Qpl(el, . . ., en), . . ., Q p j (el, . . ., en)} then we have proved that N

N

N

-

5.5.15.

K* U DOI-

N

.

Q p l ( e l ,. . en)A . . . A 3 Q(e1, ...,en)

Letq (yi, . .., ~ n = } pi

(~i,

N

Qp,(e1, . . . , e n )

.. . , Y ~ } P z( ~ i , . . .,yn} . . . p j ( ~ i ,. .. , ~ n } .

Then 5.5.15. entails and so

K U Do

I-

N

Q p ( e i , ..., en) 3 Q(e1,

...,en)

5.6.1

5.5.16.

137

PROBLEMS

K* U Do k (V-YI) .

-.

[

(v~nN ) Qp(vi,

. - .,~

n 3 )

Q ( ~ 1 , - .,~

.

n ) ]

This is equivalent to the conclusion of 5.5.14. 5.6. Problems 5.6.1. Derive Beth's theorem 5.2.1. directly from the consistency lemma

5.1.6. (Robinson 1956a).

5.6.2. Prove the following generalization of Beth's theorem.

Let the sentences X(F1, .. .,K , GI, .. ., G,) be formulated in terms of relations FI, . . .,Fz, G I , . . .,Gm, and possibly, in terms of a number of individuals which do not appear explicitly in this notation. Let X(F;, . . . FI, G I , . . ., Gm)be obtained from the original sentence, by replacing the relations FI, ..,Fl respectively by new relations F;, . . .F; with the same number of variables. Suppose that k Z 3 Y where

.

..

...

Y = @ X I ) . ( V X ~ )[FI( X I , ,x k ) fI;; (xi, ...,xk)] 2 = X(F1, . . ., Fz,G I , . . ., Gm) A X ( F i , ., FI, G I ,

..

. . ., Gm) .

Then there exists a wff Q ( x i , . .., xk) which contains only individuals and relations from X,excluding FI, . . ., Fz such that tX(F1, ...,Fz,Gi, (Craig 1957a). 5.6.3. Let

...,Gm) 3 [ ( x i ) . . . ( x k ) [ F i ( ~.--,xIE) , 3 Q(x1,...,xd>]]

+ + + ... + xj d + + . + xn ~ ~ - 3 - l

p ( z ) = xo X I z XQ22 q(2) = ~ j + l xj+z z ..

where j and n are specified positive integers. Formulate a predicate Q* ( X I ,...,xn) in terms of the relations E, S, P,which states that there exists a z such that p ( z ) = 0, q(z) # 0. If K is a set of axioms for an ordered field and K* is a set of axioms for a real closed ordered field, use standard real Algebra in order to construct a predicate Q (xi, . ..,x,) which satisfies the conclusion of Theorem 5.3.4. 5.6.4. Prove that the concept of a formally real field does not possess any

model-completion.

References. The consistency lemma 5.1.6. was given in Robinson 1956a in order to provide a semantical proof of Beth's theorem 5.2.1. Beth first stated his theorem in Beth 1953, compare also Beth 1959a. Craig proved his lemma by a syntactical method in Craig 1957, 1957a and also applied it to the proof of Beth's theorem. The fact that 5.1.6.

138

DEFINABLITY

15.6.

entails 5.1.8. has been pointed out to the author by several logicians from Warsaw and Berkeley. The theory of relative definitions is taken from A. Robinson 1957 which includes also the analysis of the Nullstellensatz in 5.4. 5.4.6. was proved previously in Robinson 1955. It shows that PO, POmay be chosen as general recursive functions of n and of the given bound onfl, ...,fi,f. This was improved by Kreisel who proves that the functions in question may be made primitive recursive (Kreisel 1958). The theory of model-completion and its application to differential fields is in A. Robinson 1959. The basic work on Differential algebra is Ritt 1950, compare also Kaplansky 1957. For some other algebraic results used in this section see Seidenberg 1952 and Seidenberg 1956.

CHAPTER VI

GENERALIZATION OF ALGEBRAIC CONCEPTS

6.1. PoIynomiaIs in General Axiomatic Systems. There are various concepts in Algebra which are based on certain fundamental ideas whose scope, though vague, would appear to be more general than appears from the concrete definition. Among these we may mention in particular the concept of a polynomial ring of n variables adjoined to a given commutative ring; the concept of a free group; the concept of an algebraic number; the concept of an ideal. Moreover, it happens quite frequently, that a single concrete concept is the embodiment of several fundamental ideas. Such a concept may then split into several notions in a more general situation, or may be given different interpretations within different frameworks. In the present chapter we shall consider some of the concepts mentioned above, as well as some others, and we shall formulate corresponding definitions within the framework of general sets of axioms of the Lower predicate calculus. The concept of an ideal, which is particularly important, will be considered in a separate chapter. There is another theory which provides a natural setting for the generalization of certain algebraic concepts. This is the theory of universal algebras, also called Birkhoff algebras. In fact, the concept of a group comes under this heading, and the notion of a free algebra is the most natural generalization of the notion of a free group. On the other hand, it can be shown that the field concept cannot be subsumed under the theory of universal algebras. Accordingly, various notions of field theory which can be generalized within the framework of our present work have no counterpart in the theory of universal algebras. However, even in cases where both methods provide reasonable generalizations, the special nature of of the metamathematical approach is apparent in that it makes explicit use of the notion of deducibility. This is true regardless of the fact that deducibility can be redefined in model-theoretic terms. The first concept which we shall formulate for a general set of axioms in the Lower predicate calculus is that of a polynomial. We shall assume throughout this section that a relation of equality is included. Note, that

140

[6.1.

GENERALIZATION OF ALGEBRAIC CONCEPTS

such a relation can be added to any given set of axioms if it was not there from the outset. The concept of a polynomial in Algebra, e.g. of a polynomial in one variable, p (x), in a commutative ring R is fundamentally different from the corresponding notion in Analysis. The natural starting point in Analysis is the general concept of a function as a correspondence between the independent and dependent variables. By narrowing down the class of functions under consideration gradually, one finally arrives at the polynomials as the elements of the class of entire analytic functions whose singularity at infinity, if any, is a pole. It is true that even in Analysis this characterization of polynomials is usually preceded by a defining equation of the form 6.1.1.

y = p ( x ) = a0

+ a1x +

a29

+ ... + anxn

Nevertheless, the fundamental concept is that of a function as a correspondence, and equation 6.1.1. is only auxiliary to it. It may be looked upon as a rule for finding the value of the dependent variable, given the independent variable. On the other hand, in Algebra, the polynomial is taken to be given by an expression such as the right hand side of 6.1.1. and the expression as such is of cardinal importance while the function defined by 6.1.1. carries less weight. For instance, the Polynomials p ( x ) = 1 x2 and q (x) = 1 - x have identical functional values for all argument values in the prime field of characteristic 2, yet they are regarded as different polynomials. The question then arrives how to bring polynomials within the accepted order of mathematical objects. One method of doing this is to conceive them as vectors (i.e. ordered sets) of the elements of the original ring of arbitrary finite length. Thus, p (x) is represented by (ao, al, . . .,an),the indeterminate x is the vector (0, l), and there are suitable rules for operating with these vectors. This method, inasmuch as it is successful is of course unexceptionable, but nevertheless, it would appear to be a deviation from the original algebraic idea of a polynomial as a rule of construction, or operation. Such a rule of construction belongs, properly speaking to the formal language within which the ring is discussed. Accordingly, we shall represent them by certain predicates R (x,y ) which give rise to descriptions, i.e. which are such that the sentences (Vx)(3y) R ( x , y ) and (Vx)(Vy)(Vz) [ R ( x , y) A R ( x , z ) =) E ( y , z)] are deducible from the given set of axioms.

+

6.1.1

141

POLYNOMIALS IN GENERAL AXIOMATIC SYSTEMS

Let K be a consistent set of axioms in the Lower predicate calculus, which includes a relation of equality, E ( x ,y). We shall consider predicates of n 1 variables xi, . . ., x n , y, n 2 1, R ( X I , . . .,Xn, y). The particular position of the variables is important, in the present context, e.g. it is not permissible to interchange R ( X I XZ, , y ) and R ( X Z , XI, y ) . R (XI, . . .,Xn, y ) will be called a prepolynomiul in K if the sentences

+

6.1.2.

(VXI . .. ( V x n ) ( 3 ~ R) (XI, . . .,Xn, y ) (Vxi) . . (VX,) (VY) (VZ)[ R(XI, . x n , y ) A R (xi, = E ( Y , 41

-

-,

-9

xn, Z)

are deducible from K. For n = 0, there are no quantifiers preceding that which involves y. From among the prepolynomials in K, we select a subset whose elements will be called polynomials. They are defined as the prepolynomials which are of the form 6.1.3.

(321)

.

@zm)

Q (zi,

..

a ,

zm,

xi,

. . .,Xn, Y )

where the matrix Q does not contain any quantifiers, and involves the connective of conjunction only. It is understood that the ZI, . . .,z m are arbitrary variables other than the specific XI, . .,Xn, y, which were selected in advance. For n 2 1, the set of polynomials (and hence the set of prepolynomials) is not empty since it contains the elements

.

6.1.4. R ( x ~.,. ., x n , y ) = [E(xl, XI) A E ( x ~ ~, 2 A) A . . . A E ( x n , x n ) ] , i = 1, 2, . .., n .

-

A E(X$,y )

In fact, for a predicate of this kind, the sentences 6.1.2. are deducible from the axioms 2.1.1. and hence, from any set of axioms which includes a relation of equality. For n = 0, the set of prepolynomials may be empty but we shall exclude this possibility from our considerations. For a given n, n 2 0, let C n be a set of individuals which are in one-toone correspondence with the prepolynonlials R ( X I , . . .,Xn, y ) in K, R t)CR. We define a structure M n whose set of individuals is the set C n in the following way. For any relation A ( y i , . . .,yk), k 2 0, which is contained in K , and for any set of prepolynomials in K, R I , .. ., Rk, with corresponding individuals ci, . . ., C k in C n , the relation A (ci, .. ., ck) shall hold in M n if and only if 6.1.5. (Vxl) . . . (VXn) (VJ’l) . . . (vyk) [Ri (Xi, . . ., Xn, J’i) A Rz (xi, . . .,Xn, Yz)A . . A Rk (Xi, .. ., Xn, yk) 2 A ()’I, . * yk)]

.

-

142

[6.1.

GENERALIZAnON OF ALGEBRAIC CONCEPTS

is deducible from K. M will be called the prepolynomial structure of order n over K. By restricting the relations of Mn to the set C i of individuals which correspond to polynomials over K, we obtain a substructure M i of Mn which will be called the polynontial structure of order n over K. We may also introduce the prepolynomial structure M,, by putting C , = U Cn where the Cn are supposed to be disjoint. We then define n

.

that A (c1, . .,c k ) holds in M,, where cz E C, corresponds to Ri ( X I , xn,, y ) , if and only if

6.1.6.

.-

[RI(xI,

- 9

xlt,, ~

1

( V X I ) . . . (Vxn)(Vy~) .. . (Vyk) A ) . . A &(XI, . ., xnk, yk) 2 A ( y 1 ,

-

.. .,

- . ., yk)]

is deducible from K, n = max (nl, . . .,nk). M L is obtained, as before by restricting M , to the set of individuals which correspond to polynomials in K . Similarly, it is not difficult to define prepolynomial and polynomial structures for a set of variables of arbitrary cardinal number. For n > m, Mn includes a substructure M z which is isomorphic to Ma. The individuals c* of M: are obtained as follows. Let c E m correspond to the prepolynomial R (XI, . . .,X m , y), then we map c on c* E Cn which corresponds to the prepolynomial E(xn+l, X n + l ) A . . . A E(xnl, X m ) A R ( X I , . . .,Xn, y). Similarly M i includes a substructure which is isomorphic to MA. It is evident that if K and K' are two equivalent sets of axioms containing the same relations and individuals then their corresponding polynomial and prepolynomial structures coincide. For given n, the relations of K are all contained in Ma.We may ensure that the individuals of K, if any, are also contained in M by proceeding as follows. For any individual a in K, consider the predicate &(XI,

...,Xn, y ) = E ( x I , XI) A . .. A E(xn, xn) A E ( a , y ) .

It is not difficult to verify that this is a polynomial in K . We shall suppose that the corresponding individual in C is precisely a. On this assumption, the sentences of K are all defined in Ma. The question now arises whether M nis actually a model of K. If this is the case for all n then K will be said to be pretransitive. Similarly, the sentences of K are defined in M i and if they all hold in M i , n = 0, 1, 2, . . . then K will be called transitive. It is not difficult to construct examples which show that not every K is transitive or pretransitive. For example K = KCF,a set of axioms for the concept of a commutative field is not transitive or pretransitive (See

6.1.1

143

POLYNOMIALS IN GENERAL AXIOMATIC SYSTEMS

example 6.6.1. at the end of this chapter). However, it is possible to specify important classes of sentences which hold in M n or M i provided they are deducible from K. 6.1.7. THEOREM. A sentence X which is defined in, and deducible from K and which belongs to one of the classes specified below, holds in M. 6.1.8. X is a sentence of order one (a bracketed atomic formula), or the negation of a sentence of order one, or a (repeated) conjunction of sentences of order one.

X=qlqz . . . qm Q ( ~ 1 ,...,zm),

6.1.9.

m=0,1,2,

...

where qi is a quantifier involving zt, i = 1, . . .,m yand where the matrix Q does not involve any quantifiers, and is either a formula of order one or a (repeated) conjunction of such formulae, or an implication in which both implicans and implicate are formulae of order one or are (repeated) conjunctions of such formulae. Moreover, for any qt which is an existential quantifier, the following sentence also is supposed to be deducible from K, 6.1.10.

[Q (z1,

,

--

9

---

-

41-1 (VzJ Wz;’)qt+1 . qm X = q1qz ,, zt-I, z;, ~ t + l , - ,zm) A Q (z1, . .,zt-1, zd ,zt+1, . 3 E(z{,z;‘)] I

.

.

-

..,zm)

In this sentence, the quantifiers qt+1, , ., qm are defined as before while the qj, j = 1, . .., i - 1 though still quantifying the respective zj, are all universal. We note that the classes 6.1.8. and 6.1.9. overlap. The proof of 6.1.7. for some of the cases mentioned is somewhat laborious and we shall give it here only for the first two cases detailed in 6.1.8. Thus, suppose that X=A(a1,

...,a,)

where al, . . ., am are individuals and A is a relation of order m, all contained in K. The individuals al, . . .,a m correspond to prepolynomials Ra,(XI,. . .,xn,y), . . . Ram(XI, . . .,Xn, y ) as defined above. In order to show that X holds in M n we have to verify that the sentence (VXI)

... (Vxn)(VYI) - - (Vysn)[Ral(XI, - Xn, ~ m 3 )

A (YL

9

xn, YI) A

- - .,ym)]

- - - A Ram(XI,- -

is deducible from K. This sentence is, written out in more detail

9

144

[6.1.

GENERALIZATION OF ALGEBRAIC CONCEPTS

6.1.11. (VXl) . . . A E(ai, A [ ... A

..

and is clearly deducible from K , since it is deducible from A (al, .,am) together with the axiom of substitutivity 2.1.2. On the other hand, if A (al, . . ., am) is deducible from K, then 6.1.11. cannot be deducible from that set since K is supposed to be consistent. It follows that A(ai, . . ., am) holds in Mn. N

N

6.1.12. THEOREM. A sentence X which is defined in and deducible from K holds in MA if it belongs to the class of sentences 6.1.8. X also holds in M’ if it is of the form

x = (V.21) . .. (Vzm)

6.1.13.

Q (ZI,

. . .,zm)

where Q is of the same description as in 6.1.9., or if it is of the form 6.1.14. X

= (Vzl)

.. . (Azm-l) (3zm)Q (ZI, . . .,zm),m

=

1,2,

.. .

provided Q is of order one, or a conjunction of formulae of order one and the sentence

. . . (Vzm-1) (Vz;) Zm-l,~,) = E(z;,z;)]

6.1.15. (Vzl)

(VZJ

[Q(zi, . . ., zm-1,

z;)

A

Q(Z1,

. . .,

also is deducible from K. It will be seen that this result is somewhat weaker than the corresponding result for prepolynomial structures. Again, the proof will be omitted. Although the classes of sentences defined by theorems 6.1.7. and 6.1.12. are of a rather special nature, a simple check shows that they establish transitivity and pretransitivity for the sets of axioms for groups and for abelian groups, and for general rings and for commutative rings. Moreover, if K is one of the sets of axioms just mentioned, and M is a model of K with diagram D,then K u D is transitive and pretransitive. Indeed, the elements of D all belong to the class 6.1.8. and so M n and M i are actually extensions of M for all n. More precisely, let K = KZR, a set of axioms for the concept of a commutative ring with unit element 1, and let M be a model of KZR with diagram D.K:B may be obtained by adding to KCR,the set of axioms for a commutative ring defined in section 2.2., the sentence ( V 4 p (1, x, 4 *

6.1.1

145

POLYNOMIALS IN GENERAL AXIOMATIC SYSTEMS

Then we propose to show that for n 2 1, M i is isomorphic to the polynomial ring of n variables adjoined to M , M [XI, . .,xn], as defined in ordinary Algebra. Let q(x1, . . ., xn) be an arbitrary element of M[xl, . . ., x n ] , i.e. a polynomial with coefficients in M , in the algebraic sense. It is then not difficult to select a predicate R , (XI, . . ., X n , y ) which is defined in K and which is a formal statement of the relation y = q(x1, . . ., x,). To see this it will be sufficient to consider the example y = q (XI) = ao UIXI azxf where ao, al, a2 belong to M. A suitable choice for the corresponding R , ( X I , y ) then is

.

+

6.1.16.

+

(321) (322) (323) (324) [P(xI,XI, ZI)A P(az, ZI,z2) A A P (ai, XI,z3) A S (ZI,23, ~ 4 A ) S ( ~ 4 ,ao, Y ) ]

R, (Xi, y ) =

R , (XI, y ) is a prepolynomial in K U D,more particularly R , (XI,y ) is a polynomial in K u D in the metamathematical sense. We denote the corresponding element of M i by c,. The same procedure can be adopted for all other elements of M [XI,. ..,x,]. We may suppose that the constant polynomials in M [ x I ,. . ., x,] - which may be identified with the elements of M - correspond to themselves as elements of M i . Conversely, we propose to show that every element c of M i is equal, within M i , to some c,, q E M [ x l , ..., xn]. Indeed, c corresponds to a metamathematical polynomial R (XI, . . ., Xn, y ) in K. Let R (XI, . . .,~

n y ,) = (321)

.. . (32,)

Q (ZI, . . ., Zm, XI, . . .,~

n y) ,

where Q is free of quantifiers and is obtained by the repeated conjunction of formulae of order one,

. . , z n , y ) AAz(z1,. . . , ~ n , y ) A ...~ A . ~ ( z I ,. ,.z n , y ) where the variables z1, . . ., xn, y need not however be contained in all

6.1.17. Q = A I ( z I , .

the At. Since R is a (metamathematical) polynomial in K U D,it follows that the sentence (VXI) . . . (VXn)(3y) R (XI, . . ., x n , y), i.e.

. . . (Vxn) ( 3 ~(321) ) . . . ( 3 z m ) Q (21, - . zm, XI, . . ., Xn, y) is deducible from K u D.Let M' = M [ d ~.,. ., dn]be the ring obtained by adjoining the indeterminates d ~.,. ., d n to M. Then M' (which is isomorphic to M [ x l , . . ., x,]) is a model of K u D and hence, satisfies 6.1.18. (VXI)

a ,

6.1.18. It follows in particular that

(3~)(321) . . . (32172) Q (ZI,. . .,zm,

d1,

. . .,dn, Y )

146

[6.1.

GENERALIZATION OF ALGEBRAIC CONCEPTS

holds in M'. That is to say there exist elements of M' , i.e. polynomials of d1, . . ,, dn with coefficientsin M yy = qo (di, . . .,dn), zj = qr(di, . ..,dn), such that

Q (qi (di,

. . .,dn), . . .q m (di, . . .,dn), di, .. .,dn, 4 0 (&, - - -

9

dn))

holds in M'.Referring to 6.1.17., we see that this implies that

. . .,dn),di, ...,dn,qo(di, - - . y d n ) ) i = 1,2, . . . s. At is one of the relations

6.1.19. Aa(qi(d1,. ..ydn), ...,qm(d1,

holds in M [ & , . .., dn] for E, S,or P, and in general only some of the dt, 40, and qt are contained in a given At. In addition, At may contain one or more of the elements of

M.

We are going to prove that c is equal to cq, according to the relation of equality in M'. By definition, this will be the case if K u D t ( V X l ) .. . (Vxn) (Ay)( V Z ) [ R h , . . .,xn,u) A Rq, (xi, . . .,x n , 4 3 E(u, 41

Since R and Rqoboth are (metamathematical) polynomials it is sufficient to prove instead that 6.1.20. (Vxi)

.. . (Vxn)( 3 ~[ )R(XI, . . .,X n , u) A Rg, (XI, . . .,Xn, y ) ]

holds in all models of K u D. Now 6.1.19. holds for indeterminate d1, . ..,d,, and so, by the principle of substitution in rings, it holds also if we replace d1, . .., dn by arbitrary elements of any ring which is an extension of M . Thus, if M' is a model of K U D and bi, . .., bn are elements of M ' , then Ae(qi(b1, . .., bn), .. ., qrn(h, .. ., bn), bi, . . ., bn, qO(bi, . . ., bn)) holds in M' for i = 1,2, . . ., s. It follows that

.

( 3 ~ 1.). . ( 3 ~ m Q) (21, . ., zm, bi,

-.

- 9

bn,

qO(bi,

.-

- 9

bn))

holds in M',and this is R (bl, ...,bnyqo (bi, ...,bn)). At the same time Rqo(bl, . ..,b,, qo(b1, . ..,bn)) holds in M', by the definition of R,. Hence . . . bn, Y ) ] ( 3 ~ [R ) (bi, . . bn, V ) A Rqo

-

holds in M'. But bl, . . .,bn are arbitrary elements of M' and so 6.1.20. holds in M', as required. This proves that E(c, cq,) holds in M i . Now let q1 (xi, . . .,x,) and q 2 ( X I , . . .,x,) be two dzrerent elements of the polynomial ring M [ X I ,. . .,x,]. Then the sentence

6.1.I

147

POLYNOMIALS IN GENERAL AXIOMATIC SYSTEMS

(Vxd

---

---

(Vxn) (VY)(VZ) [Rql(XI, A Rqa(X1, -,Xn, z) = E(Y,

--

$1 7

xn, Y )

is not deduciblefrom K u D sinceit does not hold in the ring M [XI, .. .,x,] which is a model of K u D. It follows that E(cq1,cqJ does not hold in M i , cql and cq, are different in M’. Finally, it is not difficult to show that if q1 and q2 are any two elements of M [XI, . . .,xn] and Rql, Rqa,Rql+qn, Rqlql are the metamathematical polynomials corresponding to 41, q2, q1 q2, q1q2 respectively, and cql,cga,cql+q,,cglg,are the corresponding elements of M i then the relations S(cgl, cq,, cql+q,)and P(cqI,cp,, cglq,) both hold in M i . This shows that M i is isomorphic to M [XI, . . .,x,]. By contradistinction M , is not related directly to any standard mathematical construction. The investigation of its structure might be of mathematical interest. Now suppose that K is a set of axioms for the notion of a general (non-commutative) ring with unit element, and that M is a model of K and D its diagram. Then the structures M , and M i defined with respect to K U D are rings with unit elements which are extensions of M. However, if as is done sometimes, we define the ring of polynomials M[xI, .. .,xn] over M as the set of terms

+

-where it is supposed that the variables X* commute with the elements of M and with each other - then M i is not isomorphic to M [XI, . . .,x,]. In fact it may now be quite difficult to characterize MA algebraically. The assumption that c1, . . ., C n belong to the center of the ring is certainly not satisfied in all cases, where c1, . . ., c, are the elements of M i which correspond to the metamathematical polynomials E(x1, XI) A

. .. A E(xt, y ) A ... A E(x,,

x,),

i = 1,

. .., n.

Whether the two elements bl, and bz of M i , which correspond to the metamathematical polynomials P (a, XI, y ) and P (xi, a, y) are properly to be regarded as equal or different in M‘, for a in M depends on whether or not.

6.1.21. K U D i- (Vxi) (Vy) (Vz) [P(a, X I , ~ A ) P(xi, a, z)

2

E ( y , z)]

.

This is not necessarily the same as ax1 = x l a identically for all XI in M , and it may be difficult to decide by elementary means whether or not 6.1.21. holds. bl and b2 equal the products acl, and cla in M i , and so

148

[6.1.

GENERALIZATION OF ALGEBRAIC CONCEPTS

acl = cla if and only if 6.1.21 is true. Although this shows that the validity of ac1 = cla in M i depends on rather complicated considerations, M i and more generally M i appears to be a very natural realization of the notion of a non-commutative polynomial ring. Another interesting case is obtained by taking K = K& for the notion of a commutative ring with unit element, without reference to any particular model. For this case, M i turns out to be isomorphic to the ring of integers. Indeed, as a metamathematicalpolynomial corresponding to 0 we may take S ( y , y , y ) while for a positive integer we have the predicate Sn (1 ,y ) as defined by 2.2.10. and transformed into normal form in a suitable way. For any negative integer, - n, n > 0, we may take the predicate (W(W [S” (LZ) A S ( y , ZY 4 A S(w, w,

4

again transformed into normal form in the obvious way. It is then not difficult to show that the elements of M i which correspond to the above metamathematical polynomials constitute a ring which is isomorphic to the ring of integers and, moreover, that any other element of M i is equal to some element of this ring. We might even define the ring of integers as the ring M i . Taken with respect to the same set K, the structures M i , n 2 1, are isomorphic to the rings of polynomials of n variables adjoined to the ring of integers. Returning to the general case, let Mn and M i be prepolynomial and polynomial structures over a set K,and let M be a model of K. We may then specify certain substructures of M which are homomorphic to Mn or to M i in the following way. For given n, select elements a, ..., a , of M arbitrarily, and let R ( X I , . . .,X n , y) be a prepolynomial in K.Then the sentences (3y) R (a,. . ., an, y ) and

(b)(VZ) [R(ai, . . ., an, Y ) A R ( ~ I. , - ,an, 4 3

E(yYz ) ]

both hold in M. Thus, except for equal objects there is a unique b in M which satisfies R(a1, . . .,an, b). Now to every element c of Mn there corresponds a prepolynomial, and to every prepolynomial, there corresponds an element b of Myas defined just now, c +-R -+ b, or omitting R,c +b. Let M * be obtained by restricting the relations of M to the elements b which appear in this correspondence. Then we claim that M* is a homomorphic image of Ma. Indeed, suppose that A (a,. . ., c), holds in M for some relation A and let R1, . ..,R , be the prepolynomials

6.2.1

149

BOUNDED PREDICATES

which correspond to c1, in M*. Then

. ..,Cm, and 61, . . .,b ,

the images of c1,

. . ., cm

Rl(a1, . . ., an, bl) A . . . A A Rm (al, . . ., an, bm) 3 A (bl, . ..,bm) holds in M . But the implicans of this sentence holds in M , by the choice of the bt, and so A(b1, . . ., bm) holds in M , showing that M* is a homomorphic image of Mn, as required. It follows that a certain substructure M’ of M * , i.e. the image of M i in the correspondence defined above, is a homomorphic image of M i . If n = 0 , the preliminary choice of elements al, . . ., an is redundant. Now suppose that K is transitive so that M i is a model of K. Define the (metamathematical) polynomials Rx, ( X I ,. .,Xn, y ) by

.

Rx,(xI, . . -,xn, y ) = E(X1, X I ) A E(xz, ~ 2 A) .. . A A E(Xk, y ) A . . A E(xn, Xn), k = 1,2.. .,n

.

This notation is consistent with the definition of the metamathematical polynomials R,(xl, . ., X n , y ) introduced earlier (compare 6.1.16.). Let Ck be the element of M i , and of Mn, which corresponds to Rxk.Now take M i as the structure M of the procedure outlined above, and choose the Ck, as elements of M = M i as the a k of that procedure. This yields a homomorphic mapping of Mn into M i . Moreover, it is not difficult to check that the mapping is onto and that it maps the elements of M i as elements of Mn on themselves.

.

6.2. Bounded Predicates. The vanishing of a polynomial, in particular of a polynomial of one variable imposes a condition on its argument. Thus, if a polynomial of one variable, which is not identically zero, with coefficients in a field M vanishes for a particular value a of the argument then a is algebraic with respect to M. In the following sections we shall analyze some metamathematical aspects of this condition. Let K be a consistent set of axioms which includes a relation of equality, E (x, y ) . A predicate Q ( x ) which is defined in K will be called bounded in K if for some positive integer m the sentence

150

I6.2.

GENERALIZATION OF ALGEBRAIC CONCEPTS

..

6.2.1. X m = (Vxl) . (Vxm+l) [ Q ( x ~A) . . . A Q ( x ~ + I3 ) E ( x I ,XZ) V E(x1, x3) v v E(x1, X m + l ) v E(xz, x3) v VE(xm,xm+r)]

...

...

is deducible from K. The smallest rn for which K k Xm will be called the degree of the predicate, m = deg Q(x). However, if (Vx) [- Q(x)] is deducible from K we put deg Q = 0. We may also define the concept of boundedness for predicates of n variables, n > 1. A predicate Q (XI, . . .,x,) which is defined in K is said to be bounded in K if for some positive integer rn

.

...

.

6.2.2. K I- Xm = ( V X ~.). . (VXA)(V’X;) . . (Vx,Z) (VX?+~) . .( V X ~ + ~ ) , xz+1) 1 ,?(xi, x t ) A [Q (xi, . ., x;) A . . A Q (xy+1, A E(xi, xi) A . . . A E(xA, xz) V E(xi, x!) A . A E(x:, x i ) V V . V E ( x y , xy+1) A A E(xE, x?+l)]

.

.

..

...

...

..

The smallest rn for which K I- Xmis again called the degree of Q, deg Q, and if @XI) .. . (Vx,)[- Q(x1, . . ., x.)] is deducible from K we put deg Q = 0. In this case also, we regard Q as bounded. It is not difficult to verify that the conjunction and disjunction of two bounded predicates over a given K are again bounded, and 6.2.3. deg [Ql<x1,

. ..,x,)

6.2.4. deg [Ql ( x .,

. ..,x,)

A Q~(x1,. ..,x,)] < min (deg QI ( X I , deg Qz(x1, . . ., x.)]

. . .,xn),

V Qz (x1, . . .,x”)] < deg Q l (XI, . . .,x d deg Qz(x1, ..., x,) .

+

Also, the conjunction of a bounded predicate Ql and an unbounded predicate Q2 with the same variables is again bounded and its degree cannot exceed the degree of Ql. Now consider the following model-theoretic property for predicates Q(x) of one variable which are defined in K. We shall say that such a predicate Q (x) is saturable in K if for any chain of models of K, 6.2.5.

hf1 C kf2

C

hfs

C

... C

Mk C

...

and any sequence of individuals {a,} where an belongs to Mn,and Q(ak) holds in Mn for all k < n, there exists an no such that for all n > no, a, equals some am, in < no, in other words, the number of different elements in the sequence {a,} is finite. 6.2.6. THEOREM. A necessary and sufficient condition for a predicate Q ( x ) to be saturable in K is that it be bounded in K.

6.2.1

151

BOUNDED PREDICATES

PROOF.The condition is sufficient. Let m

= deg Q. If Q is not saturable then there exists a chain 6.2.5. such that for sufficiently large n, the set (al, . . ., a,} contains at least m 1 different individuals. But the sentences Q (US) hold in Mn, for i = 1, ...,n. and this yields a contradiction. To prove that the condition is also necessary, suppose that Q is not bounded in K. Then we have to show that it cannot be saturable in K either.

+

6.2.7.

H

=K

u {-

Xl,

-

x2,

-

...) x,, ...}

where the X , are defined in 6.2.1., must be consistent. For otherwise, K u {- Xl, ... X,} would be contradictory for some m. But K t XZ3 X , for 1 Q m and so K k X , 3 Xi. It follows that K u { N X,} would be contradictory. Hence, K t X , and Q(x) would be bounded of degree m ywhich is contrary to assumption. Let M I be a model of H. Then MI contains an infinite sequence of individuals (at}, t = 1,2, ... all different within M such that Q ( q ) holds in M I . Let K1 be the set of all sentences which are defined and hold in M I , b2 an individual which is not contained in M I , and HI the set of sentences {- E(b2, at)}, i = 1, 2, . . . Let H2 = K1 U H I U Q(b2) and let Hi be any finite subset of H. Hi can contain only a finite number of at, and will not contain any at for i > j , where j depends on Hi. Since all at are different within M I there must be a particular ar such that E(ak, at) holds in M I for i = 1, .. .,j. At the same time ar satisfies Q(x), Q(uk) holds in M I by the definition of the sequence US. It follows that if we replace b2 in Hi everywhere by ak then we obtain a set which holds in M1 and hence, must be consistent. Thus, Hi is itself consistent and the same applies to H2. Let M2 be a model of H2. MZcontains the individuals al, b2, a2, a3, . . . all of which satisfy Q (x) in M2. Moreover, M2 is an elementary extension of M I , and hence, is a model of K. Also Q (b2) holds in M z . Proceeding as before, we may construct a model M3 of K which is an elementary extension of M2 and which contains an individual b3 that satisfies Q (x) in M3 and is different from al, b2, u2, a3, . . . Continuing in this way we obtain a chain of models of K, M I c M2 c M3 c . . . such that a1 is in M I , b~is in Ms, i = 2, 3, ..., Q(a1) holds in Mj for all j , Q(bj) holds in MI for 2 < i Q j , and the individuals al, a2, b3, . . . are all different. This shows that Q ( x ) is not saturable and proves the theorem. The concept of a saturable predicate Q(x1, . . ., x,) in K may be generalized to arbitrary n, as follows. Let M I c M2 c .,. be an infinite Y

IY

N

-

-

152

[6.2.

GENERALIZATION OF ALQEBRAIC CONCEPTS

chain of models of K. Let M: be the n-dimensional Cartesian space over Mt, and let Vg be the variety of Q(x1, ..., x,) in Mi, i.e. the set of points (al, ... , a n ) with coordinates aj in Mt such that Q(a1, . . ., an) holds in Mg. Let Wt = Vt n Vt+l n Vt+2 n . . . Then Q(x1, . . ., xn) is saturable if for any such chain of models Wt = Wt+i= Wt+2 =.. . . for all sufficiently high i. For n = 1, this reduces to the definition given above. On the other hand, the fact that Q is bounded of degree m means that in no space Mn over a model M of K does the variety of Q contain more than m different points, although there exists a model M of K for which V contains exactly m different points. As before, we may prove 6.2.8. THEOREM. A predicate Q (XI,

saturable.

...,xn)is bounded if and only if it is

A somewhat simpler result, which follows from one of the arguments used in the proof of 6.2.6., is 6.2.9. THEOREM. If a predicate Q(x1,

. .., xn) which is defined in K is

such that for any model M of K, the variety V of Q in Mn over M is finite then Q is bounded. We shall now show that under certain conditions bounded predicates can be decomposed somewhat after the fashion of polynomials in field theory. In the following discussion all predicates are supposed to belong to a set JQ of bounded predicates of n 2 1 variables which are defined in K,where K is consistent and includes a relation of equality, as before. A predicate Ql(x1, . . ., x,) divides a predicate Qz(x1, ..., x,) and we write QI 1 Q2 if 6.2-10.

K

(VXI)

. . . (Vxn)[Qi(XI, . . ., x n )

3

Q2(~1,

.. ., x,)] .

Two predicates will be called associated, and we write QI M Q2, if QI I Q 2 and Q 2 I QI. If QI I Q2 then deg QI < deg Q2, if QI M Q2 then deg Ql = deg Q2. A bounded predicate of positive degree is called prime if it is associated with all its divisors of positive degree. Thus, the degree of a divisor of a prime predicate Q is either 0 or it equals the degree of Q. Even if this condition is satisfied for all divisors of Q, it is possible that Q is not prime. All predicates of degree 0 are associated. Let QI and QZbe prime in a set JQ which is conjunctive. Then if for a particular model M of K, the varieties V1 and VZof Q1 and Q2 in M , over M are not disjoint we may conclude that QI and Q2 are associated. More generally, if QI is a prime

6.2.1

153

BOUNDED PREDICATES

predicate whose variety V1 in Mn over some model M of K has a point in common with the variety V2 of a predicate QZ then Ql I Q2. In fact, consider the predicate Q 3 E JQ which satisfies K ~(VXI)

. . . CJxn) [&(xi, . . .,x n ) = Ql(x1, - -

- 9

xn)A QZ(x1,

. ..,xa)].

Such a predicate Q 3 exists since JQ is conjunctive with respect to K. The degree of Q 3 is positive since it holds at all points of V1 A Vz. Also, Q3 1 Q1, and since Ql is prime it follows that Q 3 M Ql, and hence Ql I Q3. On the other hand Q 3 I Qz and so QI I Qz, as asserted. If Ql and QZ are both prime we may therefore conclude that Ql M Qz. Suppose now that JQ is also disjunctive. A predicate Q will be called reducible if there exist predicates QI and Q2 such that and

0 < deg QZ

0 < deg Ql < deg Q

< deg Q

Q w QiVQ2.

A predicate of positive degree which is not reducible is called irreducible. A prime predicate is irreducible. One shows by induction that every predicate of positive degree is associated with the disjunction of a finite number of irreducible predicates

6.2.11.

Q m

QlvQzv... V Q m

The representation is irredundant if by omitting any one of the predicates on the right hand side we destroy the validity of the formula. The representation of a predicate by an irredundant disjunction is not necessarily unique. However, essential uniqueness, in a sense to be explained, is ensured if the following additional condition is satisfied.

6.2.12. If Ql I QZ and deg QI = deg

Q2

> 0,then

Ql M Q 2 .

On this assumption, let

. . . V Qm M Q; V Qi V . . . V QAt where Q is of positive degree and Ql, . . ., Q m , Q;, . .., QA, are irreduc-

6.2.13.

Q

M

Ql V QZV

ible. Then, for i = 1,2, Q6

so that

w

Qi

M [Qi

...,m

A Q M Qi A [Q; V Qi V . . . V QA] A Q;l V [Qi A [QL V . . . V QAjI]

deg [Qt A Q;l < deg Qi deg [Qg A [QiV . . . V QA.1

and

< deg Qi .

154

[6.3.

GENERALIZATION OF ALGEBRAIC CONCEPTS

Since Qt is irreducible this shows that deg [Qt A Q;] is equal either to 0 or to deg Qg. But Qt A Q; I Qz and so the second possibility yields Qt A Ql M Qf, by 6.2.12. It follows that in this case, Qt I Qt A Q; and hence Qg I Q;. Similarly, deg [Qt A [Qi V . . . V el,]] is equal either to 0 or to deg Qt, and in the latter case we may conclude that Qt I Qi V . V QA. Moreover, the possibility that

..

deg [Qt A Q;] = deg [Qz A [Q; V

. . . V Q;,l]

=0

may be ruled out because of 6.2.4. Thus, at least one of the two relations QtI Q;, or Qa I Qi V . . . V QA, applies in all cases. If the first of these does not apply, we consider M

Qa M Qa A [Q; V . . . V QA,l r u ~ [QfA Qil V . . . V [Qt A [QiV .. . V QA,l]

and conclude as before that at least one of the two relations Qt I Q; or Qt I QJ V .. V QA, applies. Continuing in this way, we find that Qt divides at least one Q;, 1 < k < m'. Similarly, every Q; divides at least one Qj. Thus, for any specific i, we have Qt I Q;, Q; I Qj, and hence Qz I Qj. But if so then Qz V Qj M Qj, and this shows that Qj is redundant in the representation of Q unless i =j . Hence i =j and Qt w Q;. It follows that for every Qt there is a QL with which it is associated, and vice versa. But since the two representations of Q are irredundant, no two terms of the same disjunction can be associated with each other. Hence,

.

6.2.14. THEOREM. Given two representations of the same predicate by irredundant disjunctions of irreducible predicates,

Q

M

Qi V

... V Q m M

Q; V

. . . V QA,.

we have m = m' and - after a suitable rearrangement of the second representation, Qt M Q;, i = 1,2, . . ., m. 6.3. Algebraic Predicates. Although the concept of a bounded predicate

of one variable shows certain similarities with the concept of a polynomial equation, q ( x ) = 0, the former is still too wide to be a genuine generalization of the latter. To illustrate this let K = KCF,a set of axioms for the notion of a commutative field, and let A4 be the particular field obtained by adjoining a real root of the polynomial 6.3.1.

x4

- 2x2 - 1

6.3.1

155

ALGEBRAIC PREDICATES

to the field of rational numbers. The roots of 8.4.1. are f 2/1 + 4 2 (real) and & 41 - 4 2 (imaginary). Thus, M contains the two real roots of 6.3.1. and does not contain the two remaining roots, which are imaginary. Now let q ( x ) be the polynomial

6.3.2.

q ( x ) = x2

- 2x - 1

and let Q , ( x ) be a predicate which states that x is a root of the polynomial q ( x ) . It is not difficult to formulate Q a ( x ) in terms of the relations E, S, and P of KCFand without individuals. Now define the predicate Q (4by 6.3.3.

QW = Q , < X ) A [ ( ~ Y ) P ( Y , Y , A X)] A [ ( z ) [ Q p ( z ) h[ ( 3 w ) f ‘ ( w , w , z ) ]

3

E(x,z)]]

read - “ x is a root of q (x) and possesses a square root; and any element which is a root of q (x) and possesses a square root is equal to x”. Then it is not difficult to verify that Q (x) is a bounded predicate of degree one in KCF. Moreover, for a = 1 - .d’,Q ( a ) holds in M although a is algebraic of degree 2. Q ( a ) does not hold in the field M‘ which is obtained from M by the adjunction of 4 1 -1/2. Thus we might say that Q ( x ) is an accidental property of a, and we may wish to confine ourselves to predicates which are more genuinely inherent in the individuals by which they are satisfied. Now an individual as such has no property at all attached to it in our set-up except with reference to a structure, or to a set of axioms, in which it is contained. Accordingly, in order to give a precise meaning to the notion of an inherent property, or predicate, we go back to the concept of persistence which was formulated in section 3.3. above and we introduce the following definitions. A predicate Q ( X I , . . .,x,) will be called algebraic in K is it is bounded in K and persistent under extension with respect to K. ( ? ( X I ,.. ., x,) will be called strongly algebraic in K if it is algebraic in K and persistent under restriction with respect to K. Combining the known properties of boundedness and persistence, we see without difficulty that conjunctions and disjunctions of algebraic predicates are algebraic, and conjunctions and disjunctions of strongly algebraic predicates are strongly algebraic. Now let JS be the set of strongly algebraic predicates of n variables in a given K, and suppose that the following condition is satisfied for all elements of Js.

156

t6.3.

GENERALIZATION OF ALGEBRAIC CONCEPTS

6.3.4. If the degree of Q(x1, .. ., X n ) is m, and if the variety V of Q in the n-dimensional space Mn over a model M of K contains exactly k < m different points then there exists an extension M' of M which is a model of K such that the variety V' of Q in M'n over M' contains m different points (at least and hence, exactly). We are going to show that the condition 6.2.12. is a consequence of 6.3.4., for JQ = Js. Let Ql and Q2 be two elements of JS such that Ql I Q2 and deg QI = deg Q2 = m. We have to show that Ql is associated with Q2, i.e. that Q2 M Ql, or more particularly, that Q2 I QI. In other words, we have to show that if Q2 holds at a point P in Mn over a model M of K then QI also holds at P. Let M be any model of K and assume that the variety V1 of Ql in Mn contains exactly k different points while the variety V2 of Q2 contains exactly I different points, I < m. Since QI I Q2 we have VI c V2, and hence k < 1. To prove our assertion we have to establish that k = 1. Suppose on the contrary that k < I < m. By 6.3.4. there exists an extension M' of M which is a model of K such that the variety V; of Qi in M'n over M' contains exactly m different points. Also, since Ql is persistent under extension, VI c V;, and so V; contains exactly m - k different points in addition to the points of V1. But QI I Q2 and so V; c Vi where V i is the variety of Q2 in M'n. On the other hand, Ql is persistent also under restriction and so the points of V; - V1 cannot belong to VZ- V1, regarded as a set of points in M'n. Thus V i includes the disjoint sets V1, V; - V1,V2 - V1 which, together, contain m Ik > m elements. But this is impossible since QZ is of degree m. Hence k = I, QI 1 Q2, and so Ql w Q2, as asserted. Next, we prove

+

.

6.3.5. THEOREM. Let QI(x), Q z ( x ) , . ., Qm(x) be a set of algebraic predicates of one variable, and let R (XI, ..,Xm, y) be a metamathematical polynomial in Kin the sense of section 6.1. Then the predicate

A

.

Q ( x ) = ( 3 ~ 1 ) ( 3 ~.2. .) ( h n ) [Ql(x1>A .. A Q m (xm) A R ( X I , ~ 2 .,. ~ r n x)] ,

Q2 ( ~ 2 A )

.

-

< deg QI(x) deg @ ( x ) ... deg Qm(x). Let deg Qs(x) = kt, i = 1,2, . . ., m. Given any model M of K suppose that Qr (x) holds for j g different individuals of M so that j t < kz. Since R is a polynomial in K, Q ( x ) cannot hold for more t h a n j h . . . j m < klkz .. . km different individuals in M. Hence Q(x) is bounded in is algebraic in K and deg Q(x)

6.3.1

157

ALGEBRAIC PREDICATES

K and its degree does not exceed k1k2 . . . km. To prove that Q (x) is persistent (under extension), we recall that by the definition of a metamathematical polynomial, R is an existential predicate and hence, is persistent. Now suppose that Q (a) holds in a model M of Kfor an element a of M. Then there exist elements bl, .,bm of M such that

..

Ql(b1) A Qz(b2)A

. . . A Qm(bm)A R(b1, - .

a ,

bm, a)

holds in M. It follows that this sentence holds also in every extension M' of M which is a model of K. This shows that Q(a) holds in M', Q ( x ) is persistent as well as bounded and hence, is algebraic. 6.3.5. is proved. Let M be any model of K. An element a of M will be called (strongly) algebraic if it satisfies a (strongly) algebraic predicate in K. The lowest degree of any algebraic predicate in K which is satisfied by a will be called the degree of a. A model all of whose individuals are algebraic (with respect to the given K ) will be called algebraic. A model M of K will be called algebraically complete if every algebraic predicate Q (x) of one variable in K, is satisfied by some individual of M , provided it is satisfied in any model of K which is an extension of M. A model M of K with diagram D is called algebraically closed if every algebraic predicate in K u D which is of positive degree is satisfied by some element of M. A model M of K which is algebraically closed also is algebraically complete. If K is complete (in the sense of Chapter 4 above) and A4 is a model of K then M is algebraically complete (with respect to K). For let Q ( x ) be any algebraic predicate of degree one. Then the sentence ( 3 x ) Q (x) is either deducible from K - so that Q (x) holds for some element of M - or it is inconsistent with K so that Q ( x ) cannot hold for any individual in any model of K. 6.3.6. THEOREM. Let M be an algebraically closed model of K with diagram D and let M' be an extension of M which is also a model of K. Then any element a of M' which satisfies an algebraic predicate in H = K U D must be equal, in M', to an element of M.

PROOF.Let Q (x)be an algebraic predicate in H such that Q(a)holds in

M . Since M is algebraically closed, there exists an element bl of M such that Q (61) holds in M. Since Q (x) is bounded, there exists a finite set of elements of M, bl, . . ., bn say such that (VX)

[Q ( x ) 3 E ( x , bl) V E(x, b2) V

holds in M. Consider the predicate

.. - V E ( x , bn)]

158

-

GENERALIZATION OF ALGEBRAIC CONCEPTS

Q‘(x) = Q ( x ) A

-

E(x, bi) A

E ( x , bz) A

.. . A

-

[6.3.

E(x, bn) ,

This predicate is algebraic in H, but is not satisfied in M . Accordingly, it cannot be consistent in H . Hence and so

H I- (VX) [Q(x)

This proves 6.3.6.

H

3

E ( x , bi) V

E(u, bi) V

.. . V E(x, bn)]

... V E(u, ba) .

6.3.7. THEOREM. Every model M of K possesses an extension M’ which is a model of K and which is algebraically complete.

PROOF.Let JA be the set of all predicates of one variable which are algebraic in K. For every Q(x) in J A introduce a distinct individual UQ which is not contained in K. Consider the set Q(uQ) = H. It is not difficult to see that any monotonic set of subsets of H which are consistent with K possesses a union which is consistent with K. Hence, by Zorn’s lemma, there exists a maximal subset H of H which is consistent with K. Let M be a model of K u H‘, then we claim that M is algebraically complete. For suppose it is not. Then there exists an element Q‘ ( x ) of J A such that (3x) Q ’ ( x ) does not hold in M although it holds in some extension M ’of M which is a model of K. By the construction of M, the sentence Q’ (UQ) belongs to H - H‘. It follows that for some finite number of elements of H’, Qi (uQ,), . . Q m ( u ~ JK t [QI(uQ,) A . A

A

Qm (QQ_)

and so and hence

A

Q’ (QQ’)]

K t Qi(aQ,)A

-

.

.. . A Q ~ ( Q Q 3J

K t X = [ ( 3 x i ) Q i ( ~ i )A ] A [ (34Qm (xm)] 3 [ WY) [

N

-

..

Q’(QQ’)

... A

Q’ ( Y ) ] ] .

But the predicates Qt ( x ) are algebraic, hence persistent (under extension) with respect to K. It follows that the implicans of Xholds in all extensions of K which are models of M. Thus, ( 3 x ) Q ( x ) cannot hold in any such structure. This yields a contradiction and proves that M is algebraically complete. The consideration of the special case K = KCF(a set of axioms for the concept of a commutative field) will justify the use of the word “algebraic” in the above terminology. For this set K, we show, first of all -

6.3.1

ALGEBRAIC PREDICATES

159

6.3.8. THEOREM. If the predicate Q ( x ) is algebraic in the set K u D where D is the diagram of a particular commutative field M , and a is an element of an extension M' of M which is a model of K such that a is transcendental with respect to M - then Q (a) does not hold in M'.

PROOF.Suppose that Q ( a ) holds in M' for an element which is transcendental over M. Since Q (x) is persistent, Q (a) holds also in the algebraic closure of M', M*. Let b be any element of M * which is transcendental with respect to M . Then there exists an automorphism of M * which carries a into b while leaving the elements of M unchanged. It

follows that Q (b) holds in M*. But the number of different elements of M * which are transcendental with respect to M is infinite, and so Q (x) holds for an infinite number of elements in M * . This contradicts the assumption that Q ( x ) is bounded in K u D and proves the theorem.

6.3.9. THEOREM. Let q ( x ) be a polynomial with coefficients in a particular M, not all zero, and let D be the diagram of M . Then there exists a predicate Q ( x ) which is strongly algebraic in KCF U D such that for any element a of any field extension M' of M,Q (a) holds in M' if and only if q(a) = 0. Conversely, for every predicate Q ( x ) which is strongly algebraic in K U D there exists a polynomial with coefficients in M , not all zero, such that for any element a of any extension M' of M , Q (a) holds in M' if and only if q(a) = 0.

PROOF. In order to prove the first part of the theorem, we only have to find a formal expression for the equation q ( x ) = 0, for given q (x), in terms of E, S, P and of the coefficients of q (x), and this is clearly possible. Conversely, given the predicate Q ( x ) which is strongly algebraic in KCF u D, we have to find a corresponding polynomial q(x). Let M* be the algebraic closure of M . For any two elements a, a' of M* which are conjugate over M (roots of the same irreducible polynomial with coefficients in M )there exists an automorphism of M which carries a into a' while leaving the elements of M unchanged. It follows that Q (a) holds in M if and only if Q (a') holds in M . Thus, for any polynomial p ( x ) with coefficients in M and irreducible over M , either Q (x) holds in M* for all roots of p ( x ) or for none. Let p l ( x ) , . . . , p j ( x ) be the set of monic polynomials of positive degree with coefficients in M and irreducible over M such that Q (x) holds for all roots of p~(x), i = 1, . ..,j . The number of these polynomials is finite since Q ( x ) is bounded. Put q ( x ) = p1 ( x ) . . . p j (x). Then for any element a of M * , q (a) = 0 if and only if Q (a) holds

160

GENERALIZATION OF ALGEBRAIC CONCEPTS

[6.3.

in M*. The same is true, if we replace M * by any extension of M * , in view of the fact that Q ( x ) is strongly algebraic, and by 6.3.8. Hence, hally, if M’ is an arbitrary extension of M y and a is an element of M’ then q (a) = 0 if and only if Q (a) holds in M’, since M’ can be embedded in an extension of M* and Q (x)is persistent under extensionand restriction. Note that the second part of 6.3.9. does not necessarily hold if Q ( x ) is algebraic, but not strongly algebraic. Consider the predicate

Q

6.3.10.

= P(xyxy4

A [ ( 3 ~ P) 0, Y, x)]

- read “ x is a square root of 2 and a square”. Taking M as the field of rational numbers and M’ = M(4\/3, where 4d1/z is the real root in question, we see that Q ( x ) is satisfied in M’ only by @although the conjugate of 1/27 which is - d y i s also contained in M’. Thus, there is no polynomial q ( x ) as required by the second part of 6.3.9. It is not difficult to check that Q ( x ) is algebraic but not strongly algebraic. Accordingly, we see that the notion of a strongly algebraic predicate may be regarded as a natural generalization of the condition imposed by a polynomial equation. For the particular case now under consideration, two strongly algebraic predicates QI ( x ) and QZ( x ) are associated if and only if the corresponding polynomials have the same irreducible factors over M. Condition 6.3.4., and hence, 6.2.12., is satisfied for the strongly algebraic predicates in KCF U D. This is not true for the corresponding domain of algebraic predicates. Thus, the algebraic predicate 6.3.10. divides the algebraic predicate P ( x , x, 2) but is not associated with it, in spite of the fact that both predicates are of degree 2. Irreducible strongly algebraic predicates correspond to powers of irreducible polynomials. If for a given M , we construct an algebraically complete extension M I of M y where the set K mentioned in 6.3.7. is replaced by KCF U D then all polynomials of positive degree with coefficients in M posses roots in M I . Replacing M by MI and constructing an algebraically complete extension M Zof M I with respect to the set KCF u DIwhere D Iis the diagram of MI, we find that all polynomials with coefficientsin MI have roots in M2. Continuing in this way, we obtain a chain of fields M c M I c Mz c . . . whose union will be seen to be algebraically closed in the standard algebraic sense, and hence also in the metamathematical sense defined earlier in this section. Thus, we have here another metamathematical procedure for the construction of an algebraically closed extension to any given field.

+

+

6.4.1

ALGEBRAIC PREDICATES AND CONVEX SYSTEMS

161

If instead of KCF U D, we consider a set of axioms KZp for the notion of a field of characteristicp 2 0, then the strongly algebraic predicates for the case correspond to the polynomials with coefficients in the prime field of characteristicp. In all these cases, the algebraic predicates of more than one variable correspond to sets of polynomials since, generally speaking, single polynomials of several variables yield unbounded predicates. 6.4. Algebraic Predicates and Convex systems. As in the preceding sections, let K be a consistent set of axioms with a relation of equality, E ( x , y). Suppose in addition that Kis strongZy convex in the sense defined in section 3.5. and let M be a core model of K. 6.4.1. THEOREM. Every element of M satisfies an algebraic predicate in K.

PROOF.Let a be an element of M which is equal to an element c of K.

For such a, Q ( x ) = E(c, x ) is the required algebraic predicate. Thus, it remains to prove the theorem for elements a of M which are not equal to individuals contained in K. Let D be the diagram of M and let M' be a model of K which is isomorphic to M such that only the individual constants of K (if any) are common to M and M'. Let D' be the diagram of M'.We claim that the set H = K U D u D' is consistent. Indeed, if H were contradictory, then for some conjunction Y of elements of D, and for some conjunction Y' of elements of D', we should have K l- N [Y A Y].Let Y' = Y'(a;, . . .,a;) where we have displayed the individuals of Y' which are not contained in K, and let al, . . .,a m be the individuals of M which correspond to a;, . . ., a;, regarded as elements of M in some fixed isomorphism between M and M'. Note that in such a case it is always assumed that the individuals which are contained in K are invariant under the isomorphism. Since a;, . . ., a; are not contained in K or Y, we may conclude that (for suitable z1, . . ., z,)

. . . (Vzm) [- Y'(z1, . . .z,)] But K, Y and ( 3 ~ 1 ). . . (3Zm) Y' (z1, . . .,zm) all hold in M. In particular,

6.4.2.

K u { Y } t (VZl)

this applies to the last sentence since Y'(al, . . .,a,) holds in M. Thus, 6.4.2. is impossible and H i s consistent. Now let a be any element of M, and let J be the set of sentences E(a, by)where b, varies over all elements of M'. We propose to show that the set H* = H U J is inconsistent. Supposing that the contrary is true, let M * be a model of H*.Then M* is an extension of M and of M' N

162

OENERALIZATION OF ALGEBRAIC CONCEPT3

[6.4.

and a model of K. The intersection of all normal substructures of M * which are models of K is not empty, by assumption, and, accordingly, is a normal substructure N of M and M'. M is a core model of K and so M = N . But this shows that M' contains an element b which is equal to a in M*,contrary to the construction of M*. We conclude that J is contradictory. It follows,that there exists a finite number of elementsof M', bl, . ..,br, say I 1 such that H I- E(a, bl) V . .. V E(a, bl). This in turn shows that there are conjunctions of elements of D and D', which will be denoted by Z and 2' respectively, such that K t- 2 A Z'

3

E(a, bl) V

... V E(u, bl)

Let CI, . .., cg and c;, ...,c; be the individuals of 2 and 2 which do not coincide with any individual of K , nor with bl, . . ., bl, or a. Displaying these, we write z = Z(c1, . . ., Ci), 2' = 2' (c;, . ..,c;) and we put

w = (3Wl) . . . (h) Z(w1, . .., wt), W' = (3wJ . .. (3w;)zyw;, ..., w;) Then 6.4.3.

K t- W A W'

2

E(a, bl) V

... V E(u, bl) .

Since a and bl, ,..,bl are contained in M and M' respectively, we may suppose that these individuals are contained also in 2 and Z', and hence in Wand W', respectively. Indeed, if this is not the case from the outset, we may achieve it by including additional elements of the diagrams D and D'.Also, as explained earlier, we may suppose that a is different, within M , from all individuals of K. Moreover, we may suppose that bl, . ..,bl also are different, within M', from all individuals of K. Indeed, suppose that to begin with, one of the bk e.g. bl is equal, in M ' , to an individual c of K. Then E(a, c) belongs to D and E(b1, c) belongs to D' and since E is a relation of equality, N

K I-

N

E(u, C ) A E(b1, C )

2

-

E(a, b1).

Hence, taking into account 6.4.3., K t-[WA E ( u , c ) ]A [ W ' A E ( b i , c ) ] 3 E(u, bz) V , V E(u, bi) N

..

.

By extending the scope of the existential quantifiers in W to include

6.4.1

-

163

ALGEBRAIC PREDICATES A N D CONVEX SYSTEMS

E(a, c), with a similar modification for W ' A E(b1, c), we obtain sentences by which we may replace Wand W' in 6.4.3., while omitting E(a, bi) in the implicate. Continuing in this way, we finally obtain an implicate, such that the bk which occur in it are different from all individuals of K. It is impossible that bi, . . ., bz should all be exhausted in this procedure, since in that case E(a, bi) V ... V E(a, bl) would indicate that a is equal, in M * and hence in Myto an individual of K and this is contrary to assumption. Displaying a in W and bl, . . ., bz in W , so - W = W(a), W' = W'(b1, . . ., bl) we may rewrite 6.4.3.as 6.4.4.

..., bz) 2 E(u,bi) V ... V E(u,bz)

K t W(U)A W (bi,

We conclude that 6.4.5.

K 1(VX) (Vyi) . . . (Vyz) [ W ( X A ) W' (yi, . . .,yi) 2 E(x,yi) V

.. . V E(x,yl)]. Substituting the variables x i , . .. xz in turn for x in 6 . 4 5 , it is then not V

difficult to show that

.

6.4.6. K t. (Vxi) . . (VXZ)(VX)(VJQ) . . . (Vyz) [W(xi) A . . . A A W ( X ~A) W ( X )A W' (yi, ., rz) 2 [E(xi,yi) V . . V E(xi,yz)]A.. . A [E(xl,yi)V...VB(xr,yz)]V[E(x,yi)V...vE(x,yz)]].

..

This in turn entails

.

.. .

..

6.4.7. K 1(Vxi) . . . (VXZ) (VX) (Vyi) (VJQ) [ W ( X ~ A) .A W(X,) A W ( X )A W'( ~ 1 , ,yz) 3 E (xi, XZ) V E(x1, ~ 3 )V . V V E ( x i , x z ) V . . V E ( ~ ~ - ~ , X ~ ) V E ( X VE(x,xz)]. ,X~)V..

.

.. .

.

..

...)cz which do not occur in K, (Vxi) .. . (VXZ) (VX) [ W ( X ~ A ). . . A W ( X ~ A )W ( X )

Hence, for arbitrary constants c1,

W ' ( C ~. ., .,CZ) 3 2 E(xi, XZ) V E(x1, ~ and so

K

K t

... (@z)

[(~JJI)

W'(y1, ..., JQ)]

. . . A W ( X ZA) W ( X )

2

3 V )

2

.. . V E(x, xi)

Vxi)

E(xi, X Z ) V E ( x i , ~

... ( V X Z ) ( V )[W(xi) A 3 V ) . . . V E ( x , XI).

It follows that if we define Q (x) by then Kt(Vxi)

Q (4= [ W ( x )V [ ( 1 ~ 1 ). . . (W W ' b i , .. .,YZ)]]

... ( ~ x z ) ( V X ) [ Q ( ... X ~A) A Q ( X Z > A Q ( X ) ~ E ( ~... ~,X~)V V E(xz-1, XZ) V E(x, X I ) V . . . V E ( x ,XI)] .

Thus, Q ( x ) is bounded in K,of a degree not exceeding 1. Moreover, by

164

GENERALIZATION OF ALGEBRAIC CONCEPTS

[6.4.

shifting quantifiers it is not difficult to replace Q ( x ) by an existential sentence which is equivalent to Q (with respect to K). Hence, Q(x) is persistent with respect to K, under extension, hence algebraic. Since Q (a) holds in M , the theorem is proved. The following example shows that the predicates Q ( x ) which exist according to 6.4.1. are not necessarily bounded of degree one, Suppose that K contains a relation of equality E(x, y), and another two place relation F ( x , y ) , so that E satisfies the axioms of equivalence and of substitutivity with respect to F (see 2.1.2.). In addition, K is supposed to contain the following axioms 6-48.

-

('0)[ [ J " ( ~ O , Y3) B ( a o , ~ )A] [ ( V 4 [E(ao,Z) V [ (30)[FG, U) A

-

A

[w, 4 A - E ( x , 4 A - W Y ],I). [w, v) A

E(z, v)]]]]]

(W (W [ w ,A A E ( x ,u) = [ (Ez) ( V 4 WY)(W vu A F(x, z) A F(x, v)

and 6.4.9.

E ( x , y ) V E ( x ,z) V E ( x ,v) V E ( y , z) V

v E ( y , d v E(z, 741

(W W

Y x)

(VX)(VY)[

W

Y

(W@A(W [

r) = F(Y, 41 F h u) A

m, = F(x, 4 2)

The set of axioms 6.4.9. states that F,like E, is a relation of equivalence (though not a relation of equality). The first axiom of 6.4.8. states that there exists a unique element ao (unique - in the sense of the relation of equality E) such that F(x, y) does not hold for any y which is different from ao. The second axiom of 6.4.8. ensures that if F(x, y ) holds between two different x and y then it holds also between x and some z which equals neither x nor y. Finally, the last axiom of 6.4.8. states that the number of different elements in an equivalence class with respect to the relation F ( x , y) does not exceed three. Thus, any model M of K can be described as follows. M consists of a number of equivalence classes with respect to F. Let the set of these classes be {Ny). There is just one of them, NOsay, which contains only a single element QO. All the other classes, if any, contain precisely three different elements, by the second and third axiom of 6.4.8. Any normal substructure M' of M which is also a model of K must include NO and if it contains one element of another N y it must contain all elements of that Nv. It follows that K is strongly convex. K is consistent since the following structure, M * , is a model of K.

6.4.1

ALGEBRAIC PREDICATES AND CONVEX SYSTEhiS

M* contains four individuals, uo, al, U Z , hold between these individuals, as follows 6.4.10.

u3.

E ( q , a#)

i = 0, 1,2, 3

F(ut, aj)

i, j = 1, 2, 3

W o , a01

165

The relations E and F

and not in any other case. M * contains two equivalence classes with respect to F, one, NO,which contains only ao and the other, N1 = { a l , az, as). The core of M * is the structure MO which contains the single element a0 such that both E(uo, UO) and F(uo, ao) hold in Mo. Now let K = K U { E(a0, a ) } .K is still strongly convex, its models being in fact the models of K that contain, in addition to ao, an element a1 different from ao. Then the above M * is a model of K'. More particularly, M* is a core model of K since it contains no proper (normal) substructure that satisfies K'. It follows that every element of M* satisfies a predicate which is defined in and algebraic with respect to K'. It is in fact not difficult to specify such predicates. Thus, uo satisfies E(uo, x), a1 satisfies E(a1, x ) and a2 and a3 both satisfy F(u1, x ) A E(u1, x). Of these the first two are algebraic of degree 1 and the last is algebraic of degree 2. al, and a2 cannot satisfy any predicate which is algebraic of degree 1 for the automorphism of M * which is given by uot)ao, a1 t+ al, a2 t)u3, a3 c)a2 transforms a2 and a3 into one another, so that a2 and a3 satisfy precisely the same predicates with respect to K'. Coming back to the general case, let M' ( S ) c M be obtained by the adjunction of a subset S of a model M of a convex set of axioms K to a substructure M' of M which is a model of K (section 3.5.). Let K = K U {E(uv,uv)}where avvaries over all elements of M' and of S. K' is strongly convex since it contains at least one individual. (The sentences E(uv, a,,) are added to K only in order to introduce the individuals uv into the set of axioms, they obviously do not add to the number of sentences deducible from the set). Then M ' ( S ) is the core of M with respect to K and so every element of M ' ( S ) satisfies a predicate which is algebraic with respect to K . We recall that M ' ( S ) is the intersection of all normal substructures of M that include M' and S and are models of K. If, for given K and Af, every element of M' (9satisfies an algebraic predicate of degree one in K', for any M c M which is a model of K and for any subset S of M , then M will be called deJinite with respect to K. A convex set K will be called definite if all its models are definite with respect to it.

-

-

166

GENERALIZATION OF ALGEBRAIC CONCEPTS

[6.4.

By an automorphism g of a structure M which contains a relations of equality E we mean a one-to-one mapping from the equivalence classes defined by E onto the same set such that the same relations are satisfied by corresponding elements. However, if a is an element of M then we shall use ga also in order to indicate an arbitrary element of the equivalence class on which the equivalence class of a is mapped by E. Let M be a model of the strongly convex set K, M definite with respect to K, and let MObe the core model of M. Any automorphism of M which leaves the (equivalence classes of the) elements of MOfixed will be called admissible. The admissible automorphisms of M constitute a group G. 6.4.11. THEOREM. Let g be an admissible automorphism, then the set of elements of M which are left fixed by g constitute a substructure of M

which is a model of K.

PROOF.Let S, be the set of elements of M which are left fixed by g, and let M g be the structure which is obtained by restricting the relations of M to S,. Then Mg 1 Mo. Let K‘ = K U @(av,av)},where av vanes over all elements of MB (i.e. of Sg),and let a be any element of M such that Q (a) holds in M for some algebraic predicate, Q, of degree one in K . Since Q ( x ) is defined in K all the individuals of Q ( x ) are left fixed by g. Accordingly, the fact that Q(a) holds in M entails that Q(gu) also holds in M , where ga denotes the element (“the”-up to equality) on which a is mapped by g. But Q ( x ) is of degree one and so ga must be equal to a, a also is left fixed by g. Taking the core model, Mo, for M‘ and the set Sgfor S i n the definition of the notion of a model which is definite with respect to K, we find that the elements of MO(i.e. of S,) are all left fixed by g and so are included in S,. Hence, M0(Sg)= M,, M , is a model of K. 6.4.12. THEOREM. With the same assumptions on K and M as before, suppose moreover that there exists a predicate Q ( x ) which is algebraic of degree n in K, n 2 1, such that M is generated by n different elements U I , . . ., an which satisfy Q(x). (Thus, M = M o ( S ) where S = {al, U Z , . . ., an} and Mo is the core model in M). Then G is finite, of order m , m < n!.

PROOF.For any admissible g, the sequence of elements gar, i = 1,. .., n must be a permutation of al, U Z , ..., an. Indeed, since the individuals which are contained in Q (x) (if any) are all left fixed by g, Q (gat) holds

6.5.1

167

SEPARABILITY

..

in M for i = 1, .,n. Since the gai are all different it follows that they constitute a permutation of al, a2, . . ., an. Now suppose that E(glag, gzat), i = 1, 2, ., n holds in M , for two admissible automorphisms, gl and g2. Then E(gi1g1at,ai) holds in M for i = 1,2, ...,n. Let a be any element of M, and let Q'(x) be an algebraic predicate of degree one in K U (E(a1, Q I ) , .. .,E(a,, am)>. Such a predicate Q' exists, for any given a, by one of the assumptions of 6.4.13. Since the individuals which occur in Q' ( x ) are left fixed by gzl g l , it follows that Q' (g-tgla) holds in M , and so g-igla is equal to a, gla is equal to gza. This shows that if two admissible automorphisms coincide on {al, az, . . ., an> then they coincide on the entire structure M. Since the number of permutations of n elements is n! it follows that there are at most n! elements in G. This completes the proof. If Kis a set of axioms for the notion of a commutative field of characteristic 0, and Q(x) denotes the property of being a root of a particular polynomial q ( x ) of positive degree IZ with rational coefficients then the roots of q(x) generate a field M which may serve as an illustration for 6.4.1 1. and 6.4.12. In this case, G is the Galois group of M.

..

6.5. Separability. Let K be a set of axioms with equality and let M be a model of K with diagram D. We may ask whether every (strongly) algebraic predicate of degree one in K is satisfied by some individual of M . We may also ask whether every (strongly) algebraic predicate of degree one in K u D holds for some individual of M. Selecting one of the various concepts inspired by these questions we introduce the following definition. A model M of a set of axioms K with equality will be called separable (with respect to K ) if for every substructure M' of M , with diagram D', which is a model of K, any algebraic predicate of degree one in K U D' can hold for some individual of M' in M', only if it holds in M for some individual of M . A set of axioms Kwith equality will be calledperject if all its models are separable with respect to it. It is easy to see that if a set of axioms K with equality is such that for every model M' of K with diagram D' any algebraic predicate of degree one in K u D' holds for some individual of M' then K is perfect. However, this condition is also necessary. For suppose that there exists a model M' of K with diagram D' and a predicate Q ( x ) which is algebraic of degree one in K u D' such that Q ( x ) is not satisfied by any individual of M'. Since Q(x) is algebraic of degree one there exists a model M of

168

GENERALIZATION OF ALGEBRAIC CONCEPTS

[6.5.

K U D',i.e. an extension of M', such that Q ( a ) holds in M for some individual of M. Hence, if K is perfect Q (a') holds in M' for some a' in M', which is contrary to assumption. We shall now establish the connection with the corresponding terms in standard Algebra. Let M be a commutative field, and let M* be a field extension of M. M * will be said to be a separable extension of M in the algebraic sense if every element of M* satisfies a non-zero polynomial without multiple factors and with coefficients in M. According to this definition M * must be an algebraic extension of M. 6.5.1. THEOREM. Let M be a commutative field with diagram D and let M * be an algebraic extension of M. Then M * is a separable extension of

M in the algebraic sense if and only if M * is separable with respect to K = KCF U D.

PROOF.Suppose that M* is a separable extension of M in the algebraic sense, and let M' be any substructure of M * which is a model of K. Thus, M' is an extension of M. Let Q ( x ) be an algebraic predicate of degree one in K' where K = K u D',D' being the diagram of M', and suppose that Q (a) holds in M' for some element a of M*. a is algebraic with respect to M and hence with respect to M'. It follows that there exists an irreducible polynomial p ( x ) of degree n 2 1 and with coefficients in M' such that p(a) = 0. Let A 7 be the algebraic closure of M * . contains n different roots off(x) including a. Q (a) holds in M * since Q ( x ) is persistent with respect to K . Suppose now that n > 1. Then Q ( x ) holds in i@ also for all the other roots ofp (x) since there exists an automorphism of i@ which carries a into any given root of p ( x ) while leaving the elements of M' invariant. But Q ( x ) was supposed to be bounded of degree one, and so we may conclude that n = 1, p ( x ) is linear, and a belongs to M'. Conversely, suppose that M * contains an element a which is not separable with respect to M. Thus, a is a root of an irreducible non-zero polynomial p ( x ) of degree n with coefficients in F which is of the form p ( x ) = q (xp) where p is the characteristic of the fields under consideration, p # 0. Now let M' = M(a*) where a* = u p , so that M' is of degree n/p over M. It follows that the element a, which is of degree n over M, does not belong to M'. Now consider a predicate Q (x) which formalizes the statement, "a* is the pth power of x", i.e. xp = a*. Clearly, we may express Q ( x ) within the vocabulary of K u D', where D' is the diagram of M', and Q (x)is then persistent (and even invariant) in K u D', and it holds for a in M. Moreover, the equation x p = a* possesses not

6.5.1

169

SEPARABILITY

more than one root in any extension of M' since x p - a* = x p - ap = ( x - a)p. It follows that Q ( x ) is bounded of degree one and hence, algebraic of degree one. Thus a satisfies an algebraic predicate of degree one in K u D' although this predicate does not hold for any individual of M'. This shows that K is not separable in the metamathematical sense and proves 6.5.1. Given any predicate R(x1, ..., X n , y), n 2 1, we consider the predicates which are obtained from R by permuting the subscripts of the variables xi, x 2 , . . .,x n (including R itself). Thus, let

be any permutation of order n. We then defme the predicate R, (xi, X n , Y ) by Rz (XI, ~ 2 .,..,XB, y) = R (xk,, xk,, * * 9 Y)

. . .,

-

for example if and K = (k:)

R (xi, x2, y ) = (32) [S(xi,Y , Z) A P ( x 2 , xi, 41 then

Rn (xi, x 2 , Y ) = (32) [ S ( x 2 , Y , 2) A P ( X i , xz, z)] .

Clearly, R, = R where z is the identical permutation. A predicate R(x1, ...,x,, y ) , n 2 1 will be called symmetrical if 6.5.2. @xi)

... @ X n ) (Vy) [ R(XI, - - -

Xn,

Y) 3 Rn (xi, - - -

9

Xn,

u)]

for all permutations, K, of order n. R will be called symmetrical relative to a given set of sentences, K, if 6.5.2. is deducible from K. Let M be a separable model of a strongly convex set of sentences K (with equality), and let Q ( x ) be an algebraic predicate of nth degree in K which is satisfied by n different elements of M , al, a2, . . ., an. Let Mo be the core model of K in M. Let R(x1, x 2 , . . .,X n , y ) be a predicate which is persistent and symmetrical relative to the set K. Suppose that R ( x i , ..., X n , y) defines y as a partial function of x i , ...,x n in the sense that 6.5.3. K t (Vxi)

. . . (Vxn) (VY) (VZ) [ R(xi, - - .,X n , y) A R ( x i , - . ., X n , 2) 3

E ( Y , z)]

In these circumstances, we have 6.5.4. THEOREM. Suppose that R(a1, . . ., an, a) holds in M for some element a of M. Then a belongs to Mo.

170

PROOF.Consider the predicate

-

Q*(y) = (3x1) . . . (3xn)[ E(x1,xz) A E(Xi,x3) A . . . A E ( x ~ - 1 x,) , A Q(x1) A Q ( x z ) A .. . A Q(xn) A R(x1, . . ~ n , y ) ] . N

N

[6.5.

GENERALIZATION OF ALGEBRAIC CONCEPTS

- 9

It is not difficult to see that Q* (y) is persistent relative to K. Moreover, Q* (y) is bounded of degree one in K,for if M* is any model of K such such that Q*(a*) holds in M for some element a* of M*, then Q(ap*) holds in A4 for rz different elements a; of M , and these elements are determined uniquely in M*, except for order. Moreover, R (a*,,. . .,a t , a*) holds in M * , and a* is determined uniquely by the set {a;, . . ., u t } , even disregarding its order, since R is symmetrical. It follows that Q* (y) is bounded of degree one, at most, in K. By the hypothesis of the theorem, moreover, the degree of Q*(y) is exactly one. Thus, Q* (y) is algebraic of degree one in K. Now let DObe the diagram of the core-model Mo. Then Q * ( y ) is algebraic of degree one in K U DO.Since M is separable with respect to K, it follows that Q* ( y ) holds for some element ul of MOwhich is therefore equal to a*. But MOis supposed to be normal in M , by the definition of a core-model and so a* also belongs to Mo. This proves the theorem. It will be seen that if K = KCF U D, where D is the diagram of a particular field of characteristic 0 while Q ( x ) states that x is a root of a particular non-zero polynomial which is irreducible and of degree n, and R(x1, . . .,xn, y ) expresses the fact that y is a rational symmetric function of X I ,. . ., xn, then 6.5.4. proves the fundamental theorem on symmetric functions for this case. The algebraic resources used in proving this theorem are involved in 6.5.1. They express the fact that if a predicate Q ( y ) is algebraic of degree one in a set K = KOCF u D,where D is the diagram of a field M of characteristic zero, then Q ( y ) must be satisfied by an element of M. Finally we observe that the sets K & are not perfect for any given p # 0, since there exist fields M of characteristicp in which some polynomial xp - a is irreducible. Any polynomial of this kind gives rise to an algebraic predicate Q ( y ) of degree one in K & u D (where D is the diagram of M) such that Q ( y ) is not satisfied by any element of M. This shows that K & is not perfect. On the other hand, the set of axioms for the concept of a field of characteristic zero, K&, is perfect since all fields of characteristic zero are separable in the algebraic sense, and hence, in the metamathematical sense.

6.6.1

PROBLEMS

171

6.6. Problems 6.6.1. Verify that the predicates E(x1, y) and E(x1, XI) A S ( y , y, y) belong to the polynomial structure MI', with respect to KCF.Show that KCFis neither transitive nor pre-transitive. 6.6.2. Show that the notion of a free algebra (e.g. of a free group) is included in our definition of a polynomial structure. 6.6.3. Investigate the properties of the prepolynomial and polynomial structures Mi, M i , with respect to K = K& a set of axioms for the notion of a commutative ring with unit element. 6.6.4. Formulate a generalized notion of algebraic dependence. References. Chapter VI is taken from A. Robinson 1951 with minor modifications and corrections. For the theory of universal algebras see Birkhoff 1935 and Malcev 1954.

C H A P T E R VII

METAMATHEMATICAL THEORY OF IDEALS

7.1. Introduction. If we survey the properties of an ideal Jin a commutative ring M which can be formulated without reference to the particular operations-addition and multiplication - defined in the ring, the most characteristic fact which emerges is undoubtedly that J gives rise to a quotient ring M/J = M’. M’ consists of sets of elements of M such that every element of M belongs to just one element of M’. Moreover, there is a natural homomorphism between M and M‘, under which every element of M corresponds to the element of M‘ in which it is contained. The ideal J then coincides with the zero of M’. In a non-commutative ring M a (left and right) ideal is still similarly associated with a homomorphic ring M’ and a corresponding fact holds for normal subgroups in general groups. Accordingly, when trying to dissociate the concept of an ideal from the specific operations defined in a ring one might be inclined to identify it with a homomorphism, as described above. This has the advantage of great simplicity and does not require the extensive use of a formal language. On the other hand, there are various concepts concerning ideals which cannot be expressed directly in terms of the corresponding homomorphisms. Such is the concept of an ideal basis which is of considerable importance in the theory of ideals. However, one way of looking at a homomorphism as described above is as follows. Let D+ be the positive diagram of a commutative ring M , J an ideal in M , and M’ the quotient ring MIJ. Suppose, as usual, that the individuals of M denote themselves as elements of the language. Let D* be the set of sentences obtained by adding to D+ all sentences of the form E(a, b) where a and b are elements of M , which correspond to the same element of M’ under the homomorphism i.e. which are contained in the same element of M’. In other words, we add to D +a set of sentences which “identify” (make equal) any two elements of M whose difference belongs to the ideal J. Then M‘ is a model of D*.Thus, an ideal is associated with a certain set of sentences (i.e. sentences of the form 172

7.2.1

METAMATHEMATICAL IDEALS

173

E(a, b)) which are additional to a given set (in the present case D+ together with a set of axioms for the notion of a commutative ring). In the following sections we shall elaborate this idea and shall show how certain branches of ideal theory can be developed in terms of it. 7.2. Metamathematical Ideals. Let K and JO be two sets of sentences within a language L of the Lower Predicate Calculus. A subset J of JO will be called an ideal in JO over K if S ( K U J ) n JO c J, i.e. if all sentences of JO that can be deduced from the union of J and K are included in J. We shall refer to K as the axiomatic system of the theory and to Jo as its domain. The concept of an ideal as given above coincides with the concept of a relativesystem defined by Tarski for differentpurposes in his theory of systems in case K is a subset of JOand is regarded as the set of “logical” or tautological sentences of the theory. We shall assume that the sets K and JO are fixed throughout the remainder of this section so that the term ideaZ will always refer to an ideal in JO over K. Let C be the set of these ideals; then C includes Jo. It can be verified without difficulty that the intersection of any number of ideals is again an ideal. Following the conventions of algebra, we denote the intersection of a finite number of ideals J1, . . ., Jm, by [Jl, ...,J,]. There will be no occasion for confusing this use of square brackets with their use in the language L. The sum of any number of ideals is defined as the intersection of all ideals which include the given ones. The sum of a finite number of ideals, J1, . .,Jm, will be denoted by (51, . . .,Jm). Moregenerally, if A1, . . .,A , are sets of sentences in Jo, we denote by (A1, . .., Am) the intersection of all ideals that include A1 U . . . U A,, and we say that (A1, .. .,Am) is generated by A1, . . ., Am. With a slight modification of this notation, we write (XI, . . ., X,), for the ideal J which is generated by the set { X I , . . .,X,}, and we say that {XI, . . .,Xm}is a basis of J. More generally if B is a subset of an ideal J such that J is the intersection of all ideals that include B then B will be said to generate J or, to be a basis of J. Thus, B is a basis of J if J C S ( K U B). If J and J’ are two ideals such that J’ 3 J then we say that J’ divides J or, is a divisor of J ; if, in addition, J’ # J then J’ is said to be a proper divisor of J. An ideal J is irreducible if [Jl, Jz] = J, for any two ideals J1, Jz, entails either J1 = Jor JZ = J(or both). Similarly, Jis indecomposable if (J1, Jz) = J, for any two ideds JI, JZ entails J1 = J or JZ = J. We shall be concerned with irreducibility rather than indecomposability.

.

174

[7.2.

METAMATHEMATICAL THEORY OF IDEALS

We shall say that the ideals of C satisfy the maximum condition if any non-empty set of ideals C' c C includes at least one ideal which is not a proper subset of any other ideal of C'. An equivalent condition is that every ascending infinite chain of ideals,

J C Jz

C 53 C

... c Jn

+ ...

c

is such that all ideals of the chain are equal from some positive integer n onward, Jn = Jn+l = Jn+z = . . . Yet another equivalent condition is that every ideal have a finite basis. The proof that the three conditions are equal is precisely the same as in standard ideal theory. For classes of ideals which satisfy these conditions we have the following principle of induction. If a property applies to JO and if from the fact that it applies to all proper divisors of an ideal J it follows that the property applies also to J then the property applies to all ideals (in JOover K). We shall say that the minimum condition is satisfied if every non-empty set of ideals C' c C includes at least one ideal J c C' such that J i s not a proper divisor of any other ideal in C'. An equivalent condition is that for every descending chain of ideals, J1 2

JZ 2 J3

...

3

Jn

2

...

.. .

there exists a positive integer n such that Jn = Jn+l= Jn+1 = For classes C which satisfy the maximum condition, we have the fundamental theorem 7.2.1. THEOREM. Every ideal is the intersection of a finite number of irreducible ideals.

The proof is based on the principle of induction and can be taken from Algebra almost in its entirety. The theorem holds for all irreducible ideals, in particular for Jo. Suppose that it applies to all proper divisors of an ideal J. If J is irreducible then we have finished. If not, then J = [J', J"] where J' and J " are proper divisors of J. Since the theorem is true for J' and J", by assumption, we have = [J;, J i , ..., J ' ] m :, J " = [J';, J i , . .. ., J , ] , n

J'

where the J;, J i are all irreducible. Hence,

>1 >1

7.3.1

175

CONNECTION BETWEEN IDEALS IN DIFFERENT DOMAINS

can be represented as the intersection of a finite number of irreducible ideal. This proves the theorem, by the principle of induction.

7.3. Connection between Ideals in Different Domains. Let K, K*, and JObe three sets of sentences in a language L such that K c K*. Then all ideals in JOover K* will be seen to be ideals in JOover K while the converse is not true in general. Again, let K, Jo, J*,be three sets of sentences in L such that JO c J*,. We propose to investigate the relations between the class C of ideals in JOover K and the class C* of ideals in J*,over K. Let J be any ideal in Jo.We denote by J * the set of sentences of J*, which are deducible from K U J, J * = S ( K U J ) n J*,.J * is an ideal in J*,; it will be called the closure of J i n J*,. Conversely, if J ’ is any ideal, in C* then the intersection of J‘ and JOis an ideal J in Jo.If J; and Ji are two different ideals in J*,, we may find that nevertheless the corresponding ideals J1 and JZin JO are equal, J; n JO= Ji n Jo.On the other hand, if J * is the closure of J E C in J*,then J = J * n Jo. If C* satisfies the maximum condition, so does C. In fact, the existence of an infinite strictly ascending chain of ideals in Jo, J1 c J 2 c J 3 . . . implies that the chain of closures of these ideals in JOalso is strictly ascending. A similar argument applies to the corresponding statement concerning the minimum condition. It is not generally true that if the maximum (minimum) condition is satisfied in C then it is satisfied also in C * . However there are certain particular cases in which the conclusion holds and two of these will now be considered. Given Jo, we denote by the set of sentences obtained from J, by the repeated use of the operation of conjunction, including the sentences of JOas such. Thus, if X,Y, Z, are elements of JOthen the sentences X,X A Y, X A [ Y A Z ] , [XA Y ] A Z all are elements of Similarly, we denote by JT the set of sentences obtained from JO be repeated use of the operation of disjunction. We denote the classes of ideals in Jo, and TJ over a fixed set of axioms K, by C, C A and C respectively. The connection between C and CAis quite simple. Let J A be any ideal in C A and let J be the intersection of JOand JA.We propose to show that JA is the closure of J in Indeed, if J * is the closure of J in J t then J * contains all sentences obtained by repeated conjunction from sentences of J since all such sentences can be deduced from J. Now suppose that J * also contains a sentence X which cannot be obtained by repeated conjunction from J (although, by the definition of J t , it can be obtained

Jt

Jt.

”,

Jt.

Jt,

176

[7.3.

METAMATHEMATICAL TJ3EORY OF IDEALS

by repeated conjunction from the sentences of Jo).Assume in particular that Xis a sentence of this description which is obtained from sentences of JO by a minimum number of operations of conjunction. X cannot belong to JO for in that case it would belong also to J, by definition. Accordingly, X can be written in the form X = Xi A XZwhere Xi and XZ both are obtained from JO by the application of a smaller number of conjunctions than X. But Xi and XZboth are deducible from X and so belong to J*. Hence, by the maximum property of X both Xi and Xz are obtained from elements of JO by repeated conjunction. Since X is the conjunction of Xi and XZ the same then applies to X,contrary to the definition of that sentence. It follows that J* coincides with JA, as asserted. Thus, every ideal of C* is the closure of its intersection with Jo. This establishes a one-to-one correspondence between the ideals of C and the ideals of CA, Jt, JA,such that J1 c JZfor any two ideals in C if and only if J t c J$ for the corresponding ideals in C A . It follows that C A satisfies the maximum (minimum) condition if and only if C satisfies the same condition. The connection between C and C V is less simple. However, we shall prove the following.

7.3.1. THEOREM. If the maximum condition holds in C then it holds also in CV.

PROOF.Suppose that Cv does not satisfy the maximum condition. We propose to show that in that case, C cannot satisfy the maximum condition

either. Since C V does not satisfy the maximum condition it contains an ideal JV without finite basis. From Jv we may select a sequence of elements { Y I , Yz, Y3, . . . } such that the ideals Jl = ( Yi, . . ., Yk)in . I : constitute an ascending chain with the property that for any positive integer k there exists an integer m > k such that JI is a proper subset of J:. In other words, an infinite number of the Jk are different from each other. All the Yi are disjunctions of sentences in Jo. Since the order of association of these sentences is irrelevant (i.e. sentences of the forms X V [YV 21 and [XV Y ] V Z are interchangeable) we may write

Yi = xy

v

v .. , v X 2 , i = 1, 2, 3, ...

where the X t ) belong to Jo. We are going to show how to select from the set of X ( i ) an infinite subset Y I = X$:,Y z = X(Lt, . . ., y* = X(Z) kn’ * * ~

-

7.3.1

CONNECTION BETWEEN IDEALS IN DIFFERENT DOMAINS

177

such that in the ascending chain of ideals JT c J*, ... c JZ = . . . an infinite number are different from one another where J : = (YT, . . ., YE) is the ideal generated by Y ; , . , ., Y*,in Jo over K. If mi = 1 so that Y1 itself belongs to JO then we define Y? = Y1. If ml > 1, we propose to show that we can replace Yi by a sentence Y; which contains less than rnl disjuncts, all selected from among the disjuncts of Y1, i.e. from X ( q , . . ., Xg)l, such that the resulting sequence {Y;, Yz, Y3, . .., Y,, .. .1 still gives rise to an ascending sequence of ideals an infinite number of which are different from each other. In fact, put Z; = X $ ) , 2; = X ( i ) V .. . VXpl, then Yi E Z; V Z i isa theorem. Consider the two sequences of ideals in J:, { J i } and { J i } , where JL = (Z;, Yz, ..., Yk) and J;L = (Z';, Yz, ..., Yk). We have J I = [JL, J;]. Indeed, 2; V Zy, can be deduced from both 2; and 2; and so belongs to [JL, J i ] . Hence, J x c [JL, J i ] . On the other hand, any sentence that can be deduced from K together with Yz,Y3, . . . Y k , and Z;, as well as from K together with Yz,Y3, .. Yk and Z';, can also be deduced from K together with Yz, . ., Yk, and 2; V Z';, by a simple application of the calculus of propositions. Hence, [JL, J i ] c Jk and so J I = [J;, J i ] as asserted. It follows that at least one of the two chains

.

.,

and contains an infinite number of different ideals ("is effectively infinite"). For, otherwise, we should have JL = JL+l = JL+z = . . . for some k and ,, J , = J I , , 1 = JY+z = .. . for some 1. Hence, for m = max (k,1) , I ,

[JAYJ k ] = [JA+l,Jm+l] = [JA+z,J;,,] = . .. i.e. JZ = JZ+l = JZ+a = . . . contrary to the assumption that an infinite number of JY are different from each other. Thus, either the sequence {Zi, Yz,Y3, . ..} or the sequence {Z';, Yz, Y3, .. .} (or both) gives rise to an effectively infinite chain of ideals. If only the second possibility applies, and ml > 2, we apply the above procedure to Z'; in order to reduce the number of disjuncts of Z';, which is mi - 1, still further. Continuing in this way, we finally determine a disjunct X$! of Y1 such that the sequence { X ( i ; , Yz, Y3, . ..I gives rise to an effectively infinite chain of ideals in J", 0 J (1l ) c J $ ) c . . . where 5';) = ( X ( 9 , Yz, . . ,, Yk).From the sequence {A'';), Yz,Y3, ...} we now remove all sentences which are included in ,,

178

17.3.

hiETAMATHEMATICAL. THEORY OF IDEALS

.

J ( i ) = ( X ( i l ) except X(k! itself. The resulting sequence { Y ( l ) ,Y$), . .} still gives rise to an effectively infinite chain of ideals .I1 c JIZ t J13 c ... c Jik C ... . Now the sentence Y ( t )is of the form Y'" =

1

v x'c,"'v ... v xg: .

If mk, = 1 so that Yet) belongs to JOthen we define simply Y ( i )= Y ( q . If me,> 1, we put 2; =

q), 2; = x($) v . . . v xy . "'a

Then Y(;) Z i V Z i is a theorem. Similarly as before we now consider the two sequences fV;} and {ZJL} where

where we recall that all ideals are taken in JI. Again, J(k) = [2Ji,2J'J, and we may conclude as before that at least one of the two sequences {ZJ;} and {? isI effectively ;} infinite. Repeating this procedure if necessary, we finally select a sentence Xi:) such that the sequence {Xi?, Xi!), Y$), Y$),. . .} gives rise to a chain of ideals that is effectively infinite. The chain in question is J(:) c J $ ) c J(:) c

where

. ,.

Moreover, J(i) is still a proper subset of J$), for since Y$)is deducible X:) E would simply Y ( i )E 5':) = J11, and this is contrary from,):'A to construction. Once again, we remove all J ( y , k Z 3, which are contained in J ( 3 . We obtain an infinite sequence

{Y(?),Y y , Y(i),. ..}

where Yc:)= Ycl) 1 = X(') ki Y f22 )= Xf:),such that the chain of ideals .I(: c )J $ ) C . . . is effectively infinite and J(:) # J(') 2 , J(') 2 # J(i), the ideals .If)being defined by J$? = ( Y( f ) ,... Y$)). Continuing in this way we finally obtain a sequence of sentences 3

7.4.1

179

DISJUNCTIVE IDEALS

such that every Y $ ) is a disjunct of one of the sentences of the original sequence { Y1, Yz, Y3, . . .} and such that no Y(\) = i = 2,3,4, . .. can be deduced from K U {Y(i)7. . ., Y$:;)}. It follows that the chain of ideals J : c J*, c J : c ... in JO is strictly ascending where J : = (Y(:),. . . Y(,k))is the ideal generated by Y ( ; ) , . . ., Y ( t )in Jo. This is contrary to the assumption that the maximum condition is satisfied in C and proves the theorem. 7.4. Disjunctive Ideals. The domain Jo will be said to be disjunctive with respect to K if for any X I , XZin JOthere exists a sentence X in JOsuch that [Xi V XZ]3 X is deducible from K. For any Jo, the domain JZ is disjunctive. The theory of ideals in disjunctive domains which is given below is of a particularily transparent nature. We shall refer to the ideals in question briefly as disjunctive ideals. The following two distributive laws hold for arbitrary disjunctive ideals J, Ji, Jz. 7.4.1.

[ ( J , J l ) , (J, J z ) ] = (J, [ J l , J Z I )

7.4.2.

( [ J ,Jil, [J, Jz1) = [J, (Ji, J2)]

PROOF OF 7.4.1. Let X be any element of the ideal given by the left hand side of 7.4.1. then X is deducible from the set K U J u J1 and at the same time, from K U J U Jz. It follows that there exist elements Y1, . . ., YZof J1 such that Y1 A ... A YZ3 X is deducible from K U J and at the same time, there exist elements Z1, . . .,Zm of JZsuch that 2 1 A . .. A Zm 3 X is deducible from K U J. Hence, by the rules of the calculus of proposition the sentence [Yi A

. .. A Yz]V [Zi A ... A Z m ] 3 X

is deducible from K u J and the same then applies to 7.4.3.

[Yi V Z i ] A [Yi V Z Z ]A ... A [Yt V Za] A [YiA&] 3 X .

... A

But all sentences Yt V Ze are deducible from both J1 and Jz. Also, JO is disjunctive with respect to K and so there exist sentences X t k in JOsuch that Xtk f [ Yt V Zk] is deducible from K. Hence, X t k is deducible from K U J1 and belongs to J1; similarly X4k is deducible from K U JZ and belongs to 5 2 . It follows that the Xtk: all belong to [Jl, Jz], and that X

180

[7.4.

hlETAMATHEMAllCAL THEORY OF IDEALS

is deducible from K u J u [A,Jz]. This shows that X belongs to the right hand side of 7.4.1. Conversely, suppose that X belongs to the right hand side of 7.4.1. Then Xis deducible from K u J u [Ji, Jz] and so it is certainly deducible from K u J u J1. This shows that X belongs to (J, JI). The same argument proves that X belongs to (J, Jz), and hence, to [ (J, JI), (J, Jz)]. This completes the proof of 7.4.1. PROOFOF 7.4.2. Suppose X belongs to the right hand side of 7.4.2. Then X E J and at the same time X is deducible from K u JI u Jz. Thus, there exist sentences Y I , ..., Yl E J1, and Z I , . . ., ZmE JZ such that Y l A ... A Y z A Z I A... A Z m 3 X is deducible from K. The same is then true of 7.4.4.

[XV Y I ]A

.. . A [XV Y z ]A [XV Z , ] A .. . A [XV Zm]

3

X.

Now, we are dealing with a disjunctive domain, so there exist sentences Y;, . . . Y ; , Z;, . . .,Zkin JO such that the sentences Y; [ ( X V Yt)], 2; [X V Zt] are all deducible from K. Hence, Y; E [J,J1], i = 1, . . .,Z, Zr’ E [J, Jz), i = 1, ..., m and so X E ( [ J ,Ji], [J, J z ] ) , by 7.4.4. Denote the left hand side of 7.4.2. by J’. Then [J,J I ] c Jand [J,Jz]c J and so J’ c J. Also [J, 511 c JI and [J,5 2 1 c JZ and so J‘ c (51, Jz). Hence, J’ c [J, (Ji, Jz)],every sentence that belongs to the left hand side of 7.4.2., belongs also to its right hand side. This proves 7.4.2. Except where a statement is made to the contrary, we shall suppose throughout the remainder of this section that we are dealing with disjunctive ideals. 7.4.5. THEOREM. Let BI and Bz be bases for two ideals,

J1 and Jz, respectively. For every Yr E BI and Z k E B2 determine a sentence Xik E JO such that [YsV Zk] X t k is deducible from K. Then the set {Xik} = B constitutes a basis for [A,Jz].

PROOF. Since Yi 3 X t k and Zk 3 XIKare deducible from K we have B c [Ji, J23. Now let X be any element of [A,Jz]. Then there exist sentences Yi, . . ., Yz E BI,and Z1, . . .,Zm E B2 such that the sentences and and hence

Y i A ... A Yi 3 X

ZiA

... h Z m 3 X

7.4.1

DISJUNCTIVE IDEALS

[Yi V 211 A [Yi V Z Z ]A

... A

181

[Yt V ZK]A

. . . A [YzV Zm]

3

X

are deducible from K. It follows that the sentence XiiAXi2h

... A X i k A

. a -

At1X

also is deducible from K. This shows that B is a basis for [JI,Jz]. 7.4.6. THEOREM. A necessary and sufficient condition for an ideal J to be irreducible is that for any X , XI, X Z E JOsuch that XI V X2 X is deducible from K, X EJ entails that at least one of the two sentences X I or X2 belongs to J.

PROOF.The condition is necessary. For suppose there exist X , XI, X2 in JO such that K t X I V XZG X and X E J while neither XI nor XZbelongs to J. Then we have to show that J is reducible. Put J1 = (J,(Xi)), JZ= (J,(XZ)). Then J1 # J a n d JZ# J. On the other hand, by 7.4.1. and 7.4.5.

[Ji, Jzl = [ (Ji (Xi)), (‘Ji(xz))] = (J, [(Xi), Cx2>]) = (J, (XI) = J so that J is indeed reducible. The condition is also sufficient. For suppose J i s reducible, J = [JI,Jz] where Ji # J and JZ# J and let Xi E Ji - J, X2 E 5 2 - J. Let X E JO be such that X EZX i V XZis deducible from K. Such an X exists since JO is disjunctive with respect to K. Then X I 1 X and X Z 3 X are both deducible from K and so X E Ji, X E J2, and hence X E J. Thus, the condition of the theorem is not satisfied for reducible J. 7.4.7. THEOREM. Let J be an ideal in JOand let

Y EJO- J. Then there exists an ideal J‘which divides J and which does not include Y, and which is maximal with respect to the property of including J and excluding Y. In general, we say that an ideal J is maximal with respect to a particular property if among all ideals which satisfy the property there is none that contains J a s a proper subset; and that J i s minimal with respect to a given property if it does not contain any ideal with that property as a proper subset.

PROOF.Let T be the set of ideals that include J and exclude Y. T is not empty since it contains J. It is partially ordered by the relation of inclusion. Let {Jv> be a subset of T which is monotonic under this relation (i.e. Jvc JL( or Jflc Jvfor any Jv,Jflin the set). Then {Jy}has an upper bound in T,i.e. the union Jvwhich is also an ideal that includes J

u V

and excludes Y. 7.4.7. now follows immediately from an application of Zorn’s lemma.

182

METAMATHEMATICAL THEORY OF IDEALS

[7.4.

7.4.8. THEOREM. Let J be an ideal in Jo, let Y E JO - J and let J be maximal with respect to the property of including J and excluding Y. Then J is irreducible.

PROOF.Suppose that J is reducible. By 7.4.6. there exist sentences X , X I , X2 in JO such that K I- X X I V X2 and such that X EJ but X I # J, X2 # J. .By the maximal property of J, the ideal (J,(XI)) includes Y and so K u J t XI Y, and similarly K U J t XZ 3 Y and hence, f>

K U J t X l V X 2 3 YandKU J t X 3 Y . B u t X E J a n d s o K u J k Y , and hence, Y E J. This is contrary to assumption and proves 7.4.8. Combining 7.4.7. and 7.4.8.,we obtain

7.4.9. THEOREM. Let J be an ideal and let Y EJO - J. Then there exists an irreducible ideal which includes J and excludes Y. Indeed, the ideal J’ which exists according to 7.4.7. is maximal also with respect to the property of including itself and of excluding Y. Hence J’ is irreducible, by 7.4.8. 7.4.10. THEOREM. Every ideal is the intersection of all irreducible ideals in which it is contained. This is obvious for J = JOand followsimmediately from 7.4.9.for J # Jo.

7.4.11. THEOREM. Let J be any ideal and let J’ be an irreducible ideal that includes J. Then there exists an ideal J“ c J’ which is minimal with respect to the property of being irreducible and of including J and being included in J’.

PROOF.Let T be the set of all irreducible ideals that include J and are included in J‘. T is not empty since it contains J’. It is partially ordered by the relation of inclusion. Let (J,,} be a subset of T which is monotonic (J,,) belongs to T. with respect to this relation. We claim that J* =

n Y

Indeed J* clearly is an ideal which includes J and is included in J’. Suppose that J* is not irreducible. Then there exist sentences X , X I , X2 in JOsuch that K k XE X I V X2 and such that X E J* but XI # J * , XZ# J*. If so, then X E J y for all ,, but there exist p and p such that X I # JD and X2 # Je. Now, by assumption either JD c Je or Je c Jb, and we shall consider the former possibility. Then XI # JD and XZ # JD although X EJb. This is contrary to the assumption that Jb is irreducible and proves that J* is indeed irreducible. 7.4.11. now follows from a direct application of Zorn’s lemma.

7.4.1

183

DISJUNCTIVE IDEALS

7.4.12. THEOREM. Every ideal J i s the intersection of all irreducible ideals which are minimal with respect to the property of including J and of being (at the same time) irreducible.

PROOF.For given J, let T* be the set of all irreducible ideals which are minimal with respect to the property mentioned in the theorem. T*is not empty, by 7.4.9. and 7.4.11. More particularly, let Y EJO - J. By 7.4.9. there exists an irreducible ideal J' that includes J and excludes Y, and by 7.4.11. there exists an irreducible ideal J" which is included in J' and hence excludes Y, and which includes J and is minimal in the sense of 7.4.11. and hence of 7.4.12. This proves 7.4.12. for the case that J # Jo. The theorem is trivial if J = Jo. Suppose now that JO satisfies the maximum condition, in addition to being a disjunctive domain. In this case we have 7.4.13. THEOREM. Any ideal J can be represented in one and (except for order) only one way as the irredundant intersection of a finite number of irreducible ideals. In this connection, an intersection of ideals J = [JI, ..., Jk] is called irredundant, if the omission of any Jt from the intersection leads to a result different from J.

PROOF.Suppose that J = [JI, .

. .,Jk] = [J;,...,JA]

where both representations are irredundant, and the ideals Jt, Ji are all irreducible. Consider the ideals (Ja, J;) and (Ja, [Ji, .. ., J A ] ) , i = 1, 2, ... k.By7.4.1.,

[ (Jt, JJ, (JB

[Ji, J;,

* * - 2

= (Jt, [J;,

JAj)] = (Jc

[J;,

[Ji, - JAI]) *

.. ., J m ] )= (Jt, J ) = Jg .

- 9

But Jz is irreducible and so either (Ji, J;) = Ji, and hence Jt 3 J;, or (Jt, [Ji,. . .,J A ] ) = Jt, and hence Jt 1 [Ji,. . .,JA]. If the first alternative does not hold, then the second alternative holds. In that case, consider the expression

[ (Jt, Ji), (Jt,

[J;,

* * - 9

JAI)] = ( J t , [Jk [J;, . . ., J;]) = Jt .

= (Ji,[Ji,

* *

- 9

JAI])

Since Ja is irreducible this again leads to the conclusion that Jt 3 J i or else Jc 3 [J;,. ., J' 1. Continuing in this way, we finally infer that Jt m, divides at least one Jl, 1 < I < m. Conversely, we may show that every

.

184

METAMATHEMATICAL THEORY OF IDEALS

[7.4.

Ji divides at least one Jg. Now, for a specific Jt, suppose that we have found an appropriate Ji such that Jt =I 5;. Interchanging the roles of the two representations,we find that there exists an i' suchthat Ji =) Jg'. Hence Jg =) Jt' But if i # i', then we can omit Je' in the representation J = [JI, . . ., Jm] without affecting the equality. This is contrary to the assumption that the representation is irredundant and so i = i', Jg = Jf'. = Ji. Thus for every i, 1 < i < k there exists an 1 = Z(i) such that Jt = Ji(a),1 Q l ( i ) < m. That is to say, all ideals which appear in the first representation appear also in the second representation. Since the second representation is irredundant it follows that it cannot contain any other ideals. This shows that there cannot be more than one representation of J as an irredundant intersection of irreducible ideals. In order to establish that there exists at least one representation of this kind we consider any representation of J as an intersection of a finite number of irreducible ideals. Such a representation exists, by 7.2. l., J = [A, ., .,J,], say. Now consider a sequence of integers of minimum length ii < i 2 < .. . < ij, j < n, such that the intersection [Jt,,Jc,, . . ., Jr,] is still equal to J. This is the required representation of J a s an irredundant intersection of irreducible ideals. The proof of 7.4.13. is now complete. We observe that the ideals Jt which appear in the representation J = [A, . . .,J,] of an ideal as an irredundant intersection of irreducible ideals are minimal with respect to the property of being irreducible ideals that include J. Indeed, consider any particular Ji which is included in this set. Then Jt 3 J. Hence, by 7.4.1 1. there exists an irreducible ideal JI which is included in Jg and includes J and which is minimal in this sense. Then J = [J, Ji] = [Ji, . . ., Jt, . . ., Jn, Ji] = [Ji, . . ., JI-1, JI, Jg+l,

...,Jn] .

It is not difficult to see that the last-mentioned representation of J must be irredundant, and hence that JI = Jt, so that Jt itself has the required minimal property. Now let Jo be any domain, not necessarily disjunctive, and let JT be the domain of sentences obtained from JOby the repeated use of the connective of disjunction as introduced previously. J r is a disjunctive domain. 7.4.14. THEOREM. Let JV be an irreducible ideal in JZ and let J = JOn J v

Then J v is the closure of J in JZ.

= [Yl V . . . V Yn] be any element of JV, where Yt E Jo, . . ., n. The repeated application of 7.4.6. shows that at least one

PROOF.Let X i = 1,

7.5.1

185

IDEALS A N D HOMOMORPHISMS

of the Yt belongs to Jv and hence to J. Since YZ=I X is a theorem, it then follows that X belongs to the closure of J i n JZ. Accordingly, J V is included in the closure of J in J Z . On the other hand, any sentence which is deducibIe from K U J c a n also be deduced from K u J V and so J V includes the closure of J in J : . This proves 7.4.14. Thus, every irreducible ideal in JZ is the closure of an ideal in Jo. 7.4.15. THEOREM. Suppose that JO satisfies the maximum condition. Let J be an irreducible ideal in J o and let Jvbe the closure of J in JX. Then J V is irreducible.

PROOF.If Jv is not irreducible then it can be represented as the irredundant intersection of a finite number of irreducible ideals in JZ, J V = [Jy, ...,J',]. Since the J Y are irreducible they are the respectiveclosure of the ideals Ja = JOfl J Y in J:. It follows that if Jt = J for some i, then JY = J v , so that Jvis irreducible and we have finished in this case. Suppose then that Jt # J, i = 1, .- .,n, n > I. Then

. . .,Jm]. But JOn Jv = J since any sentence of JOwhich is deduciblefrom K u JV is deducible also from K U J. Hence J = [JI, . . ., Jn] so that J is redu-

JOn J v =JO

n

[JT, . . ., Jx1 = [Jor~J y , . . ., Jon J',]

= [J1,

cible, and this is contrary to assumption.

7.5. Ideals and Homomorphisms. Let H be a consistent set of sentences which includes a relation of equality, E(x, y), and let M be a model of H.

Let D f be a positive diagram of M, and suppose for simplicity that the relations and individuals of M are denoted by themselves in both D+ and H in so far as they occur in the Iatter set. Let K = H U D+ and let JO be the set of all atomic sentences which are constituted by the individuals and relations of M. We propose to establish a connection between the homomorphic images of M which are models of H and some of the ideals in JO over K. Let then M' be a model of H which is a homomorphic image of M (see section 2.1.) such that the relations and individuals which occur in H denote themselves also in M'. Let J be the ideal in JOwhich is generated by the sentences E(u, b) E JOsuch that in the homomorphic correspondence from M to M', a and b correspond to equal individuals in M'. Then we claim that, conversely, any sentence E(a, b) belongs to J only if u and b correspond to equal individuals in M'. To see this, we define a structure M* which contains the same individu-

186

METAMATHEMATICAL THEORY OF IDEALS

L7.5.

als and relations as M such that any relation holds in M* whenever it holds between the corresponding individuals of M‘. It then follows from the fact that M* is homomorphic to M‘ under the given correspondence that whenever a relation holds between certain individuals as individuals of M i t also holds between them as individuals of M * . Similarly, since M‘ is a model of H,M * also is a model of H and since M is a model of D+, M also is a model of D+.Thus, M * is a model of K. Now we suppose that E(c, d) belongs to J. In order to prove our assertion that c and d correspond to equal individuals in M’, we have to show only that E(c, d ) holds in M*. Assume on the contrary that E(c, d ) holds in M*. Then M * is a model of K, and of the sentences E(a, b) which constitute a basis of J but, at the same time, satisfies E(c, d). It follows that E(c, d ) cannot be deducible from K u J and hence, cannot belong to J. This is contrary to assumption, and proves our assertion. In this sense, therefore, there exists an ideal J in JO for every homomorphism of M. It is not difficult to see that if two homomorphic images of M , M’ and M”, correspond to the same ideal Jin JOthen they can be reduced to isomorphic normal structures. In particular, if H = KR,where KR is a set of axioms for the notion of a general ring (see section 2.2.),then we can not only find an ideal J for any given homomorphism from a ring M onto a ring M’ but, conversely, for every ideal J in JO we can construct a model M’ which is homomorphic to M and which bears the above mentioned relation to J. This follows from a one-to-one correspondence which will now be established between the ideals J i n JO and the “ordinary” two-sides ideals in M . To distinguish the latter from our metamathematical ideals, we shall refer to them briefly as arithmetical ideals. Given an ideal J in Jo, we define J* as the set of all individuals a of M such that E(a, 0) E J where 0 is a neutral element with respect to addition in M . (Unless M is normal, there may be more than one neutral element with respect to addition in M y but all these elements are equal in M. Accordingly, the particular choice of the neutral element 0 is irrelevant). J* is an arithmetical ideal in M . Indeed, if E(a, 0) E J and E(b, 0) E J and S(a, b, c ) holds in M - and hence belongs to D + and to K - then E(c, 0) E J since E(c, 0) can be deduced from E(a, 0), E(b, 0), S(a, byc) and KR, and so can be deduced from the union of J and K = KR u D+. Similarly, if E(a, 0) E J and E(b, 0) E J as before, and S(a, b, c) holds in M and, hence, belongs to D+,then E(d, 0) E J. It follows that, together with any two elements of My J* contains also their sum and difference N

7.5.1

IDEALS AND HOMOMORPHISMS

187

Again, if E(d, 0) and P(h, a, f) and P(a, h, g) hold in M and hence, belong to D+, then E ( f , 0) and E(.q, 0) also belong to J. Thus, together with any element a, J* contains also the elements ha and ah, where h is an arbitrary element of M. Finally, J* cannot be empty since E(0, 0) J and so 0 E J*. This shows that J* is indeed an arithmetical ideal in M. Conversely, let J* be any arithmetical ideal in M and let M‘ be the quotient ring MJJ*. Let J be the metamathematical ideal in JOwhich belongs to M’ in the sense detailed above. Then a sentence of the form E(a, 0) belongs to Jprecisely when a belongs to J,, i.e. the correspondence between J and J* is reciprocal, J t , J* . Moreover, under this correspondence, if J(l)+-+Jp)and J(’)t--)Jp) and J(l)= J(’)then JL’) = J $ ) and vice versa. Let J t , J,.Then the set { a , .. .,an, ...] is an ideal basis for J* if and only if the sentences E(al, 0), E(az, 0), . . ., E(an,0), . .. constitute a basis for J. Indeed, let . I ; be an arithmetical ideal which is generated by a single element ax, Ji = (a+ Then the corresponding metamathematical ideal J’ contains the sentence E(a1,O). Let J”be the metamathematical ideal generated by E(a1,O). Then J” c J ‘ and so the corresponding arithmetical ideal J’; is a subset of Ji. But J;’ contains UI and so J;’ =Ji and hence J” = J‘. Conversely, if a metamathematical ideal is generated by E(a1,O) then the corresponding arithmetical ideal is generated by al. Finally, if the set {al, a2, . ..,a,‘, ...} forms a basis of J*, then J* is the intersection of all arithmetical ideals that include the ideals (al), (a& . . ., (a,), . . It follows that J* corresponds to the ideal J in Jo,which is the intersection of all ideals in JOthat include the ideals (E(al,O)),(E(az,0)), . . . (E(a,, 0 ) ) , . . . Thus J is generated by the set E(a1, 0), E(az,O), . . . E(a,, 0), .., as asserted. In this sense, the present generalized concept of an ideal preserves the idea of the basis. Similarly, a maximum condition in JOcorresponds to a maximum condition in the related set of arithmetical ideals, irreducible ideals correspond to irreducible arithmetical ideals, and the representation by intersections of such ideals correspond. All our conclusions still apply if commutative rings are considered in place of general rings, H = KCRin place of H = KR. Taking H = KCR, we shall now consider ideals in the disjunctive domain JZ obtained from the domain JOdefined above. Thus, the elements of JT are the sentences

.

.

where Xk,

x = x 1 v x z v ... v x , , n = l , 2 , . . . k = 1,2, ...,n stands for an atomic sentence of the form

188

METAMATHEMATICAL THEORY OF IDEALS

[7.5.

E(a, b) or S(a, 6, c ) or P(a, b, c). It will simplify the discussion without restricting it in any essential way if we assume that M is normal. We may then denote by a b, a - b, ab the individuals which are equal to the sum, difference, or product of any two elements a and b of M. We wish to characterize the sets of disjunctions X which constitute disjunctive ideals J V in JZ, by purely arithmetical conditions. To begin with, it is clear that a sentence X = Xi V . . . V X , in which Xk, say, is of the form E(a, b) is contained in a disjunctive ideal J v if and only if the sentence X’which is obtained from X by replacing E (a, b) by E (a - b, 0) is also contained in J V . Similarly, if Xk is of the form S(a, b, c) then X belongs to P if and only if the sentence X‘ which is obtained from X by writing E(a b - c, 0) in place of Xk, belongs to J V . Finally, if Xk is of the form P(u, b, c ) then X belongs to J V if and only if the sentence X’ which is obtained from X by replacing Xk by E (ae b - c, 0), also belongs to J V . Accordingly, we may confine ourselves to the consideration of sentences of the form

+

+

7.5.1.

X = E(ai, 0) V E(uz,0) V

.. . V E(an, 0)

where we disregard brackets, as usual. Thus, if we denote by GOthe set of sentences of JZ which are of the form 7.5.1. there is a one-to-one correspondence between the ideals G in GOover K and their closures J v in J: over K. 7.5.2. THEOREM. In order that a non-empty set of sentences 7.5.1. constitute an ideal in GOit is necessary and sufficient that the following conditions be satisfied. 7.5.3. If we change the order of the atomic sentences in a sentence of G in any way then we obtain a sentence of G;

I f X e G and Y E G then X V Y E G ;

7.5.4.

7.5.5. If X E G contains an atomic sentence more than once then by omitting all except one of its occurences we obtain a sentence of G. For instance, if E(a, 0) V E(b, 0) V E(a, 0) belongs to G then E(a, 0) V E(b, 0) also belongs to G ; 7.5.6. Finally, if

and

X = E ( a i , O ) V E ( a z , O ) V ... V E ( U , , O ) E G 2, ~1

.

Y = E ( h , O ) V E ( b n , O ) V . . VE(bm,O)EG,m2 1

7.5.1

IDEALS

189

A N D HOMOMORPHISMS

then Z E G, where Z is the disjunction (in any order) of the nm atomic sentences E(CtkUr

f p i k ai

+

dikbk

&

qtkbk,

--

= 1, 2, ., n k = l , 2 ,...,m .

01, i

In this expression, Cfk and d i k are arbitrary elements of M and PIk and are non-negative integers, indicating continued addition, Oa = 0, l a = a, 2a = a a, etc. In particular, by 7.5.5. and 7.5.6.,E(0, 0) c. G. Other special cases of 7.5.6.are the following

qgk

7.5.7. then

+

If E ( u ~0), V E ( u ~0) , V

... V E(an, 0)

E(Q, a1,O) V E ( ~ z e~, 2 0) , V

E

G

. .. V E ( c ~ u0)~ E, G

where the Ck are arbitrary elements of M or are non-negative integers regarded as operators, as before. 7.5.8. If E(a1, 0) V E(az, 0) V

... V

E(an, 0) E G then the sentences an, 0) also belong to G for

.. . V E(f + -.

E(+al, 0) V E ( k a2, 0) V arbitrary distributions of and

PROOFOF 7.5.2. The conditions 7.5.3.-7.5.6. are necessary. 7.5.3. is

obvious, 7.5.4.and 7.5.5.follow immediately from the fact that X 3 X V Y and XV X =I Xare theorems of the Lower predicate calculus for arbitrary sentences X and Y. In order to prove the necessity of 7.5.6. we have to show that Z can be deduced from KCRand D+ together with X and Y as detailed in 7.5.6.In other words, we have to show that if Xand Y hold in a commutative ring M' which is a model of D+,then Z also holds in M'. But X and Y only hold in M' if E(ag, 0) holds in M' for some i, 1 Q i < n, and E ( b k , 0) holds in M' for some k, 1 < k < m. Hence, E ( C i k a t fp t k a r d k b k f q t k b k , 0)

+

holds in M' for these specific i and k and this entails that Z holds also in M'. The conditions 7.5.3.-7.5.6. are also sufficient. Given a non-empty set G c Go which satisfies these conditions, we have to show that any sentence X E GO which can be deduced from G in conjunction with KCR u D+,is contained in G. Suppose that X = E(b1, 0) V ... V E(bn, 0). The assumption is that there exists a sentence

190

METAMATHEMATTCAL THEORY OF IDEALS

Z = YiA

[7.5.

... A Y j z X

where YZE G, i = I , ...,j , which is deducible from KCR U D+ = K. Let Yi = E(a(:),0) V E(a(i),0) V . . . V E(a(;),0), i = 1, .. .,I. Then Z is deducible from K if and only if the sentence

[E(a(i:,O)A E(a(gi,O) A

. . .] V [E(at',O) A E(a\t),O) A . . .] V . . .

3

X

is deducible from K where the implicans contain all possible combinations for the subscripts. In that case also, the sentences 7.5.9.

E(a(;{,0) A E(u(;!, 0)A

...

3

X

can be deduced from K for all possible combinations of the subscripts kl, k2, . . . 7.5.9. shows, for a particular set kl,k2, . . . that Xis contained in the ideal G* in GOwhich is generated by the set {E(u(i;,0), E(a(E!,0), , . . It follows that Xis certainly contained in the closure J* of G* in Jo. On the other hand, let J be the ideal generated by the set {E(u(i;,0), E(di$O), ...I in Jo. By the procedure explained at the beginning of this section, we may associate J with a homomorphism of M. More particularly, we may choose the homomorphic image M * of M which corresponds to J so that its elements coincide with the elements of M and such that M * is a model of D + as well as of J. It follows that X = E(b1,O) V E(be, 0) V . .. holds in M*, i.e. E(bz, 0) holds in M * for some i. This shows that br belongs to the arithmetical ideal generated by a(;:,a(gi, ... Thus, bc is of the form C(CZU%{ & pza%{)where the ca are elements of M

>.

1

and the pi are non-negative integers. There is such an expression for every a(;!, . . . which may occur in the implicans of 7.5.9. combination It follows that Xis identical with a disjunction whose terms are 7.5.10.

varying over all the combinations a(;:, a(;!, . . .,which may occur in 7.5.9., possibly together with some expressions E(bk, 0) which cannot be represented by 7.5.10. It is also quite possible that 7.5.10. represents the same E(br, 0) for different sets a(;:,a(;!, . . . and a(;:, a(;:, . . ., Let X' be the disjunction of all sentences 7.5.10. which occur for the different combinaa(Ei, . . . . The repeated application of 7.5.6. shows that X' tions belongs to G. But Xis obtained from X ' by the possible omission of some terms which coincide with a surviving term and by the possible addition to the disjunction of some more terms E(bk, 0). Hence X E G, in view of 7.5.4. and 7.5.5. This completes the proof of the theorem.

7.6.1

191

PROBLEMS

A slight modification of condition 7.5.6. is required in order to adapt 7.5.2. to general (non-commutative) rings, H = KR.The condition becomes 7.5.11. If and

E(a1, 0) V E(a2,O) V

. . . V E(u,, 0) E G

E(bi,O) V E(b2,O) V

. . . V E(bm, 0) E G

then Z E G, Z being the disjunction of the nm sentences E(ctkat atcik fpixat dtk bt+ f qtkbr, 0) where the ctk, cik, d t k , dt'k correspond to arbitrary elements of M while the p t k and q t k are non-negative integers, as before. We may remove all traces of Metamathematics from the concept of a disjunctive ideal by replacing every sentence of the form E(a1,O) V E(a2,O) V . . . V E(ak, 0) by a finite set (al, . . ., ak). Thus a disjunctive ideal becomes a set of finite sets of the elements of a given ring, obeying certain conditions which correspond to 7.5.4.-7.5.6. for a commutative ring or to 7.5.4., 7.5.5., 7.5.1 1. for a general ring.

+

+

7.6. Problems. 7.6.1. Correlate the normal subgroups of a given group with certain metamathematical ideals. 7.6.2. Correlate the left ideals of a given ring with certain metamathematical ideals. (Hint. Make use of a modified set of axioms for substitutivity with respect to multiplication requiring only (VX)WY)(\dZ)(\dV)(\d'w)

[m, A E(z, 0)

'w)

= [w, Y , z> EE P(X, v, 'w)]])

*

7.6.3. Let K = KCF U D, where KCFis a set of axioms for the notion of a commutative field, as before, and D is the diagram of a model M of KCF. For each polynomial q (x) with coefficient in M , construct a sentence X , which states that q ( X ) possesses at least one root. Let JObe the set of all these sentences. Discuss the theory of metamathematical ideals in JO over K. 7.6.4. Which of the theorems of 7.4. (i) remain valid, (ii) fail completely, (iii) apply in a modified form, if we drop the assumption that JO is disjunctive with respect to K?

References. Chapter VII follows A. Robinson 1951 and A. Robinson 1955 with some simplifications. The theory of systems mentioned at the beginning of section 7.2. is given in Tarski 193511936.

C H A P T E R VIII

METAMATHEMATICAL THEORY OF VARIETIES 8.1. Varieties of Structures. For the particular case that K is empty while Jo includes all sentences of a language L, the ideals of JO over K coincide with the T-systems of section 1.6. In that case, JO is clearly disjunctive.

We shall now develop a theory of the sets of models which correspond to the ideals of JOfor the general case of a domain JOwhich is disjunctive over a given K. Suppose that in addition to Kand JOwe are given a set YOwhich consists of models M of K such that all sentences of JO are defined in M, and such that for every set H which is consistent with K (i.e. H u K is consistent) and which consists of a subset of JOtogether with the set of negations of another subset of Jo, there exists an element of VO,which is a model of H. A set VOof this kind will be called a full system of structures for JOover K. Let A be any subset of Jo. Then the set of structures of the given VO which are models of K u A will be called the variety of structures of A, V = V ( A ) . The relation between A and its variety will be indicated also by A --f V ( A ) in agreement with the notation introduced in 1.6, above. If J = (A) is the ideal generated by A in JO then it will be seen that V ( J ) = V ( A ) . Accordingly, when investigating the class of all varieties of structures we may suppose that the varieties are determined by ideals in Jo. 8.1.1. THEOREM. Let J1 and JZ be two ideals in Then V(J1) 3 V(J2).

JO such that

J1

c Jz.

PROOF.Obvious. 8.1.2. THEOREM. Let J1 and JZ be two ideals in JO such that J1 # J2. Then V(J1) # V(J2).

-

PROOF.By assumption on of the two ideals, say J1 contains a sentence X which is not contained in Jz. It follows that K u JZ u { X}is consistent and hence, possesses a model M in VO.Thus, M E V ( A ) but M f V(J1). This proves the theorem. 192

8.1.1

193

VARIETIES OF SlRUCWRES

In particular, if JZ is a proper subset of

8.1.1., V(J1) is a proper subset of V(Jz).

J1

then taking into account

Since every variety is the variety of some ideal, 8.1.2. shows that the correspondence between ideals and varieties is one-to-one.

8.1.3. THEOREM. Let V be the variety of an ideal J in Jo.Then J consists of all sentences of JOwhich hold in all models of V. PROOF. Let J1 be the set of all sentences of $0 which hold in all structures of V. J1 includes J since all sentences of J hold in all structures of V. Suppose that J1 # J. Then there exists an element X of J1 - J such that K u J u {- X >is consistent. Let M be a model of this set. Then M E V although M does not satisfy all sentences of J1. This contradicts the definition of J1, and proves the theorem. As an immediate consequence of 8.1.3. we have 8.1.4. THEOREM. V(J1) J1, JZ in Jo. Next, we prove

3

V(J2) entails

J1

c J2 for any two ideals

8.1.5. THEOREM. V ( ( J I , JZ)) = V ( A ) n V ( J Z ) and V((J1, J2)) = V(J1) u V(J2)

for any two ideals J1, JZ in Jo. PROOF.Any model of

(51,Jz)is

a model of both J1 and Jz.Hence

v ( ( ~ 1JZ)) ,

= V(JI)n

~ ( ~ 2 1

On the other hand, any structure M which is a model of K and of J I , and also of Kand of JZ satisfiesall sentenceswhich are entailed by K U J1 U Jz and which are defined in M. Hence, V ( J I ) n v ( J ~ )= V (( J ~JZ)) ,

.

This proves the first formula of 8.1.5. In order to prove the second formula, we observe that V(J1) c V ( [Jl, Jz])and V(Jz) C V ( [A,Jzl) and so V(J1) u V(J2) = V ( [Jl, JZI) Now suppose that there is a structure M E V ( [Jl, Jz])which is neither a model of J1 nor of J2. Then there exist sentences X I E J1, XZE J2 such that M satisfies neither XI nor XZ.Since these sentences are defined in M, it follows that M satisfies X I A XZ and hence [ X I V XZ]. But JOis disjunctive and so it contains a sentence X such that X XI V XZis

- -

-

194

METAMATHEMATICAL THEORY OF VARIETIES

[al.

deducible from K . Accordingly, M satisfies also X . But X belongs to [JI,521 since K t XI 2 X and K k X Z 3 X. This shows that M cannot be a model of [J1,J Z ] ,contrary to assumption. We infer that V(J1) U V(J2) = V ( [JI,J z ] )as asserted. We conclude that the correspondence J -+ V establishes a lattice duality between the set of ideals and the set of varieties such that the sum of two ideals corresponds to the intersection of the corresponding varieties and the intersection of two ideals corresponds to the union of the two varieties. N

8.1.6. THEOREM. Let (Vv) be any set of varieties and let (Jv> be the corresponding set of ideals. Then U {.Iv} -+ f7 {Vv}. V

V

PROOF.On one hand, all structures of of each {Jv), hence V ( U ( J y ) ) 3 v

{Vv} satisfy the sentences V

f7 {Vv}. On

the other hand, if any

Y

structure is a model of all Jv then it belongs to all Vv, so that V ( U { J , } ) c f7 {Vv}. This proves 8.1.6. - Observe that in general V

V

U {Jv)is not an ideal. V

A variety Y is called reducible if there exist varieties VI such V2 and that

8.1.7.

V=

v1

u

V2,

v#

Vl, v # V2

A variety which is not reducible is called irreducible. The above one-toone correspondence between ideals and varieties shows that a variety is irreducible if and only if the corresponding ideal is irreducible. A structure M E VOis said to be a generic structure of the variety V c VOif any sentence of JO holds in M if and only if it holds in all structures of V. Thus, if V = V ( J ) where J is an ideal in JO then any sentence of JObelongs to J if and only if it holds in M . It is an immediate consequence of this definition that M E V. M will be said to be a generic structure of J also. 8.1.8. THEOREM. In order that a non-empty variety, V, be irreducible it is necessary and sufficient that V possess a generic structure M .

PROOF.Let J be the ideal which corresponds to V , J --f V. Suppose that V, and hence J, is irreducible. If J = JOand V is not empty then every M E Vis a generic structure of V. Suppose now that J # JOand let H be the union of Jand of the negations of all sentences of Jo - J. We claim that K U H is consistent. Indeed, K u J is consistent, otherwise J = Jo, contrary to assumption. Thus, if K u H is inconsistent there exist

8.1.1

VARIETIES OF STRUCTURES

195

sentences Y I , .. ., YZE JO- J, I 2 1 such that K u J u {- Y1, N Y2, ..., Y z ) is inconsistent. Then K U J t- Y1 V YZV ,.. V YZand so Y1 V Y2 V . . . V Yz E J. But J is irreducible and so Yj E J for some j , 1 < j , < I, and this is contrary to assumption. We conclude that K u H is consistent and possesses a model M in VO.M is the required generic structure of V. Conversely, suppose that V possesses a generic structure M . Then we have to show that V is irreducible or, equivalently, that J is irreducible. Let X , X I , XZbe three sentences of JOsuch that K t- X X I V XZ,and such that X belongs to J. Then X holds in M. It follows that at least one of the two sentences X I or X Zholds in M and hence, belongs to J. Hence J is irreducible, by 7.4.6. This proves 8.1.8. N

8.1.9. THEOREM. Suppose V =

U {Vv) where

V, Vv are varieties such

V

that Vv # 0 and Vv # V for all v, and where the number of the V, may be infinite. Then V is reducible. Thus, an irreducible variety cannot be represented non-trivially even by an infinite number of varieties.

PROOF.Suppose, contrary to the assertion of the theorem that V is irreducible. By the assumptions of the theorem, V cannot be empty. Hence, by 8.1.8., V possesses a generic structure Mywhich is included in V. Since V = {Vy}, M E V, for some particular ,u = v. Let J be the

u V

ideal which corresponds to V, and let J, be the ideal which corresponds to V,. Then V 2 V, and hence J C J,. Also, M satisfies all sentences of J, since it belongs to V,, and does not satisfy any sentence which does not belong to J since it is a generic structure of J. Hence J = J, and V = V,. This is contrary to assumption and proves the theorem. Let M be any structure which is contained in VOand let JM be the set of all sentences in JOwhich hold in M. Then it will be seen that J M is an ideal. The corresponding variety VM= V ( J M )contains M . Moreover, M is a generic structure of V M ,so that VMis irreducible. 8.1.10. THEOREM. Every variety V is the union of a set of irreducible varieties. The theorem is trivial if V is empty, since it is then itself irreducible. For non-empty V, we have V = {Vv}where Vv = VM”as defined

u V

above, and Mv varies over all elements of V. 8.1.11. THEOREM. Let V

=

u {Vv} V

=

U { V i }be two representations of V

196

METAMATHEMATICAL THEORY OF VARIETIES

[8.1.

a variety as a union of irreducible varieties. Then every variety V, of the first representation is included in a variety V‘ of the second representation and vice versa. PROOF.Trivial, if the V under consideration is empty. Suppose V # 0 then V includes a generic variety M. M E V and so M E V; for some irreducible variety of the second representation. Let J i be the corresponding ideal, then the sentences of J; all satisfy M and hence, belong to the ideal J, of Vv.Thus J; c J, and so V, c V;, as required. Let V, V‘ be two varieties, V’ c V and V‘ irreducible. Then V‘ is said tb be maximal in Vif any irreducible variety V” such that V’ c V” c Vis equal to V’. 8.1.12. THEOREM. Let V, V‘ be two varieties, V’ c Vand V’ irreducible. Then there exists an irreducible variety V” which is maximal in V and such that V’ c V’‘ c V.

PROOF.Let J, J’ be the ideals which correspond to V, V‘ so that J c J’ and J’ is irreducible. By 7.4.11. there exists an ideal J” c J’ which is minimal with respect to the property of being irreducible and of including J. Let V” be the corresponding variety. V” satisfies the condition of the theorem. 8.1.13. THEOREM. Every variety V is equal to the intersection of all irreducible varieties which are maximal in V. PROOF.Let ( V y }be the set of irreducible varieties which are maximal in V. We have to show that every M E V is contained in some Vv. Now M is contained in the irreducible variety VMand VM c V. Hence, by 8.1.12., there exists an irreducible variety which is maximal in V and which includes VMand hence M. 8.1.14. THEOREM. Let V = U {V,} = U {VY’}be two representations of 1

V

a variety V as a union of irreducible varieties which are maximal in V. Then the two representations coincide, i.e. the same irreducible varieties appear in both. Indeed, by 8.1.1 l., every Vv is included in some V,. But Vv is maximal in V and so V, = V,. This proves 8.1.14. The theorem shows that the representation given by 8.1.13 is unique. The class of varieties of VOis said to satisfy the minimum condition if in any non-empty set of such varieties there is at least one which is not

8.1.1

VARIETIES OF STRUCTURES

197

included in any other variety of the set. It follows from the correspondence between ideals and varieties that the varieties of VOsatisfy the minimum condition if and only if the ideals of JO satisfy the maximum condition (7.1. above). An equivalent condition is the finite descending chain condition for varieties which states that for any descending chain of varieties V l D v2 3 V3 ... there exists an integer n such that Vn = Vn+l = Vn+z = . . . Supposing that these conditions are satisfied and making use of the correspondence between varieties and ideals, we obtain from 7.4.13., 8.1.15. THEOREM. Suppose that the minimum condition for varieties is satisfied. Then every variety V can be represented in one and (except for

order) only one way as the irredundant union of a finite number of irreducible varieties, V = VI u VZ U ... U V,. The representation V = V1 u VZ u . . . U Vn is called irredundant if the omission of any variety on the right hand side destroys the equality. It follows from a remark in 7.4. that the varieties which occur in the representation are all maximal in V. Theorems 8.1.13. and 8.1.14. now show that V1, VZ,.. ., Vn constitute the totality of all ideals which are maximal in V. To obtain an example for the theory of this section, let KIDbe a set of axioms for the concept of a (commutative) integral domain. By this we mean here a commutative ring with a unit element which is different from 0 and without zero-divisors. KIDmay be taken as the union of the set of axioms for commutative rings, KCR,and of

8.1.16.

P(1, X, 1) S(1, 1, 1) (vX>(vY)(vZ)[P(x,y,z)hS(z,z,z) = S(X,X,4WY,Y,Y)] (VX)

which contains an individual constant, 1. As usual, 1will be supposed to denote itself in any model of KID. Now let MObe a particular integral domain, i.e. a model of KIDand let Dt be the positive diagram of MOsuch that the relations and individualsof MOare denoted by themselves. Suppose that 0 is a neutral element with respect to addition in it40 (while 1 is a neutral element with respect to multiplication, by previous remarks). Let K = KID u 0: and let J o be the set of sentences E(a, 0) where a varies over all individuals of Mo. We claim that JOis disjunctive with respect to K.

198

METAMATHEMATICAL THEORY OF VARIETIES

t8.l.

Indeed, let X I = E(u, 0) and XZ= E(b, 0) be two arbitrary elements of Jo. Then there exists an individual c such that the sentence P(u, b, c) is included in Dt. Let X = E(c, 0). Then K 1 X G XI V X2. For the models of K are precisely the integral domains which are extensions of homomomorphic images of Mo. In any such integral domain P(u, byc) holds, and so E(u, 0) together with E(b, 0) entails E(c, 0), and on the other hand, since we are dealing with an integral domain E(c, 0) can hold only if one of the sentences E(a, 0) or E(b, 0) holds as well. Thus, we have proved that JOis disjunctive over K. Any homomorphic image of MOcan be realized in a structure M whose relations and individuals coincide with the relations and individuals of MOsuch that the positive diagram of Mo, D;, is a subset of the positive diagram of M. Let VObe the set of all structures which are integral domains. VOdoes not contain any model of K U JOfor this set contains both the sentence S(1, 1, 1) and the sentences S(0, 0,O) and E(1, 0). It also contains an axiom of substitutivityfrom which we deduce S(1,1, I), and this shows that K u JOis contradictory. Now let H be any consistent set of sentences which consists of K and of a subset A of JOtogether with a set B whose elements are negations of the elements of another subset of Jo. Then we propose to show that VO contains a model of H. Indeed, since H is consistent it possesses a model M . A4 is an integral domain since it is a model of KIDand it is an extension of a homomorphic image of MOsince it is a model of Di.Moreover, we may suppose that all individuals which occur in H occur also as elements of M and as such are denoted by themselves. By construction, these are also the individuals of Mo. Restricting M to these individuals, we obtain an integral domain M' which belongs to VO.Any atomic sentence which holds in M and which is defined in M' holds also in M'. Thus, M' is a model of Di such that all sentences of JOare defined in M' and such that any sentence of JOholds in M' if and only if it holds in M. It follows that M' is a model of H, as required. This shows that VOis a full system of structures for JOover K as defined at the beginning of this section. Thus, the theory of varieties developed above applies. Since K includes a set of axioms for the notion of a commutative ring, the ideals in JO over K constitute a subset of the ideals obtained for the same domain in the preceding section. Thus, any ideal J in JO over K, J = {E(av,0)} is such that the uv which occur in this set constitute an arithmetical ideal J.+ in Mo. However, not every arithmetical ideal N

8.1.1

VARIETIES OF STRUCTURES

199

J* = {ay} corresponds to a metamathematical ideal in JO over K. The problem is settled by the two following theorems. 8.1.17. THEOREM. A set J = {E(ay,0)} is an irreducible ideal in JO over K if and only if the corresponding set J* = {av}is a prime ideal in M.

PROOF.Suppose that J = {E(ay,0 ) } is an irreducible ideal in JO over K. Then J* = (ay}is at any rate an ideal. Suppose that a product a = bc belongs to J*. Then E(bc, 0) belongs to J. But Kincludes the axioms for an integral domain and so the sentence E(bc, 0)

E(b, 0) V E(c, 0)

is deducible from K. But J is irreducible and so, by 7.4.6., either E(b, 0) or E(c, 0)belongs to J. It follows that either b or c belongs to J*, showing that J* is prime. Conversely,suppose that J* is prime. If J* comprises all elements of MO then I coincides with JOand hence, is irreducible. If J* does not comprise all elements of MO then the quotient ring Mo/J, is an integral domain. We may suppose as before that MolJ, is realized in the set of individuals of Mo, and that it is a model of D i in a correspondence in which the individuals of M denote themselves. Then E(av, 0) holds in MoIJ, if and only if a, E J* and so Mo/J* is a generic structure of J = {E(a,, 0)). This shows that J i s an irreducibleideal and completes the proof of 8.1.17. 8.1.18. THEOREM. A set J = {E(ay,0) ] is an ideal in JO over Kif and only if the corresponding set J* = {av}is an arithmetical ideal which is its own radical. The radical J ’ of an arithmetical ideal J is the set of all elements b such that a positive power of b is included in J. It is not difficult to see that J’ is an arithmetical ideal which includes J. J’ = J if and only if bn E J entails b E J for all elements of the ring. In this case J’ is its own radical, a fact which is also expressed by the statement that J is a radical ideal.

PROOF OF 8.1.18. Suppose first that J = {E(ay,0)} is an ideal in JO over K. Then Jcan be represented as the intersection of a set {Jv}of irreducible ideals, by 7.4.10. Let {J*v} be the set of corresponding arithmetical ideals, then J* = {J*y}.This represents J* as an intersection of prime

n V

ideals, by 8.1.17. Now let bn E J* for some b E Mo. Then bn E J*v for all Y , and so b E J*I since JeY is prime. We conclude that b E J*, so that J* is its own radical.

200

[8.2.

METAMATHEMATICAL T H M R Y OF VARIETIES

Conversely, suppose that J* is a radical ideal. Let J = {E(av,0)) be the corresponding subset of Jo. In order to prove that J is a metamathematical ideal, we have to show that for any b which is not included in J.+,E(b, 0) cannot be deduced from K U J. Now since J* is its own radical, the fact that b does not belong to J* implies that none of the elements of the set B = {b, b2, b3, . . .} belong to J*. A familiar argument involving Zorn's lemma now shows that there exists a radical ideal J i which includes J*, and which excludes the elements of B, and which is maximal with respect to the last-mentioned property. J i is prime. For if cd E J; for two elements c and d of MOand neither c nor d belongs to J; then, by the maximal property of J i there exist elements rl, rz of MOand j1, j z of J i , and positive integers rn and n such that

+j l = bm, rzd +- j2 = bn . Hence bm+n= bmbn = (r1c + j l ) (rzd + jz) = rlrzcd + r1cjz + rzdjl + rlc

jljz E J;. But J; is its own radical and so b E J i , contrary to assumption. Hence, J; is a prime ideal and the corresponding subset J' of JO is an irreducible ideal in JO by 8.1.17. Moreover J' includes J but excludes E(b, 0). This shows that E(b, 0) cannot be deduced from K u J' and, a fortiori, cannot be deduced from K u J. The proof of 8.1.18. is now complete. The reader will have noticed that the second half of the proof involves a simple but crucial argument which is familiar in the standard theory of ideals in commutative rings. Within our present framework, this argument is required, not in order to develop our ideal theory but in order to enable us to correlate our metamathematical ideals with the ideals of commutative ring theory.

8.2. Pre-ideals and Their Varieties. Let W = {xy}be a fixed non-empty set of distinct variables with index set N = { v } , such that Wand N may be finite or infinite. We shall consider well-formed formulae whose free variables, if any, belong to W,and whose bound variables do not belong to W. The set of these wff will be denoted by Hw. Let H be a subset of H w and let Q be a wff which belongs to Hw. Then we may say that Q is deducible from H,in symbols H t Q , if there exist elements Ql, ...,Qm of H such that 8.2.1.

t (VXl)

.. . (AXn)[ Q l A - .. A Q m

2

Q]

where XI, . ..,Xn are the elements of W which occur in at least one of QL Qm, Q.

--

- 9

8.2.1

PRE-IDEALS A N D THEIR VARIETIES

201

Let K and JObe subsets of HWwhere we suppose that K U JOcontains at least one relation of positive order. Then a set J c JOwill be said to be a predicate ideal - briefly, pre-ideal - in JOover K if for any Q E JOthe fact that K u J k Q entails that Q belongs to J. In order to reduce the theory of pre-ideals to the theory of ordinary ideals considered in sections 7.2.-7.4., we introduce a set P* = {a:) of distinct individual constants with the same index set as W,N, such that the elements of P* do not occur in any of the wff of K or Jo. For any Q E K u Jo, we define Q* as the sentence which is obtained from Q by replacing its free variables, xy, by the corresponding a*,. This establishes a one-to-one correspondence between the wff Q of K u JO and certain sentences Q*.Let K* be the set of sentences which correspond to the sentences of K and let J*, be the corresponding set of sentences which are obtained from the wff of Jo. Then 8.2.1. holds if and only if kQTh ... A Q Z

= Q*

It follows that a subset J of JOis a pre-ideal over K if and only if the corresponding subset J* of J*, is an ideal in J*, over K*. This reduces the theory of pre-ideals entirely to the theory of ideals of sections 7.2.-7.4. All the definitions and results which are to be found in these sections still hold if we consider pre-ideals in place of ideals. However, the theory of varieties of pre-ideals has to be developed separately. Let H be any subset of the set HW introduced above. By a slight generalization of our previous terminology, we shall call the structure M a model of H if all objects and relative symbols of H are in one-to-one correspondence with some of the individuals and relations of M such that all sentences of H hold in M. In particular, it is possible that the individual and relative symbols of H denote themselves in M, as in previous cases. For the given index set N and for any structure M, we denote by MN the space of all arrays P = {a,} with index set N such that the avbelong to M. In particular, if Nis finite and contains precisely n elements we obtain the n-dimensional Cartesian space MB introduced previously (in section 3.3.). The arrays P are the points of the space. If M is a structure and P is a point in M N then the ordered pair (M, P) will be called a compound point.

Let M be a model of H c HW where we shall suppose for simplicity that the object and relative symbols of H denote themselves in M. Let Q

202

METAMATHF,MA”ICAL THEORY OF VARIETIES

[8.2.

be any wff which belongs to H, and let P be a point of MN. We replace all the free variables xy E W which occur in Q by the corresponding elements of P (i.e. by the av with the same subscripts). This yields a sentence QP. QP is defined in M if we adopt the (natural) assumption that the elements of P denote themselves. If Qpholds in M then we say that the wff Q holds at P in MN, or that the point P satisfies Q in M N or that Q holds at the compound point (MyP). In particular, if Q is a sentence then it holds at all points P of MN provided Q holds in M. Let VObe a set of compound points ( M , P)which satisfies the following conditions. If ( M , P) belongs to Vo for any structure M and for any point P of M N then ( M , P’)E VOalso for every other point P‘ of MN. If ( M , P) E VOthen the wff of VOare defined in M. If H i s a subset of HW which consists of K and of a subset of JO and of the negations of the elements of another subset of JO such that H* is consistent then there exists a compound point (M, P) E VOsuch that every wff which belongs to H holds at P in M N . It is not difficult to establish the existence of a set which satisfies these conditions. Indeed, since H* is consistent there exists a model M * of H*, where we may assume that all relative and individual symbols which occur in H* denote themselves in M*. By adding, if necessary, a number of sentences to H* which together include all ur and are theorems of the Lower predicate calculus and otherwise arbitrary we may ensure that all u*, are included in M* and hence, that P* is included in M*N. Then all wff of H hold at P* in M*N. Let {Mf) be obtained as described, for the set (H,}of all sets H = Hu as introduced above, and let VObe the set of all compound points {Mf, PA} where P?.varies over all points of M f N . Then VOsatisfies all the conditions that were specified above. Such a set VOwill be called a full system of compound points for Jo over K. Let A be any subset of Jo. Then the set of elements ( M , P) of VOsuch that all wff of A hold at P in M N will be called the (geometrical) variety of A , to be written as V ( A ) . The same fact will be indicated also by A -+ V ( A ) . If J = ( A ) is the pre-ideal generated by A in JOthen it will be seen that V(J) = V ( A ) . We shall suppose that JO is disjunctive over K, i.e. by definition, that J*, is disjunctive over K*. Replacing the word “ideals” in 8.1.1. by “pre-ideals” we obtain a theorem whose proof is obvious. Also two different pre-ideals give rise to two different varieties. Thus, the correspondence between pre-ideals and their varieties is again one-to-one. Theorem 8.1.3. is replaced by the following

8.2.1

PRE-IDEALS A N D THEIR VARIETIES.

203

8.2.2. THEOREM. Let V be the variety of a pre-ideal J in Jo. Then J consists of all wff of JO which hold at all compound points ( M , P) of V. PROOF.Let J1 be the set of all wff of Jo which hold at all compound points of V. Then J1 3 J. Suppose J1 # J, then there exists a wff Q of J1 - J such that K* U J* U {- Q*} is consistent. Let ( M , P) be a compound point in VOat which the wff of K u J u {- Q } hold. Such a point exists by virtue of our assumptions on VO.Then ( M , P) E VO,but Q does not hold at ( M , P) although Q E J1. This contradicts the definition of J1, and proves the theorem. Theorems 8.1.4.,8.1.5. and 8.1.6. still hold in the present case provided the word “ideals” is replaced by “pre-ideals” in both theorems. We conclude as before that the correspondence J + V is one-to-one such that the sum of two pre-ideals corresponds to the intersection of the corresponding varieties and the intersection of two pre-ideals corresponds to the union of the two varieties. The definition of an irreducible variety remains unchanged. A variety is irreducible if and only if the corresponding pre-ideal is irreducible. A compound point (M, P) E VOis said to be a genericpoint of the variety Vif any wff Q of JOholds at ( M , P)if and only if Q belongs to the ideal J that corresponds to V. In this case, ( M , P)will be called a generic point of J as well. We shall see presently that the generic points of Algebraic geometry may be regarded as special cases of the generic points just defined, hence our terminology.

8.2.3. THEOREM. In order that a non-empty variety, V, be irreducible it is necessary and sufficient that V possess a generic point, ( M , P). PROOF.Parallel to the proof of 8.1.8. Let J - t V, and suppose that V, and hence J is irreducible. Suppose first that J # JO and let H be the union of J and of negations of all sentences of JO - J. Then K* u H* is consistent, similarly as in the proof of 8.1.8. and possesses a model M * , where we may suppose that all coordinates of P* belong to M*. Then (M*, P*)E VOand all wff of K u H hold at ( M * , P*).( M * , P*)is the required generic point of V. If J = JO and V is not empty then any element of V is a generic point of K. Conversely, suppose that V possesses a generic point ( M , P). We have to show that J i s irreducible. Let Q, Ql, Q 2 be three wff of JO such that K t Q Ql V Q 2 and such that Q belongs to J. Then Q holds at ( M , P) and hence, the sentence Q p holds in M. It follows that either QIPor QZP

204

r8.3.

METAMATHEMATICAL THEORY OF VARIETIES

holds in M y i.e. either Ql or Q 2 holds at (MyP) and hence, belongs to J. This shows that J is irreducible, by the counterpart of 7.4.6. for preideals. Theorem 8.1.9. holds without any verbal change. The same applies to Theorem 8.1.10. For the proof of the latter theorem we introduce, for any compound point (M, P),the pre-ideal J(M,P ) of all wff of JO which hold at (MyP).Finally, theorems 8.1.1 1.-8.1.15. are still applicable and so are the notions introduced in connection with these theorems (e.g., irredundant representation). We observe that the theory of pre-ideals and of their varieties requires the introduction of a set VO of compound points (M, P) which may involve several structures M. However, it is also possible that the set of all compound points (MyP)with a given fixed M may satisfy the conditions imposed on VO. This happens in a classical case, which will be considered in the next section.

8.3. Metamathematical Varieties and Algebraic Varieties. Let KkF be the set of axioms for the notion of a commutative field which was specified in section 2.2., above. KhF contains individual symbols, 0 and 1, for the neutral elements with respect to addition and multiplication respectively. Let MObe a model of KhF, and let DObe the diagram of MOsuch that 0 and 1 occur in MOand DOwith the same meaning and such that all individuals and relations of KLF U Do denote themselves in Mo. Put K = KhF U DO. Let Mn= MO[xy..., xn] be the ring of polynomials of n variables with coefficients in Mo, n 2 1. For every element q(x1, . . ., xn) of M,, select a predicate Q g ( x l , . . ., x,) which is formulated in terms of the individuals and relations of K and which states that q(x1, . .., x,) is equal to 0. Let JO be the set of these predicates. We shall consider the theory of pre-ideals in J o over K. The theory of these pre-ideals is disjunctive, for if Qq,(XI, . . .,xn) and Qg,(xl, . . ., x,) are two elements of K, then

8.3.1. K t Qqlgs (XI, -

-

9

xn)

Qq, (XI,

e e

- 3

x n ) V Qq, (XI,

. .,xn)

Construct a set Vo as indicated in the preceding section. Note that U J*, = K U is contradictory since JZ contains the predicate QI (u:, . . .,u:) and this predicate asserts that the constant polynomial q = 1 is equal to 0 for the particular individuals a*,, . . .,a: as arguments. For any ideal J in JO over K, J = { Q , ( X I ,... A$) say, we define J*

K*

JZ

8.3.1

205

METAMATHEMATICAL AND ALGEBRAIC VARIETIES

as the set of the polynomials qa(xi, . . ., X n ) E Mn which appear as subscript to the predicates of J. We maintain that J* is an ideal in the ordinary sense in Mn. Such an ideal will again be called an arithmetical ideal, Indeed, suppose that Q,, (xi, ...,X n ) and Q,, ( X I , ..., Xn) belong to J. Since K includes the field axioms we then have

K

1 Qg, (xi,

where both

- .-

xn) A

Qg,

(xi,

.y

xn)

3

Qgl*qp

(XI,

--

+ and - may be chosen in the implicate, and K 1 Q q (XI,

xn)

. ..,Xn) 3 Q r p (xi, . . .,Xn)

where r(x1, .. ., xn) is an arbitrary element of Mn. This shows that J* is an arithmetical ideal in Ma.

8.3.2. THEOREM. A subset J = { Q , (xi, .. .,x,) 1 of JOis an irreducible pre-ideal in JO over K if and only if the corresponding subset of M , J* = {qP(x1,. . .,xn)} is a prime ideal in Mn.

PROOF.Suppose J is an irreducible pre-ideal in JOover K. Then we have shown already that J* is an ideal in M,. Suppose that a product q ( X I , ..., Xn) = r ( X I ,.. .,xn) s( X I , ...,xn) belongs to J* where r and s belong to Jo. Then Qg ( X I , ...,xn) belongs to J. A model of K cannot have any zero divisors and so K

Qg(x1,

--

-9

Xn) 3

Qr(x1, - .., xn) V Qs(x1, - .., xn)

But J is irreducible and so either Qr or Qs belongs to J and either r ( X I ,. ..,Xn) or s (xi, . . .,X n ) belongs to J*. This shows that J* is prime. Suppose on the other hand that J* is prime. If J* = M then J = Jo and so J is irreducible. If J* does not include all elements of Mn then the quotient ring Mn/J* is an integral domain which is isomorphic to an extension of Mo. It will be convenient to substitute for M , an isomorphic ring M z which is obtained by replacing the symbols X I , . . ., Xn which, in our language stand only for variables, by the individuals a:, . . ., a:. The same substitution transforms J* into an ideal JZ in M i such that the quotient ring M : / J f is isomorphic to Mn/J*. We may suppose, more precisely that Mz/Jx is actually an extension of MOwhich contains the individuals a:, ...,a:. Then Q,(a:, . ..,a 3 holds in M:/Jf if and only if q(aT, ...,a:) E J t , i.e. if and only if q(xi, . ..,Xn) E J* and Q g ( x l , ..., xn) E Jo. M:/JE is not in general a model of K since it is not always a field. However, we may embed this structure in its field of quotients,

206

METAMATHEMATICAL ‘IHEORY OF VARll3TIF.S

[8.3.

M’, and the compound point (M’, P*) = (M’, (u;, . . .,a*,))may then be regarded as an element of VOand as a generic point of J. In more detail, since any wff of JOholds at (M’, P*)if and only if it belongs to J, J is a pre-ideal in JOover K, and since any Qrs holds at (M‘, P*)only if either Q,. or Qs holds at ( M ,P*),J is irreducible. This completes the proof of 8.3.2. 8.3.3. THEOREM. A subset J = { Q , (XI, . . .,X n ) } of JOis an ideal in JO over K if and only if the corresponding subset of Mn, J* = {qiU(xl,..., x n ) } is a radical ideal in Mn.

PROOF.Suppose first that J is a pre-ideal in JO over K. Then J is the intersection of a set {JA}of irreducible pre-ideals. Let {J*A}be the set of corresponding arithmetical ideals, then J* = {J*A}.This represents

n A

J* as an intersection of prime ideals, by 7.8.2. Now suppose that bm E J* for some b E Mn, m > 1. Then b m E J*A for all I and so b E J*A, taking into account that J*Ais prime. Hence b E J*, J* is a radical ideal. Conversely, suppose that J* is a radical ideal. In order to prove that the corresponding subset J of Jo is a pre-ideal in JO over K, we have to show that for any q(x1, . . ., x,) E M , - J*, the predicate QP(x1, . . ., xn) cannot be deduced from K u J. Since J* is a radical ideal in Mn, the set B = {q, q2, q3, . . .} has no element in common with J*. Hence (compare the proof of 8.1.18) there exists a prime ideal J i in Mn which includes J* and which has no element in common with B. The corresponding subset J’ of JO is an irreducible pre-ideal in Jo over K, by 8.3.2. J’ includes J but excludes Q q(XI, . . .,X n ) This shows that Q@(xI, . ..,Xn) is not deducible from K U J and completes the proof of 8.3.3. By Hilbert’s basis theorem, the arithmetical ideals of Mn satisfy the maximum condition and so the same condition is satisfied by the radical ideals in Mn and hence, by the pre-ideals in JO over K. Theorems 8.3.2. and 8.3.3. are independent of this condition and can be proved equally well for a general index set N i.e. for polynomials whose variables belong to an infinite set W. However, we shall continue here to state our results for W = (XI, . . .,Xn) which is the classical case. We are going to show that the necessity part of 8.3.3. may be regarded as a reformulation of a weak version of the Nullstellensatz (compare section 5.4) which is as follows 8.3-4. THEOREM. Let

(XI,

...,Xn), 41 (XI,

,

. .,~ n ) ., .., qr (XI, ...,xn)

8.3.1

METAMATHEMATICAL A N D ALGEBRAIC VARIETIES

207

be polynomials in the polynomial ring MO[xi, . . .,x,] over a commutative field MO such that q vanishes for all joint zeros of the polynomials 41, . ., qr in all field extensions of Mo. Then there exists a positive integer e sucht hat q@belongs to the ideal (41,.. .,qr) in MO[XI,. . .,x,].

.

PROOF.The assumption of the theorem is precisely that we have in all models of K,

..(Vxn) [ Q q , (xi,. . .,x,) A . - Qp, (XI,. . .,x,) 3 Q q (xi, . ..,x,)] . Thus &(XI, . . ., x,) belongs to the pre-ideal J which is generated by QP1,. . ., Q,, in JOover K. Let J* be the corresponding arithmetical ideal in MO[xi, . . ., x,]. Then J* is the radical of the ideal (41,. . ., qr) in MO(XI, . . .,x,). Since Q, (XI, . . .,x,) E Jo, q (xi, . .,, x,) E J*, and so a positive power of q belongs to (41, . . ., qr). This proves the theorem. (Vxl).

It is evident that we may, in the formulation of 8.3.4. replace “in all field extensions” by “in all algebraically closed extensions”. For if there exists an extension M of MO such that 4.1(bl, . . ., b,) = 0, i = 1, . . ., r for certain bl, . . ., b, E M but q(b1, . .., b,) # 0 then the same is true in any algebraically closed extension M’ of M. Moreover, metamathematical considerations alone are sufficient to weaken the hypothesis of the theorem still further, as explained in section 5.4. Thus, model-completeness considerations permit us to replace the assumption that q vanishes for all joint zeros of 41, . . .,qr in all algebraically closed extensions of MOby the weaker assumption that this condition is known to hold in the algebraic closure of MOalone. While our general theory of varieties was based on a set VOof compound points ( M , P) ,where M varies over several models, we may in the present case find a set VOwhich consist of all compound points ( M , P) for a specific M. To see this, we consider the set of all pairs of subsets of Jo, (Ag,BJ such that the set K U A U B is consistent where Bu is the set of negations of elements of Bu and the asterisk indicates, as before, that we have replaced the variables xY by the corresponding constants a *, v = 1, . . ., n. (Note that in the present case K* = K ) . We then replace the a*, in the different sets A: U B*, by distinct individuals a;. The result of this substitution are sets which will be denoted by A; u 1 ; .We claim that the set H = K U U {A; U B;} is consistent. To see this, we only

z z

P

have to show that every finite subset of H i s consistent. In other words, we have to show that if the sets K u A; u B; are consistent, for finite A;, B;, p = 1, . . ., I, then the set K u U {A; u BE} is consistent. P

208

METAMATHEMATICAL THEORY OF VARIETIES

r8.3.

For each p, p = 1, .. ., I, let Y, be the wff obtained by taking the conjunction of the elements of A$ u & and by replacing the individuals u; in the resulting formula by distinct variables xt. Also let 2, be the sentence which is obtained from any Y, by the existential quantification of the x;. A familiar argument of the predicate calculus shows that in order to prove that K U U {A; U Bi} is consistent we only have to P

establish the consistencyof K u {ZI A . .. A Zl} assuming the consistency of K u {Z,], p = 1, .. .,I. Now if K U {Z,} is consistent it possesses a model M , which is an extension of Mo. Since 2, states that a certain system of equations and inequations (inequalities of the form q # 0) possesses a solution it must possess a solution also in every extension of M,, more particularly in every algebraically closed extension of M,. The model-completeness of the notion of an algebraically closed field now shows that Z, holds already in the algebraic closure of Mo. This is is a model of K and of true for p = 1, . . ., I . Since K also holds in Z1, . . ., 21, showing that K u (21, . . . Z Z ]is consistent. It follows that the set H also is consistent and possesses a model M. All the compound points (M, P) together, P tl Mn, constitute a set VOwhich is a full system of compound points for JO over K. Indeed, any consistent set A, u Bfi as above holds at some point of Mn, by construction, and VOdoes not contain a point at which all wff of JOhold, since K u Jo is contradictory, We have not shown that we may take the algebraic closure it70 of MO as the structure M , and a simple argument shows that this is indeed not possible. Thus, take n = 1 and let B* be the set of all elements Q,(u*,) of JO excepting Qo(u;), which corresponds to the zero polynomial. Let B* be the set of negations of the elements of B*. Then K U B* is consistent. For B* merely states that a* does not satisfy any algebraic equation and so B* is satisfied in every extension of Mo which is not purely algebraic. On the other hand, 8*is not satisfied in any purely algebraic extension of MO and, in particular is not satisfied in the algebraic closure of Mo. Some simple calculations in transfinite arithmetic show that we may impose a bound on the cardinal of M depending on the cardinal of Mo. Thus, if m = max (KO, I MO [ ) where [ Mo 1 is the cardinal of MOthen 1 K I and I JO I are cardinals not exceeding m and the number of pairs of subsets Ar, B, of JOdoes not exceed 2". It follows that M may be chosen so that its cardinal does not exceed 2". A better bound which need not be exceeded by the cardinal of M can be

no,no

8.4.1

209

DIFFERENTIAL IDEALS

found by algebraic considerations. More precisely, we are going to show that we may take for M a field M which is algebraically closed and of degree of transcendence KO over Mo. This is the so-called universal field of Mo. Thus, if Mo is finite, M is countable while if MOis infinite, it possesses the same cardinal as M. Let H = K U A, u Bflwhere A f l is a subset of JOand Bflis the set of negations of a subset of Jo. Suppose that H* = K U A: U B*,is consistent where the asterisk indicates as usual that the variables xi have been replaced by the constants a:. Then H* possesses a model M which is an extension of Mo. By proceeding to the algebraicclosure of M', if necessary, we may add the assumption that M' is algebraically closed. Then the set H holds at the point P* = (a;, . . .,a:) in M'. It follows that H holds at P* in the field M* which is defined as the algebraic closure of Mo (a;, . .., u:). Then M * is of degree of transcendence n over Mo, at most. Now let M be an algebraically closed field which is of degree of transcendence KOover Mo.Then M contains a subfield M" which is isomorphic to M* under an isomorphism which leaves the elements of M O fixed while Q;, . . ., u: E M * corresponds to bl, . . ., bn. It follows that H holds at P = (bl, . .., bn) in M" and hence, in M. This shows that the set of compound points (M, P),P with coordinates in M , is a full system of compound points for JOover K. We are now going to verify that the theory of metamathematical varieties developed in section 8.2. coincides with the classical theory of algebraic varieties in the space M* where M is the universal field just introduced. To see this, we match any compound point (M, P) E VO with the ordinary point P E Mn. Then the variety of any metamathematical ideal J c JOcorresponds to the algebraic variety Vof the arithmetical ideal J* c MO[XI, . , ., xn] which is associated with J. Since J* is a radical ideal it is the ideal which belongs to V(i.e. consists of all polynomials that vanish on V) and since every radical ideal of MO[ X I , . .,xn] is associated with some J c Jo, it follows that all algebraic varieties in Mn are obtained in this way. Moreover, the irreducible varieties of VO are transformed into the irreducible varieties of Mm and any generic point of a variety of VOis turned into a generic point of the corresponding algebraic variety. In this sense, the theory of metamathematical varieties developed in this chapter is a true generalization of the classical theory of algebraic varieties.

.

8.4. Differential Ideals. As was to be expected, the theory of differential

210

METAMATHEMATICAL THEORY OF VARIETIES

18.4.

ideals as developed by Ritt and his collaborators, also can be subordinated to the metamathematical theory of ideals. In the present section we shall be concerned chiefly with ideals of differential polynomials. Let KDF be a set of axioms for the notion of a differential field of characteristic 0, as outlined in section 5.5. Thus, KDF contains the relations E, S, P and A and the individuals 0 and 1. Let MObe a model of KDF and let DObe its diagram. Put K = KDF U DO. Let Mn = M O{yl, ...,y,} be the ring of differential polynomials of n variables with coefficients in Mo, n B 1. Given any polynomial q {yi, . , y,} in Mn, select a specific predicate Q,(yl, ...,y,) which is defined in K and which states that q {yi, . . .,y n } = 0. Let JO be the set of these predicates. We shall relate the pre-ideals in JO over K to certain ideals in M n . The theory of pre-ideals in JO over K is disjunctive. Introduce a set of sentences J*, as in section 8.2., then K U J*, is contradictory. Let VObe a full system of compound points for JOover K as defined in 8.2. For any ideal J in JO over K let J* be the set of the differential polynomials q { y i , . .,y,} which appear as subscripts to the predicates of J. We show as in 8.3. that J* is an ideal in M n in the ordinary sense.

..

.

.

..,y,)} of JOis an irreducible pre-ideal in JO over K if and only if the corresponding subset of M n , J* = {qN{ y l , . . ,y,} } is a differential prime ideal in Mn. An ideal in Mn is diflerential if it is closed with respect to the operation of differentiation. 8.4.1. THEOREM. A subset J = {Q,, ( y l ,

.

PROOF.Suppose that J is an irreducible pre-ideal in JO over K. Then we can show, as in the proof of 8.2.2., that J* is a prime ideal. Now let q{y1, . .,y n } be any element of J* and let r { y i , . . .,y,} be the derivative of q. Then we have to show that r {yi, .. ,yn} also belongs to J*. In other words, we have to prove that

.

.

We claim that, more precisely,

.

For suppose that bl, . ., bn are elements of any differential field M' which is an extension of M such that q {bl, .. .,b,} = 0 in M'. Differentiating this equation, we obtain r {bi, . . ., bn} = 0 showing that Qr(bl, . . ., bn) holds in M'. This proves 8.4.3. and shows that the

8.4.1

DIFFERENTIAL IDEALS

21 1

condition of 8.4.1. is necessary. The fact that the condition is also sufficient can be established by means of the same quotient ring as in the proof of 8.3.2. To continue we require two simple auxiliary results which are due to Ritt [Ritt 19501. 8.4.4. THEOREM. Let F be a differential ring which includes the field of rational numbers, and let J* be an ideal in I;. Suppose that some positive power of an element b of F belongs to J*. Then some positive power of the derivative of b belongs to J*. It follows that the same is true for all derivatives of b.

8.4.5. THEOREM. Let F be a differential ring which includes the field of rational numbers and let J* be an ideal in F. Suppose that bc is a product of elements of F such that some power of bc beIongs to J*. Then for any pair of non-negative integers i and j , some power of the product u(')b(j) belongs to J* where a(') and b") denote the ith and j t h derivatives of u and b respectively. Using 8.4.4. and 8 . 4 5 , we are going to prove

.

8.4.6. THEOREM. Let J = {Q, (yi, . .,y n ) } be a subset of JO and let J* = {qa { J Q , . .,Y n } } be the corresponding set of elements of Mn. Then J is a pre-ideal in JOif and only if J* is a perfect ideal in Mn.

.

An ideal in Mn is perfect if it is a differential ideal which is its own radical. PROOF.If J is a pre-ideal in Jo, then it is the intersection of a set {JA} of irreducible pre-ideals. The corresponding sets J*n in M n are differential prime ideals, and so their intersection J* is a radical ideal, as shown in the proof of 2.3.3. Moreover, J* is closed under differentiation, since it is the intersection of ideals which are closed under differenttiation. Thus, J* is a perfect ideal. But J* is the image of J in Mn, showing that the condition of the theorem is necessary. The condition is also sufficient. For suppose that J* is a perfect ideal. In order to prove that J is a pre-ideal in Joywe have to show that if the predicate Qg( y l , .. .,yn) E JOdoes not belong to J then it is not deducible from K U J. By assumption, q { y l , ., yn} $ J*. Let T be the set of all perfect ideals in Mn which include J* but exclude q. T is not empty since it contains J*. T is ordered partially by the relation of inclusion. The union

..

212

[8.4.

METAMATHEMATICAL THEORY OF VARIETIES

of any monotonic set of elements of T, serves as an upper bound to the set. Hence, by Zorn’s lemma, T contains a maximal element, J,’ say. J,’ is a differential ideal which is its own radical.We propose toshow that it is also prime. Suppose on the contrary that rs E J i for certain elements r and s of Mn,while neither r nor s belongs to J,’. Since J i is perfect, it follows from 8.4.4. and 8.4.5. that all products of the form r(r)s(J) belong to J l where the superscript denotes differentiation, as before. Let J1 be the set of all elements of Mn which are of the form b

= bor

+ blr’ + . . . + b d k ) + jl

where k 2 0, bo, bl, ...,bk E Mn and j l E J i . J1 is a differential ideal. Let J; be the radical of J1 then J; also is a differential ideal by 8.4.4. It follows that J; is perfect. But J; is a proper extension of J i since r E 51’ - J i . Hence, by the maximum property of J;, q belongs to J i and there is a positive power of q which belongs to J1, so qm = bor

+ blr’ + ... + b d k )+j l ,

Similarly, there is a positive power of q, ql, which can be written in the form ql = cos CIS’ . . . c(s(*) j z

+

+

+

+

where 1 > 0, i 2 0, co, . . .,ct E Mn and j2 E J i . Then qm+l = CbAc,,r(’)dp) j 3 where j3 E JI. But rs E J i by assumption and so the products r(A)s(p) also belong to J i , by 8.4.5. It follows that q W + l E JI, But this is impossible since J i is a radical ideal and q E J,’. We conclude that J i is indeed prime. Let J’ be the corresponding set of predicates in Jo, then J’ is an irreducible pre-ideal, by 8.4.1. J‘ includes J but does not include Q q ( y l , ., yn). This shows that Q q ( y l , . . .,y n ) cannot be deducible from K U J. The proof of 8.4.6. is now complete.

+

..

.

..

.. .,

8.4.7. THEOREM. (Ritt). Let q { y i , . ., yn}, 41 { y i , ., yn}, qr { y l , . . .,y n } be differential polynomials in the ring Mo { y l ,

. . ., y n }

over a differential field MO such that q vanishes for all joint zeros of 41, . . ., qr in all differential field extensions of Mo. Then some positive power q Q of q belongs to [ql, . ,qr],where [ql, .,qr] is the differential ideal generated by 41, .. .,qr in Mo.

..

..

.

PROOF.Let J* be the set of all differential polynomials of MO{ y l , . . ,yn} which have some positive power in [ql, . . ., qr]. Taking into account 8.4.4., we see that J* is a perfect ideal. We have to prove that q belongs to J*. The assumption of the theorem is that in all models of K,

8.4.1

213

DIFFERENTIAL IDEALS

8.4.8.

(VYd

[ Q q , h

-

* *

PYn)

- - - , v ~ ) A AQP,(YL. . - , ~ m ) 3 Q

g h

... ~ n ) ]

In other words, the assumption is that 8.4.8. is deducible from K. Thus,

Qq(yl, . . ., y n ) belongs to the pre-ideal J which is generated by Q,,, . . ., Q,, in JO over K. Now let J i be the set of polynomials which corresponds to J in MO {yl, ., yn}. Then J i contains 41, . . ., qy, and is the intersection of all perfect ideals which contain these differential polynomials. It follows that J i = J*. But we have just seen that QP belongs to J and so q belongs to J i = J*. This proves 8.4.7. Similarly, as in the Nullstellensatz for ordinary polynomials, we may replace the assumption that q vanishes for all joint zeros of 41, .. .,9; in all differential field extensions of MO by certain weaker conditions. Indeed, since every differential field (of characteristic 0, as usual) can be embedded in a differentially closed field, we may write “all differentially closed field extensions” instead of “all differential field extensions”. Moreover, since the notion of a differentially closed field is modelcomplete, 8.4.8. holds in all differentially closed extensions of MO if it holds in some differentially closed extension of Mo. Accordingly, we may replace “all differential field extensions” even by the phrase “some differentially closed extension” of Mo. The conclusion of Ritt’s theorem, 8.4.7. states in more detail, that there exist a positive integer p, and integers lt 2 0, i = 1, . r such that

..

8.4.9.

(4 {Yl, .

a ,

ym)e =

c c PIj {Yl, Y

k

d=1

j=o

..

* *

.,h} q({)(Yl,

*

- .,yn> -

In this form, the conclusion cannot be formulated in the Lower predicate calculus as a predicate of the coefficients of ql, ., qr, q. However, an analysis which is precisely parallel to that used in the proof of Theorem 5.4.6. leads to

..

8.4.10. THEOREM. For given bounds on n and on the orders and degrees of 41, .. .,qr, q in the formulation of Theorem 8.4.7., there exist bounds PO, Ao,,uo, on the exponent of q and on the orders of the derivatives q($)

and on the orders and degrees of the ptj, respectively. These bounds are independent of the particular choice of the differential field Mo. The reader will have no difficulty in working out the details of the proof. By choosing a suitable full set of compound points for JO over K we may apply the general theory of varieties developed in section 8.2. to the present case. It is not difficult to verify that the theory of varieties ob-

214

[S.S.

METAMATHEMATICAL THEORY OF VARIETIES

tained in this way coincides with Ritt's theory of differential varieties. It is noteworthy that Ritt's theory as originally presented by him required the consideration of points in several fields, i.e. in our terminology, the introduction of compound points. A construction in Kolchin 1948 shows that it is actually sufficient to consider the set of all points in one particular differential extension of Mo. The corresponding problem for algebraic varieties was settled at the end of section 8.3. Since we do not dispose of a counterpart to the notion of algebraic closure (as distinct from the notion of algebraically closed extension) for differential fields, the procedure adapted there does not apply directly to the present situation. However, we may still prove the relevant result, which is 8.4.11. THEOREM. Let MObe a differential field, and let P = { (A,, B p ) } be the set of all ordered pairs of subsets of MO {yl, . .,y n } such that the system of equations and inequations

.

8.4.12.

4 { y l , .. .,yn>= 0 for all q E A@ r { y l , . . ., y,} # 0 for all r E Br

possesses a solution in some extension of Mo. Then there exists an extension M' of MOsuch that for every pair (A,, Bp) in P,the system 8.4.12. possesses a solution in M'.

PROOF. Let DO be the diagram of MO and let Q,(yl, . . .,y,) be a predicate formulated in terms of the vocabulary of DOwhich states that q { y l , . . .,y n } = 0. For each pair (A,, B,) which belongs to P, introduce distinct individuals a",, v = 1, ...,n. Let A: be the set of all sentences Qq(a:, . . ., a;) for q E A p and let B*, be the set of all sentences Qr (a:, .. ., a:) for r E B,. Then any model M' of H = HDF u DO u U

-

Ir

{A*, U BE} satisfies the conditions of the conclusion of 8.4.11. Accordingly, we only have to prove that H i s consistent. But this follows from the fact that every finite system of equations and inequations 8.4.12. which possesses a solution in any extension of MOpossesses a solution in every differentiallyclosed extension of Mo. This completes the proof of 8.4.11. 8.5. Hilkrt's Seventeenth Problem. The theory of formally real fields

and of real closed fields was developed originally by Artin and Schreier in order to provide a solution for Hilbert's seventeenth problem which deals with the representation of rational functions by squares. As given in section 2, the definition of a real-closed field is independent of the

8.5.1

215

HILBERT’S SEVENTEENTH PROBLEM

notion of a formally real field although the methods of the theory of formally real fields are still useful in connection with the derivation of some fundamental results on real-closed fields. In the present section we derive Artin’s main results by relying on the fact that the concept of a real-closed ordered field is the model completion of the concept of an ordered field. We shall also prove the existence of certain bounds in the representation of positive definite functions, similarly as in the case of the theorems of Hilbert and of Ritt on the zeros of (differential) polynomials (see 5.4.6. and 8.4.10). Although this is not essential for our proof of Artin’s theorems, we fipd it natural to introduce the ordering problem for a commutative field within the framework of the metamathematical theory of ideals. Let KA3 be a set of axioms for the notion of a commutative field of characteristic 0 formulated in terms of the relations E, S, P and in terms of the individuals 0 and 1 as in section 2.2. and let M be a model of KCi with diagram D. Let H be the union of the sets of axioms 2.2.5., 2.2.6., and 2.2.16. which introduce the relation Q ( x , y) (with the interpretation x < y). Then KAF = K&! U H is a set of axioms for the notion of an ordered field. Let Jo be the set of all sentences of the form Q (0, a) where a belongs to M and is different from 0. We shall study the theory of ideals in JO over the set K = KAP U D. To every subset J of Jo, J = {Q (0, av)}there corresponds a subset J* of the set M O= M - (01,i.e. the set of all individuals avthat occur in J. If J is an ideal in JO(over K ) we shall call J* an order ideal in M. We wish to characterize the subsets of JO which are ideals or equivalently, the subsets of M which are order ideals. Let then J and J* be an ideal in JO and the corresponding order ideal in M , respectively. Since all models of K are ordered fields we see immediately that the following conditions are satisfied.

+

8.5.1. J* is additive and multiplicative, i.e. a, b E J* entails a b E J* and ab E J* ;and J* contains all squares of elements of Mo, in particular

1 E J*. Moreover, we are going to show

8.5.2. THEOREM. If the order ideal J* contains an element a # 0 as well as its inverse with respect to addition, - a, then J* = Mo. PROOF.Put a’ = - a. By assumption, D contains the sentence S(a, a’,O)

216

[SS.

MFTAMATHEMATICAL THEORY OF VARIETIES

and J contains the sentences E(0, a), Q (0, a), and Q(0, a'). It follows that K U J is contradictory since a a' = 0, 0 < a, 0 < a' is possible in an ordered field only if a = 0. Accordingly K u J t X for all X EJo and so J = Jo, J* = Mo. On the other hand, J* = MOis indeed an order ideal since the corresponding J is equal to JO and this is an ideal in JOover K. J* = MOwill be regarded as an improper order ideal, all the remaining order ideals are proper. Let @ be the set of subsets J* of MOwhich satisfy 8.5.1. as well as the following condition. N

+

8.5.3. For any a E Mo, a and - a are not both contained in J*. The elements of @ contain all sums of squares of elements of Mo.

+

+

Conversely, the set JS of all sums of squares a: .. . a:, n 2 1, ...,n, satisfies 8.5.1. If 0 can be written in this form, 0 =at . . . u i then n 2 1 since a1 # 0. But then the element a = a: belongs to Js and so does its inverse with respect to addition - a = af . . . a:. This contradicts 8.5.3. and shows that in this case @ is empty. On the other hand, if 0 cannot be written as a sum of squares as indicated then @ is not empty since it contains Js. Thus, @ is not empty if and only if @ is formally real. Clearly, M O is not contained in @.

ar E Mo, i = 1,

+

+

+

+

8.5.4. THEOREM. Let J' E @ and a E MO- J'. Then there exists an element J" of @ such that J" 3 J' and (- a) E J".

PROOF.Let J" be the set of elements of M which are of the formp (- a) where p ( x ) = bo bl x ... bnxn, bn # 0,n 2 0 is a polynomial whose non-vanishing coefficients belong to J'. J" includes J' (take p ( x ) = bo); J" contains the element - a(take p(x) = x ) ; and J" satisfies 8.5.1. since it is additive and multiplicative and contains all squares in Mo.Suppose now that J" contains both b and - b for some b E Mo. Then b (- b) = 0 also is of the form p (- a) for some polynomial p (x) as defined. By dividing p (x) into even and odd powers of x we may write

+

+

+

+

8.5.5.

p (x) = q (x2)

+ xr (x2)

where q ( y ) and r ( y ) are polynomials whose non-vanishing coefficients belong to J'. By assumption 8.5.6.

p(- a) = q(a2) - ar(a2) = 0 .

8.5.1

HLLBERT’S SEVENTEENTH PROBLEM

217

Now if r ( y ) is the zero polynomial (of y) then 8.5.6. yields q(a2) = 0 . Suppose that 4 w = CjY’

+ + -+ CZY

* *

CmP,

... < m , c j # O , c ~ # O , ..., cm # O . q(a2) = cja2j + cla21 + . . . + xmu2m = 0 implies that j
Then the nonzero element qu2j and its inverse - qa2Jboth belong to J’. It follows that q(y) also is the zero polynomial. But this is impossible since in that case p (x) also would be the zero polynomial (of x). Thus, r (y) is not the zero polynomial and r(a2) is an element of J’, and hence r(a2) # 0 . Similarly, q ( y ) cannot be the zero polynomial and so q(&) E J’ and q(a2) # 0, 8.5.6. now yields

But the factors q(a2), r(&, and (r(u2)-1)2 on the right hand side of this equation all belong to J’ and so a E J‘, contrary to assumption, We conclude that p ( - a) # 0 for all ( x ) as defined and so J” E @, 8.5.4. is proved. If @ is not empty then it is ordered partially by the relation of inclusion. Moreover, the union of any monotonic set of elements of @ belongs to @. Using Zorn’s lemma, we conclude that every element of @ is included in a maximal element. 8.5.7. THEOREM. Let J’ be a maximal element of @ and let a be any element of Mo. Then J‘ contains either a or - a. Conversely, if J’ E @ and if for every a E MOeither a E J’or (- a) E J’ then J’is maximal in @.

PROOF.Suppose that J’ is maximal in @ and that a E M O- J’. Then (- a) E J” for some J”which includes J and is contained in @, by 8.5.4. But J’ is maximal and so J” = J’,(- a) E J’ as required. The converse is obvious. It follows that every maximal element of J’ of may serve as the set of positive elements of an ordering of M . The corresponding set J in JO constitutes a maximal ideal in Jo. More generally, we are now in a position to prove 8.5.8. THEOREM. Let J* be a proper subset of Mo. Then J is an order ideal if and only if it satisfies 8.5.1. and 8.5.3.

PROOF.The assertion of the theorem is that J* is an order ideal (other

218

METAMATHEMATICAL THEORY OF VARIETIES

18.5.

than Mo) if and only if it belongs to @. We have seen already that the condition 8.5.1. is necessary and the same applies to 8.5.3., by 8.5.2. To see that 8.5.1. and 8.5.3. together are also sufficient, suppose that J* E @ and let J be the corresponding subset of Jo.We have to show that J is an ideal in J. Equivalently, if a E M O- J* then we have to prove that Q(0, a) is not deducible from K U J. Now by 8.5.4. there exists an extension J‘ of J* such that - a E J‘ and J’ E @. By Zorn’s lemma, J in turn possesses an extension J”which is maximal in @. J” defines an order in M in which a is negative while all elements of J”and in particular all elements of J* are positive. This proves that Q (0,a) is not deducible from K u J. Accordingly, Theorem 8.5.8. establishes @ as the set of order ideals other than MOwhile 8.5.7. characterizes the maximal proper order ideals. 8.5.9. THEOREM. Any proper order ideal J* is the intersection of all

maximal proper order ideals in which it is contained. Equivalently, every proper ideal in JO over K is the intersection of all maximal proper ideals in which it is contained.

This follows immediately from earlier arguments. For if a E MO- J* then there exists a maximal proper order ideal which includes J* but excludes a. 8.5.10. THEOREM. Let C be any subset of M such that 0 f C and 1

E C. In order that an element a of it4 be positive in all orderings of the field M in which all elements of C are positive it is necessary and sufficient that a can be represented in the form

8.5.11.

+

+

+

a = cia; czai . . . cnai, n > l , a g E M , a s # O , i = l , ..., n

where the cg are products of elements of C.

PROOF.Sufficiency is obvious. To prove necessity, let J* be the set of elements of M which can be written in the form of the right hand side of 8.5.11. under the specified conditions. At any rate, J* satisfies 8.5.1. Suppose that 0 E J*, then there exists an identity 8.5.12.

O = cia; + cza; +

. .. + cnai

as in 8.5.11. In this case, there can be no ordering of M in which the elements of C, and hence of J* are all positive, and we have to show that

8.5.1

219

HILBERT'S SEVENTEENTH PROBLEM

all elements of M can be represented as in 8.5.1 1. Now n since cia: # 0. We then have, for arbitrary a E M al(l

-

ClCn (------')2 &an

+. +

(

)' + (7)

an-i(1 - a)

Cn-lCe

&an

> 1 in 8.5.12. l +a

Thus

+

(!$)2.

This is a representation of a of the required kind if we omit the last term on the right hand side for a = - 1 and all the other terms for a = 1. Suppose next that 0 # J*. Then J* c Mo, and J* satisfies 8.5.3. (for a E J,, - a E J* entails 0 = a (- a) B J,). It follows that J* is an order ideal, and if a E J* for any a E MO this entails the existence of an ordering in which a is negative while the elements of J* and hence of C are positive. The proof of 8.5.10. is now complete. We observe, generally speaking, that the set JOis not disjunctive. This is suggested by the fact that there is no obvious sentence of the form 0 < c which is equivalent to the disjunction of two sentences which signify 0 < a and 0 < b. To prove our assertion rigorously, we observe that in a disjunctive domain, the intersection J3 of two ideals J1 and J2 is such that K u J3 1 XI V X Z where XI and XZare arbitrary eIements of Ji and Jz respectively. Now let M be the field obtained by adjoining a single indeterminate to the rational numbers R, M = R ( t ) . Let J; and Ji be the sets of positive elements of M which define the orderings in which t or - t are infinitely large compared with the rationals, respectively. Let J1 and JZ be the corresponding ideals in Jo, respectively. Then Ji = J; n JL consists of all elements of R ( t )which can be represented as quotients of polynomials of even degrees. Let 5 3 be the ideal in JOwhich corresponds to Ji. LetX1 and X Zbe the sentences Q(0, t l ) and Q(0, t z ) respectively

+

220

[8.5.

METAMATHEMATICAL THEORY OF VARIETIES

where tl = t - 1 and t 2 = - t - 1. Thus, XI e J1 and X Z E Jz. On the other hand, XIV XZis not deducible from K u J3 for it is not difficult to fmd a model M* of K U J3 in which [XI V X Z ]holds. Indeed, we may choose as M3 the field M i n which an order has been defined by identifying t with any real transcendental number whose absolute value is smaller than 1. We shall now apply our results, in particular 8.5.10., to the proof of certain generalizations of Artin’s theorem on definite functions. Let MObe an ordered field, M its real closure, Mn the n-dimensional Cartesian space with coordinates in My and M I = M O [ X I , . . ., x,]. Furthermore, let J* be a prime ideal in M1 and let V be the corresponding variety in Mn.(Thus, V is irreducible relative to MOalthough it may be reducible in Mn,) With this notation, we are going to prove

-

8.5.14.

THEOREM. Let

8-5-15. f(x1,

* * *3

xn), gl (XI,

- .,xn), . ..,gr (XI, .. .,

~ n ) ,r

.

2 1

be a set of polynomials which belong to M I such that gt (XI, . .,x,) E J*, .. .,r and such that for every point (al, . . .,a,) which belongs to V the set of inequalities

i = 1,

entails

gi(a1,

...,an)> 0 f(a1,

i = 1,

...,r

...,an) 2 0

Then there exists a congruence 8.5.16.

...) Xn))2f(Xly . . .) x n )

E

..

where the polynomials ho (XI,...,x,), hl (XI,. . .,xn), . . .,hm ( X I , . ,x,) all belong to M Iand ho (XI, . . ., x,) E J* and where the kc (XI, ..., x,) are products of some or all of the polynomials gt(x1, . . ., x,) and of positive elements of Mo. Before proving the theorem, we observe that we may suppose that the products kr(x1, . . ., x,) contain every factor gj(x1, . . ., x,) not more than once since all even powers of the polynomials gj (XI, . . .,x,) can be absorbed in the corresponding hi (XI, . . .,x,).

PROOFOF 8.5.14. Let p l (XI, . . .,x,), . . ., p s (XI, . . .,x,) be any basis for the ideal J*. Suppose that there does not exist any congruence of the type given by 8.5.16. Let M2 be the quotient ring M [ x l , ...,xn]/ J*. It will not

8.5.1

HILBERT'S SEVENTEENTH PROBLEM

221

cause any confusion if we use f, gl, . . .,gr not only for the polynomials of 8.5.15. but also in order to denote the elements of M2 which (when regarded as equivalence classes) contain the polynomials of 8.5.15, respectively. By the assumption of the proof there does not exist any identity of the form 8.5.17.

hi f =

m

C kfh: i=1

where ho, hl, . . ., hm are now elements of M2, ho # 0, and the kt are products of the gj and of positive elements of MOregarded as a subfield of Mz. Let M3 be the field of quotients of M Zso that M3 is an extension of MO and hence, is of characteristic 0. Then no identity of the type

f

8.5.18.

m

=

Ct=l kihf"

can exist in M3 such that the kt are products of gj and of positive elements of MOand the ht are elements of M3. Let C be the set of positive elements of MOtogether with the gg regarded as elements of M3. Then the conditions of Theorem 8.5.10. are satisfied, and the absence of an identity of the type of 8.5.18. shows, firstly, that f ( m , . . ., xn) does not belong to J* (for otherwise 8.5.18. can be satisfied by letting all ht be equal to zero) and, secondly, that there exists an ordering of the field M3 in which f is negative while all elements of C are positive (otherwise we can satisfy 8.5.11. by deleting in 8.5.18. all hi which are equal to zero in M3). Let M3 with this ordering be denoted by Mi. M*, includes MO as an ordered field since the positive elements of MObelong to C. Now let
<,,

.

...,x,, such that pi(xi, . . ., xn) = 0, i = 1, . .., s and gi(x1, . . ., xn) > 0,i = 1, . . ., r 8.5.19. There exist individuals XI,

while at the same time f(x1,

. . ., X n ) < 0.

We may formulate 8.5.19. as a sentence X whose individuals all belong to Mo. X still holds in the (ordered) real closure of M: which will be denoted by M4.But the concept of a real-closed ordered field is model-

222

[8.5.

METAMATHEMATICAL THEORY OF VARIETIES

complete relative to the notion of an ordered field and so the fact that X holds in M* which is a real-closed extension of MOentails that it holds also in A4 which is another real-closed extension of Mo. But this implies that the assumption of Theorem 8.5.14. is not satisfied, and completes the proof. We note various special cases of the theorem. For J* = 0, r = 1, gl(x1, . . ., x,) 1 we obtain

=

8.5.20. THEOREM. Let Mo be an ordered field and let M be the real

closure of Mo. Suppose that the polynomial f(x1, definite, i.e. it satisfies f(a1, . . ., a,) 2 0

. . ., x,)

in MO is

.

for all al, . . ., a, E M. Then there exist positive elements cl, . ., Cnc of Mo, and rational functions fi (XI, . . .,x,), . . .,fm (XI, . . ., x,) with coefficients in MOsuch that f(x1,

...,x,)

c cr(fr(x1, ...,x,))z Tn

=

6=1

Artin's main results are obtained by taking for M Othe field of rational numbers, or the field of real algebraic numbers, or the field of real numbers. In the second and third case every positive number is a square while in the first case every positive number is a sum of four squares. It follows that in all these cases we may absorb the coefficients ct in theft We then obtain a representation off(x1, . . .,x,) as a sum of squares, f(x1,

. .,x,) *

c (fa (x1, . . m

=

iT1

*Y

xn))2

Moreover, in the second and third case, M coincides with MOwhile in the first case the assumption that f ( a 1 , . . ., a,) 2 0 for all al, . . ., a n in M Ois already sufficient to ensure thatf(a1, . . ., a,) 2 0 for all al, ..,a n in M. Hence

.

8.5.21. THEOREM. Let MO be the field of rational numbers, or of real

algebraic numbers, or of real numbers. Suppose that the polynomial f(x1, ...,x,) with coefficients in MOsatisfies f(a1,

...,a n )

20

for all al, . . ., a, E Mo. Then there exist rational functionsf(x1, . . .,fm(x1, . . ., x,) with coefficients in MOsuch that f(X1,

...

y

x,)

c (fa (Xl, . . ., rn

=

I=1

X">)2

.

. . .,

XS),

8.5.1

223

HILBERT’S SEVENTEENTH PROBLEM

Other special cases of Theorem 8.5.14. have been proved by Artin and by Lang. Now suppose that the variety V of J* is a primal, i.e. an n - 1 dimensional irreducible variety. Then J* possesses a basis which consists of a single polynomia1,p (XI, .. .,x,). For this case, we are going to show8.5.22. THEOREM. Suppose that the assumptions of 8.5.14. are satisfied, and that the degrees of the polynomials XI, . . ., x,), gt(x1, ., xn), p (XI, . ,x,) do not exceed a specified bound v. The; 8.5.16. is satisfied

..

..

+

by m 1 polynomials ho(x1, . . ., x,), . .., h m ( x l , . . ., x,), and by m polynomials kt(xl, . . ., x,) such that the number m and the degrees of the polynomials hr and kr are below certain bounds po and ilo which depend only on n and v and are independent of the particular choice of the coefficients of the polynomialsf, gz, p and even of the choice of the particular field Mo.

PROOF. We formulate the hypothesis of 8.5.14. for general polynomialsf, gl, ...,gr, and p, of degree v with indeterminate coefficients. This yields a predicate Q* (ZI, . .,zj) of the coefficients zi of these polynomials ranged in an arbitrary but definite manner. Again, for any pair of positive integers p and A, we may formulate a predicate Q,a(zi, . . ., zm)which states that there exist polynomials ho, h ~ .,. ,h, of degrees not exceeding A such that

.

.

. ..,x,) - C kt (XI, - - ,x,) (hi (XI, . .,~ n ) =) <=I q (XI, . . .,x,) p (XI, . . .,x,) where ho (XI, . . .,x,) is not divisiblebyp (XI, . ..,x,) and the kr(x1, . . .,x,) (ho (XI, . . .,~

P

n ) XI, ) ~

are products of distinct gj(xl, . . ., x,) and of positive numbers. The predicates Q (z1, . . .,zj), Qpa (z1, .. .,zj) can all be formulated in terms of E, S, P,Q, and 0, and 1, i.e. in terms of the vocabulary of K i F . Theorem 8.5.14. states for any al, . . .,a;r in Mo, that if Q* (a, . .,q ) holds in M then Qpa (al, . . ., aj) holds in MOfor at least one pair p, A. Now by Theorem 5.3.7. there exists a predicate Q (21, . ..,zj) such that for all al, . ..,aj in Mo, Q (al, . . .,a,) holds in MOif and only if Q * ( a , .. .,aj) holds in M . It follows that for all al, . ..,aj in Mo, if Q (al, . ..,aj) holds in M Othen Qpa (al, . ..,a,) holds in MOfor at least one pair p, 1. Accordingly the union of KhF and of the set

.

8.5.23.

-

{Q @I, . . .,4 ,

Qaa @I,

. . ., 41

is contradictory, where p, ilvary independently over all positive integers

~

224

[8.6.

METAMATHEMATICAL THEORY OF VARIETIES

and where al, ...,uj are arbitrary individuals other than 0 and 1. We conclude that there exists a finite subset of 8.5.23., 8-5-24. (Q(a1, . . .,aj),

Q ~ J(a19 ,

N

--

9

aj),

--

*9

N

. . ., I

Q~,A,(~I,

aj)

such that the union of K;)F and of 8.5.24. is contradictory. Hence, 8.5.25. K i F t Q (ai, .

..,

uj) 3

Let PO = max pi, 10= max 1scgr

At.

1btCf

K i p t Qp,A,(al,. . ., uj) and so K;)F

...,

Q p l ~(a, ,

~

j

V)

... V Qp,n,(ai, . . .,U J )

Then

Qp.no(a,

. . ., aj), i = 1, . .., r

Q (al, . - .,aj)

&J,

(al,

- .,uj)

and by universal quantification

.

K i F I- (Vxl) . . (Vxn)[Q( X I ,. . .,xn) 3 Q p o l 0 ( X I , .. .,xn)] This may be regarded as a formalization of the statement of 8.5.22. and proves that theorem. 8-5-26.

8.6. Problems 8.6.1. A differencejield is a commutative field which contains a shifting operator T ( x ) . T ( x ) satisfies the conditions T(a b) = T(u) T(b), T(ab) = T ( u )T(b)for all a and b in the field, and T(1) = 1. Develop a

+

+

metamathematical theory of difference ideals and varieties by analogy with section 8.4. (Compare Ritt and Raudenbush 1939). 8.6.2. Discuss the theory of ideals in lattices and semi-lattices from a

metamathematical point of view.

8.6.3. Let K be a set of axioms for the notion of an integral domain

with unit element together with the diagram of a particular structure M of this kind. Let JObe the set of all sentences (32) P(u, z, b) (i.e. “a divides b”) where a and b vary over the elements of M. Investigatethe ideals in JO over K and in JZ over K. 8.6.4. Let K and M be given as in 8.6.3. For any polynomialp ( X I , . . .,x,) with coefficients in M define a predicate R p ( X I ,. . .,X n , y ) which states that y = p ( X I , . . ., Xn). For any pair of polynomials p ( X I , . . .,X n ) and

q(XI,

. . .,xn) with coefficients in M let

-

Q p g ( X I , , -,xn) = ( 3 ~(32) ) (3~) [Rp(x~ . - - , x n , ~ ) A R p . ( -x.~. ,,x n , z ) A P ( y , w , z ) ]

8.6.1

PROBLEMS

225

Thus, Q p q(XI, . . .,x,) states that the functional value of p (XI, . .., x,) divides the functional value of q (XI, . . .,x,) for the argument values XI, . . .,Xn. Let JO be the set of all Qpqfor a fixed n. Develop the theory of pre-ideals in JO over K,and of their varieties. 8.6.5. Investigate the varieties of structures of the metamathematical ideals introduced in 7.6.3. 8.6.6. Develop a theory of ideals and varieties for systems of polynomial inequalities of the types p(x1, . .., x n ) 2 0 and p ( x i , . . ., x,) > 0.

References. Sections 8.1-8.4 follow A. Robinson 1951 and A. Robinson 1955. For a standard proof of Ritt’s theorem 8.3.7. see Ritt 1950. The original papers on formally real fields and on the solution of Hilbert’s seventeenth problem are Artin-Schreier 1927, Artin 1927. Lang’s work on the subject is in Lang 1953.For the theory presented here see Robinson 1955b,1956, 1957. Further developments are given in Henkin 1957a, Kreisel 1958.

CHAPTER IX

SELECTED TOPICS

9.1. Introduction of Function Symbols. In section 3.1. we found it convenient to introduce certain function symbols, or Skolem functors in

connection with the interpretation of existential quantifiers. Such functors occur naturally in many or most algebraic structures, e.g., the functors of sum and product, (~(x, y ) and a (x,y ) in a ring or field. (We shall use this style of notation in preference to the familiar x y or x * y which is suitable only for functors of two variables). So far we have avoided the introduction of such function symbols into our formal language L, using instead certain relative symbols whose particular character was laid down axiomatically, e.g. S ( x , y , z) and P ( x , y , z ) for sum and product. However, for certain purposes it becomes essential to regard the functors as part of the formal language, and we shall now indicate what modifications are necessary in the theory of Chapter I1 above in order to carry out this idea. The atomic symbols of the language are, firstly, the atomic symbols introduced at the beginning of section 1.2. (individual object symbols, variables, relative symbols, connectives, quantifiers, and brackets) and secondly, the function symbols or functors. These are divided into disjoint classes Fn, n = 1, 2, 3, . . . (functors of order n or n-place functors). Functors will be denoted by x ( ), q ( ,), v/ ( , , ) (small Greek letters followed by n empty spaces - separated by commas if n > 1 - in round brackets). The classes Fn constitute well-defined sets whose cardinal numbers are specified but arbitrary. By means of the atomic symbols we now define terms inductively. These are firstly, the object symbols and variables and secondly, the result of filling the empty places of a functor with terms. A term is complete if it does not contain any variables. Atomic formulae are obtained by filling the empty spaces of a relative symbol with terms. Well-formed formulae (wff) are defined exactly as in section 1.2. beginning with atomic formulae. It is understood that a wff X will be

+

226

9.1.1

INTRODUCTION OF FUNCTION SYMBOLS

227

said to contain a variable y not only when y fills one of the empty places of a relation in X directly but also when y occurs within a term which in turn fills one of the empty places of such a relation (e.g. A ( v / ( x , y), b ) ) . The division into sentences and predicates is the same as before. There will be no change in the bracket conventions. The axioms and rules of inference of the language are as in section 1.3. except that in 1.3.2. a may be any complete term. The symbol a in 1.3.3. is still supposed to be an object symbol although this may occur within a functor which appears in a term. Rules 1.3.4.-1.3.7. are consequences of the rules of inference, as before. A structure M of the kind which will be considered here may contain in addition to the relations which are defined in it also a number of functions of n variables, k 2- 1. A function p ( ,. . .,) is given if p assigns to every n-ple (al, . . ., a,) of objects of M a definite functional value in M . Thus as in the case of relations (section 1.4.), we include the possibility that two distinct functions coincide in the set-theoretic sense. Let C be a one-to-one correspondence which maps the objects, relations and functions of a structure M on certain sets of object symbols, relative symbols and functors, such that the mapping preserves the order of the relations and functions. We say that a symbol denotes a relation, function, or object, if it corresponds to it (under the given C). A given correspondence C can be continued in a natural way so as to map all complete terms whose object symbols and functors appear in C into the set of objects of M . Thus, if al, . . .,a, are object symbols which appear in C and SP ( , . . . ,) is a functor or order u which appears in C and at --fa;, p --f p‘ by C then we define 9 (al, . . .,a,) --f p’ (a;, . . .,a;) where p’(a;, . . ., a;) is the functional value of p’ for the argument values a;, . . ., a; in C. This process can be continued following the inductive definition of a term. Thus, if we have already mapped the terms t l , . . ., t m on objects b;, . . ., bk of M and if the functor of order m, v / ( , . . . ,) is mapped on a function w’ then we map v/ (tl, . . ., t m ) on the object v/’(b;, . . ., b k ) in M . Let K be the set of wff of the language which are defined in M under C, i.e. whose object, relative, and function symbols appear in C. To every atomic formula X in K there corresponds, by C and by the continuation of C defined above, the expression X ‘ of a relation between individuals of M which either holds or does not hold in M . Starting from these

228

SELECTED TOPICS

[9.1.

atomic formulae, the definitions 1.4.1.-1.4.3. still determine, for any sentence of K, whether or not the sentence holds in M. The relations between deductive and descriptive concepts remain the same as in the case of the language without functors (section 1.5.). This is obvious for sentences which do not contain any relative symbols of positive order. For sentences which contain relative symbols of positive order but do not contain any quantifiers, we amend the procedure described in section 1.5. Thus, for a given set of sentences K, we introduce a structure M whose relations are the relative symbols of K and whose objects now are the terms, and not only the object symbols, of K. Any functor y ( , . . . ,) in K, of order m, will be regarded also as a function of it4 whose functional values are defined in the following way. Let tl, ..., tm be any objects of M, so that the t l , .. ., tm are terms of K by definition. Then the functional values of p for the arguments tl, ...,tm shall be the term p(t1, . . ., tm) regarded as an object of M . In all other ways, the procedure remains unchanged and still leads to the conclusion that M is a model of K. Finally, suppose that at least one of the sentences of K contains a quantifier.We may show, as in Chapter 1 that every sentence X is equivalent to a sentence X' in prenex normal form, X X'is a theorem of the calculus. Accordingly, we may confine ourselves to the case when all sentences of K are in prenex normal form. Proceeding as in 1.5. we obtain from K a set K* which is free of quantifiers and we show that K* is consistent and hence, possesses a model, which is also a model of K. In this way, we establish the completeness theorem and its generalizations and consequences (1.5.3.-1.5.5.) also for the case of a language with functors. The reader will have no difficulty in verifying that all the general definitions and results established previously for languages without functors still apply to the present case, sometimes with minor modifications. In particular, this is true of the notions and results connected with the prefix problem and with the theory of model-completeness,and some of these will be used subsequently without further comment. For example, the diagram of a structure still consists of (bracketed) atomic formulae, and of the negations of such formulae, but an atomic formulae now contains terms and not only symbols and variables. We have seen that in any sentence in prenex normal form functors may be used in order to eliminate the existential quantifiers. In some cases these functors are determined uniquely, in others they are to some extent

9.1.1

229

INTRODUCTION OF FUNCTION SYMBOLS

arbitrary. We shall now reformulate some important systems of axioms in terms of functors. The following system of axioms for the concept of a (general, noncommutative) group is formulated in terms of the object symbol e (for the neutral element), the relations (of equality) E ( x , y), the functors a ( x ,y ) for the product x y and p (x) for the inverse, x-1. Note that we have filled the empty places of the functors with variables, for convenience of reading only. The required axioms are, first, the axioms of equivalence (2.1.1 .), and, secondly

-

9.1.1.

(W Wy)(V.4 (W [ a x ,z) A E ( y , 4 = E(+, u), a (z, w))] (W (W [E(x,Y ) = E(@(X)> e
( X I , X)'

e) *

Note that the first two axioms of 2.2.2. have no counterpart in 9.1.1. since the very use of the functor a (x,y ) implies that the group operation is defined uniquely on the entire group. The resulting set of axioms will be denoted by HG. In order to obtain a set of axioms for the concept of an abelian group HAG,we add to H G the commutative law 9.1.2.

A set of axioms for the notion of a general ring H R can be formulated by means of the object symbol 0, the relation E ( x , y ) and the functors a(x, y ) (for x y), n(x, y ) (for x .y ) and ~ ( x(for ) - x), as follows

+

9.1.3. ( W W Y ) @ Z ) ( W [E(x, z> A E(Y, w) = E ( d x , Y ) , g(z, 4) A A E(" (x, v), 71 (z, w)) A E ( P ( X I , P (4)] (VX>WY)(VZ)E ( o ( d x , J %z O ( X , 4%2 ) ) ) ( V X ) (W @4E( a (a (x, v), z a(& a (Y,4)) W X ) E(o(O,x), x ) (W E(+(X), x ) , 0) (W(W E(" (x,Y), (u, 4 ) ( W W . Y ) W Z ) [E(+(X,Y), 2)' a(& 4, K ( Y , .I))]

i Y

Y

-

A set of axioms for the notion of a commutative field, H C F ,is obtained by adding the object symbol 1 and the functor p(x), for x-1- except when x = 0, see below, and by including the following axioms in addition to 9.1.3.

230

SELECTED TOPICS

r9.1.

9.1.4.

If we so desire, we may, for the sake of definiteness, add an axiom which determines the value of p ( x ) for x = 0, e.g. E(p(O), 0). However, this will not be essential for the sequel. A survey of 9.1.1.-9.1.4. shows that all the sets of axioms for the notions considered above consist of sentences in prenex normal form with universal quantifiers only. The following result is fundamental in the theory of such systems. 9.1.5. THEOREM. (Bernays) Let

X = (3~1). . . (gym) Q ( y l , . . .,ym), m 2 1

9.1.6.

be an existential sentence with matrix Q . Thus, Q is free of quantifiers. Suppose that Xis deducible from a set of universal sentences {XI, . . .,X k } which contain at least one object symbol. Let T be the set of complete terms which consist entirely of object symbols and functors that are contained in XI, . ., X k . Then there exist elements ti of T, i = 1, . . .,m, j = 1, . .., I 1 2 1, such that the sentence

.

. . ., t h ) V Q(t:, . . ., t i ) V . . . V Q ( t I , . . ., ?A) is deducible from {XI, . . .,Xk}. 9.1.7. Y = Q(t:,

We note that the converse of 9.1.5. is certainly true since Y 3 X is a theorem of the calculus. Proof of 9.1.5. Suppose that the theorem is not true. Let H be the set of sentences Q ( t l , . . ., tm) where tl, ., tm belong to T. Then H' = { X I , . ..,X k } u H must be consistent. For if this set were contradictory, then { X I , . . .,X k } u H" would be contradictory for some finite H" c H, which may be supposed non-empty for ease of discussion. If so, let

..

N

-

. . ., Q (ti, . . ., I;)> Then { X I , . . ., Xk, [ Q ( t i , . . ., t k ) v . . . v Q ( t I , . . ., th)] is contradictory and so the sentence XI, . . .,X k 2 Y is a theorem, where Y is as H" = {

N

Q (ti, . . ., t;),

N

in 9.1.7. This would entail the truth of 9.1.5. Thus, if that theorem is not true, H must be consistent. Now let K' be the set of sentences which are obtained from XI,. . .,Xk when the quantifiers are dropped and the variables are replaced in all

9.2.1

231

THE ELIMINATION OF QUANTIFIERS

possible ways by terms of T. Suppose that an element Z of K' has been obtained in this way from a particular Xt. Then Xt 3 2 is a theorem. It follows that K U H cannot be contradictory, otherwise { X I , ...,Xk> u H = H' also would be contradictory. Next, let K" be a set of sentences obtained from K' U H by replacing all atomic sentences which occur in K' U H by relative symbols of order 0, distinct atomic sentences being replaced by distinct relative symbols. Then K" is consistent and hence, possesses a model M". Now let M be a structure which is defined as follows. The objects of M are precisely the elements of T. The relations and functions of M are the relative symbols and functors of X I , . . .,Xk. Moreover, if v, ( , . . . ,) is such a functor, of order p , and 11, . . ., t p are elements of T then the functional value of v, at ( t l , . . ., t P ) is precisely the term v, (tl, . . .,t p ) . Finally, if R ( , . . . ,) is a relation of order q in M , and t l , . . ., t q are elements of T then we define that R (tl, . . .,t q ) does or does not hold in M according to the following rule. If R(t1, . . ., t4) occurs among the atomic sentences of K' u H then it holds in M if and only if the relative symbol of order 0 which corresponds to R(t1, ...,tq)in K" holds in M". If R (tl, .. .,t q ) does not occur among the atomic sentences of K' u H then we define (arbitrarily) that it holds in M. With these definitions, M becomes a model of K' U H and of H = { X I , .. .,X R } u H. But X , as given by 9.1.6. cannot hold in M since Q(t1, ..., tm) holds in M for all objects of M. This contradicts the assumption that Xis deducible from { X I , . . .,X k } and proves 9.1.5.

-

9.2. The Elimination of Quantifiers. Let K be a consistent set of sentences in prenex normal form with universal quantifiers only, in a language L with functors. Let Q ( X I , . . ., x,), n 2 0,be a wff which is defined in K i.e. all the object, relative, and function symbols of Q appear in K. Moreover, if n = 0, suppose that K contains at least one object symbol. A straightforward adaptation of the results of section 3.3. shows that if Q ( x i , . . .,x,) is persistent with respect to K under extension then there exists an existential sentence Q1 (XI, . . .,x,) which is defined in K such that 9.2.1.

K @'xi)

... (VX,)

[Q(xi, .. .,xB) G Qi (xi,

. ..,x,)]

while if Q (XI, ...,x,) is persistent with respect to K under restriction then there exists an universal sentence Q2 (XI, . . .,x,) which is defined in K such that

232

9-22.

[9.2.

SELECTED TOPICS

.. . (Vxn) [Q (xi, . . .,X n ) E QZ(XI, . ..,~ n ) ]

K 1(Vxi)

Now suppose that Q is persistent with respect to K both under extension and under restriction, i.e. in the terminology of 3.3. that Q is invariant with respect to K. In these circumstances, we are going to prove 9.2.3. THEOREM. There exists a WBQ‘ (XI, and is free of quantifiers such that 9.2.4.

K I- (VXI)

...,X n ) which is defined in K

... (Vxn) [Q (xi, ...,xn)

Q’(XI,

...,x.)]

PROOF.Since Q is persistent with respect to K both under extension and under restriction, there exist predicates Qi and Qz as in 9.2.1. and 9.2.2. Suppose that Ql(x, . . ., x) = ( 3 ~ 1 ) . . . (3ym) Si(x1,

. . ., X n , yi, . . ., ym), m 2 0 ... (VZZ) S Z ( X..., ~ , X n , ~ 1 ..., , ZZ), I Z 0 . Introducing object symbols (11, .. .,a n which did not occur previously, Q2(x,

...,

= (Vzl)

X)

we obtain from 9.2.1. and 9.2.2.

K

I-

3

[WZI) . - WZZ)~Z(a1,.- an, ~ 1 ,.. . ,zZ)] 3 [ ( 3 ~ 1 )- - . ( 3 ~ m~1) (ai, . . .,an, ~ 1 .,..,y m ) ]

It follows that there exists a finite subset of K, {XI, ..., Xk), k 2 0, such that the sentence

(3Yi)

- - . (3rd

S1 (ai,

.- .,an, .YI, ...,ym)

is deducible from 9.2.4. {XI, ..., Xk,((VZI)

. .. (VZZ) S Z ( U ~. .,., an, ZI, ..., ZZ)] } . Applying 9.1.5., we conclude that there exist terms 4 formulated by means of some (or all) of the object and function symbols of XI, . . .,X k , (Vz1) ... (Vza) SZ(ai, ..., an, zi, .. ., ZI) such that the sentence SI(Q, . . .,an, t 11, . ..,tk) V .. . V ~1 (al, . . .,a n , t;, . . ., tz)

is deducible from 9.2.4. Putting

..., X n ) = Si (XI, . . ., X n , TI,1 . .., T;) V . .. V S(XI,...,X n , q, . .. 7;) where the 4are obtained from the 4 by replacing al, . . .,an everywhere by xi, . . ., X n , we then obtain by one of the rules of inference, Q’(xi,

9.2.1

233

THE ELIMINATION OF QUA-

9.2.5. K k (Vx1)

. .. ( V X ~ )[ [ (Vzl) . .. (VZZ) S2(~1,. .., xn, ZI, . ..,ZI)] Q‘(x1,

.-

xn)]

.

3

On the other hand, the sentence @(GI,

.. . , a n )

2

... ( g y m ) Sl(a1, .. ., an, ~ i. .,., ~

(2~1)

m )

is a theorem of our calculus and so 9.2.6.

K t (Vxi)

.. . ( V X ~ )[Q’(xi, . . ., x,)

3

Qi(x1,

. ..,x.)]

3

Q’(x1,

. .., x,)] .

3

Q(x1,

while 9.2.5. may be rewritten as

.

(Vxi) . . ( V X ~ )[Qa(xi,. ..,x,) Combining 9.2.1. and 9.2.6., we obtain

9.2.7.

K

9.2.8.

K

(Vxi)

... ( V X ~ )[Q’(xi7 ..., xm)

Similarly, from 9.2.2. and 9.2.7.,

..

Q’ (xi, .,x,)] The two last formulae together yield 9.2.4. This proves 9.2.3. Now let K and Q (xi, ...,xn) have the properties detailed in the first paragraph of this section, and suppose moreover that Kis model-complete (section 4.2.). Then we are going to prove that 9.2.3. holds without any additional assumptions on the nature of Q.To see this we only have to show that, for model-complete K, every wff Q(x1, . . ., x,) as described is persistent under extension (and hence, every such wff is invariant) with respect to K. But this fact follows immediately from the definition of model-completeness, which postulates that if Q (al, . ,., an) holds in a model M of K then it holds also in every extension of M which is a model of K. This shows that Q is persistent with respect to K under extension. However, even without supposing that Kis model-complete, the assumtion that 9.2.3. holds for all existential wff Q(x1, ..., xn) entails that it holds for all Q.Indeed, the assumption that 9.2.3. holds for all existential wff implies at any rate that it holds also for all universal wff. For if Q (XI, . ..,x,) is universal then 9.2.9.

K k (VXI)

... ( V X ~ )[Q (XI, ...,x,)

..., x,)] .

-

3

... (Vxr) [ e (x1, ... xn) 3 Ql is provable for the existential wff Q l (XI, . .., x,) WXl)

y

(x1,

... x.)] y

which is obtained from Q by replacing the universal quantifiers by existential ones and by inserting the sign of negation between the prefix and the matrix. Then there exists a Q;which is free of quantifiers such that

234

and so

f9.2.

SELECTED TOPICS

K t- (VXI) . K b (VXI) .

. . (Vxn) [QI (XI, . .,xn) E Q; (XI, .- .,xn)]

. - (Vxn) [Q (XI, . . .,xn)

N

Q; (XI, . . .,xn)

Thus, Q;is the required wff. We exemplify the procedure for the general case by considering a wff (Vx) ( 3 ~(Vz) ) ( 3 4 Q (x, Y, z, w, 4 where Q is free of quantifiers. The formula contains a single free variable, 9.2.10.

U.

By assumption, there exists a wff QI(x, y, z, u), which is free of quantifiers such that Hence,

K I- ( V 4 OW (VY)W

4 [ [ ( 3 ~ Q) (x, Y,z, w,41 QI (x, Y , z , 4 ]

9.2.11. K I- (Vu) [ [ (Vx) (3y)(Vz) (3w)Q (x, y, z, w, u)] i~ [ (VX) (39(V4 QI (x, Y , z, 413

Also, by what has been proved already, there exists a wff Q2 (x, y , u), which is free of quantifiers such that Hence,

K

W.4 (VX)@A[ [ 0'4 QI (x, Y , z, u)]

9.2-12- K t (VU) [ [ (VX)(3.~1 (vz) QI (x, Y , Z,u)

1

Q2

( x , Y , u)]

-

[ (VX)(3.~1Q2 (x, Y , u)]]

Similarly, there exist a wff Qs(x, u) such that 9-2-13. K I- (VU)[ [ (VX)( 3 ~ Q2 ) ( x , Y , u)]

and a Q4(u) such that 9.2.14. K t (VU) [ [ (VX)Q ~ ( xu)] ,

[ (VX)Q3 (x, 411

Q~(u)]

Combining 9.2.1 l., 9.2.12., 9.2.13., 9.2.14., we obtain 9-2-15. K

( V 4 [ [ ( V X ) ( ~ Y ) ( V Z ) ( ~ Q(x, W ) Y , z, w, 43

Qa(u>]

so that Q4 (u) is the required predicate.

9.2.16. THEOREM. Let K be a consistent and non-empty set of sentences in a language L with functors, Suppose that for every existential wff Q (XI, . . .,x,), n 2 0 which is defined in K , there exists a wff Q' (XI, , x,) which is defined in K and is free of quantifiers, such that

.. .

9-2-17. K b (VXI)

... (VX~)[Q(xI, ...,x,)

Q'(xI,

..., x,)] .

9.2.1

235

THE ELIMINATION OF QUANTIPIERS

Then K is model-complete.

PROOF.Let M and M' be models of K such that M' is an extension of M . (The notion of an extension now takes into account also the functions of M and M ' , in the obvious way). Let X be a sentence which is defined and holds in M . Then we have to show that X holds also in M'. Let al, ..,a, be the object symbols of X which do not occur in K, n 2 0. Then we may write X = Q ( a , . . .,a,) where Q (XI, , ., x,) is a wff which is defined in K. By the assumption of 9.2.16. and by the remarks preceding it, there exists a wff Q'(XI, . . ,,x,) which is free of quantifiers and defined in K such that

.

.

9.2.18.

K k Q(a1,

. . ., a n ) shows that Q'(a,...,a,)

. . ., an)

Q'(a1,

Since M is a model of K, 9.2.18. holds in M. And since Q'(a1, . . ., a,) is free of quantifiers, it holds also in M'. It follows, again by 9.2.18., that Q (al, . . .,a,) holds also in M'. That is to say, X holds in M', as required. This proves 9.2.16. Theorem 9.2.16. shows that if to every existential Q which is defined in K we can find an equivalent Q' which is free of quantifiers then this establishes the model-completeness of K . Moreover, if the prime-model test, 4.2.3., applies then the procedure also proves that K is complete in the ordinary sense. This is the method of elimination of quantiJers. If the elimination of quantifiers is possible then it follows immediately that the model-completeness test, 4.2.1., applies but we did not require this fact in order to prove 9.2.16. We observe that the model-completeness test may still apply in cases where it proves difficult or impossible to eliminate quantifiers. On the other hand, Theorem 9.2.3. shows that model-completenessmay sometimes be used in order to establish the possibility of eliminating quantifiers. However, the quantifier-free predicate Q' obtained in this way for given Q may as yet be unsatisfactory. Thus, consider the notion of a real-closed ordered field. A set of axioms for this notion, ROF, has been given in section 2.2. in terms of the relations E ( x , y), S ( x , y, z), P (x, y , z), Q ( x , y ) (equality, sum, product and order). In order to apply 9.2.3., we have to replace ROFby a set of axioms in prenex normal form with universal quantifiers only. It is natural to try and obtain such a set by the introduction of Skolem functors. This requires, to begin with, the introduction of object symbols for zero and one, and of functors for sum, product, inverse with respect to addition and inverse with respect to

236

SELECTED TOPICS

I9.2.

multiplication (with a special convention for the value of the corresponding function at zero). In addition we aIso have to introduce functors corresponding to the roots of equations of odd degree and a functor for square roots (with a convention for the value of the corresponding function, e.g. 0, for negative values of the argument). The functors just mentioned do not define unique functions and, accordingly, the resulting set of axioms is not complete. Completeness can be re-established in this case by the introduction of additional axioms which make the functions in question unique. Thus, for the square root functor we may stipulate that the value of the root is non-negative. For the polynomials of odd degree, we may require that the functor in question denotes the smallest root. Even so, the new functors will in general enter into the wff Q' which may accordingly contain square roots and roots of equations of high degree with XI, . ..,X n as parameters. In actual fact, the introduction of these irrational functions can be avoided by combining 9.2.3. with 5.3.7. 9.2.19. THEOREM. Let K b e a non-empty and consistent set of sentences in

prenex normal form with universal quantifiers only, formulated in a language L which includes function symbols. Let K* be the modelcompletion of K,and let Q* (XI, .. .,x,), n 2 0, be a wff which is defined in K* and K. For n = 0, suppose that K contains at least one individual object. Then there exists a wiT Q' (XI, ...,xn) which is defined in K and is free of quant3ers such that for any 41, . . .,a, in a model M of K, Q'(al, . . ., a,) holds in M if and only if Q*(al,. . ., a,) holds in all extensions M * of M which are models of K*. The wording of the conclusion has to be modified in the obvious way if n = 0.

PROOF.Suppose that the assumptions of 9.2.19 are satisfied. It then follows from 5.3.7. that there exists a wff Q(x1, .. ., x,) such that if al, . . .,a, are elements of a model M of K then Q (al, . . .,a,) holds in M if and only if Q*(al, . ..,a,) holds in all extensions M* of M which are models of K.Q (XI, . . .,x,) is invariant with respect to K. By 9.2.3. it may therefore be replaced by a wff Q' (XI, ...,x,) which is free of quantifiers. This concludes the proof of 9.2.19. For an application of 9.2.19., let K be a set of universal axioms for the notion of an ordered field formulated by means of the relations of equality and order and of the individuals 0 and 1 and by means of the

9.2.1

237

THE ELIMINATION OF QUANTIFIERS

(functors for the) functions of addition and multiplication and of the inverses with respect to these operations, as introduced above. Also, let K* be a set of axioms for the notion of a real-closed ordered field, formulated by adding to K axioms for the existence of square roots of positive elements and of roots of monic polynomials of odd degree. These axioms can be formulated in the vocabulary of K, but with the inclusion of existential as well as universal quantifiers. Then Kand K* satisfy the conditions of 9.2.19. For example, taking for Q*(XI, . . ., x,) the predicate that the polynomial yn XI yn-1 . . X n possesses a root, we now find that we may formulate Q’(x1, . . ., x,) entirely in terms of equality, order, 0, 1, and the four functors used in the formulation of K. We may improve this result still further by taking for K a set of universal axioms for the notion of an ordered integral domain with unit element. It is not difficult to verify that K may be formulated in terms of the above relations and individuals and in terms of addition and multiplication and the inverse with respect to addition alone. The set K* is still to be taken as a set of axioms for the notion of a real-closed ordered field but formulated in terms of the same constants, relations, and functions as K. Then the conditions of 9.2.19. are again satisfied. Thus, for given Q*(XI, . ., x,) we may formulate the corresponding Q’in terms of addition, multiplication, and the inverse with respect to addition (alternatively, subtraction), together with 0 and 1 and with the relations of equality and order. A slight modification then shows that Qf is equivalent to a disjunction of systems of polynomial equations and inequalities with integer coefficients of the form

+

+ . +

.

pr(x1,

..., x,)

..

= 0,

i = 1,

..., k, @(xi, ..., x,) > 0, i = 1, ..., 1 .

If the XI, .,x, represent the coefficients of a polynomial of y as detailed above, such a disjunction is in fact given directly by Sturm’s theorem. For general Q* the existence of a Q f of the required form was established by Tarski, who showed thereby that the theory of real closed fields is (decidable and) complete. Another classical example is obtained by taking K as a set of universal axioms for the notion of an integral domain with unit element, and K* as an extension of K which is a set of axioms for the notion of an algebraically closed field. The vocabulary to be used is the same as in the previous example, excepting the relation of order. Then 9.2.19. applies. For example, suppose that Q*(XI, .,x,) states that a certain system of equations and inequations with unknowns y l , . . .,Y m and with coeffi-

..

238

SELECTED TOPICS

19.3.

cients which are themselves polynomials of XI, . . ., xn with integer coefficients -

...,X,,JJI,. ..,vm)= 0,i = 1, ...,k, qt (XI,. . .,xn)yl, . - .,ym) # 0,i = 1, . . .,I

pt(x1,

has a solution. Then 9.2.14. establishes the existence of a disjunction of systems of equations and inequations with integer coefficients of the form

. . ., xn) = 0,i = 1, . , . , j , si(x1, . . ., x,) # 0,i = 1, . . .,m such that for any set of elements m, . . ., a, in a given field or integral domain M , pz (al,. . .,an,yl, . . .,y m ) = 0,qi (ai,. . .,an, yl, . . .,y,) # 0 rt(x1,

has a solution in the algebraic closure of M if and only if for one of the systems of the disjunction, rt(a, . . ., a) = 0,st (al,. . ., an) # 0 in M . The existence of such a test follows also from an analysis of resultant theory in classical Algebra, such as has been carried out by Seidenberg. Seidenberg’s work in this field is closely related to his results in Differential algebra, which were used in 5.5. in order to establish the existence of a model-completion to the notion of a differential field.

9.3. Direct Products and Ultraproducts. In several branches of Algebra, an important part is played by certain standard procedures by which new structures are obtained from given structures of a particular kind. An example of such a procedure is provided by the passage to a quotient group or to a quotient ring, whose metamathematical significance was discussed in Chapter VII. Another procedure of this kind leads to the direct product of a given set of groups or rings. The generalization of this notion to general structures with or without functions is straightforward. Let T = { M y } be a set of similar structures (i.e. containing the same relations and functions) with index set I = {v} where I is not empty and may be finite or infinite. Mv is a structure M* which is defined as Then the direct product V

follows. The set of individuals of M * consists of all functions f = f ( v ) defined on I such that f ( v ) E M y for all v E I. In this connection, it is important to distinguish between the functions (i.e. “operations”) of a structure, such as addition and multiplication which are formally part of the structure, and any other functions which may be definable in the domain under consideration. For any function (XI,. . .,xn) which is contained in the structures of T,we define 9 (XI,. . .,x,) in M * by

+

9.3.1

9.3.1.

DIRECT PRODUCTS A N D ULTRAPRODUCTS

+(fi,

...,fn)

= +(f i ( v ) ,

239

...,f n ( v ) ) in M y for all v E Z.

+

That is to say, for any f i , .. .,f n in M*, the value of for the arguments ...,fn is the function f ( v ) which is given by the right hand side of 9.3.1. Moreover, if an individual a is contained in all My, we shall identify the constant functionf (v) E a with a. Let R (XI, . . .,X n ) be any relation which is contained in the structures of T. For any set of individuals fi, . . .,f n of M * we then define that R (fi, . . .,fn) holds in M * if and only if R (fi (v), . . .,f n (v)) holds in Mv for all v E Z. It is not difficult to verify that if the structures Mv are rings formulated for instance as models of KR in terms of the relations E, S, P, and without individuals or functions, then M * also is a model of KR and is in fact the direct product of the Mv in the usual sense. If the Mv are commutative fields then M* is a commutative ring but, for I I I > 1, is not a commutative field since it includes zero divisors. Accordingly, we see that a sentence X may well be defined and hold in all M y without being satisfied by M * . The question, what sentences have the property of remaining true under passage to the direct product has been the subject of a number of papers. We shall now describe a construction which is due to Lo$ although some of the ideas leading up to it can be found also in an earlier method of Skolem’s. For this purpose, we require a generalization of the special valuation lemma 1.5.6. To formulate it, we shall make use of the notation of 1.5.6. A non-empty set A of non-empty subsets of a set I is called a net on Z if for any JI,JZE A there exists a 5 3 E A such that J1 n JZc Js.

fi,

9.3.2. VALUATION LEMMA.Let Z = {v} be an index set for a set @ = { $ y } of partial valuations of a set S. Let A be a net on Z such that for every finite subset U of S and for every element J of A there exists a v E J for which U c DC#,,. Then there exists a total valuation I,Y of S such that for every finite subset U of S and for every J E A there exists a v E J such that U c DC#,, andy/I U = + , I U. We prove 9.3.2. by generalizing the proof of 1.5.6. A partial valuation of S will be called admissible if for any finite subset U of S and for any element J o f A there exists a v E Jsuch that U c Dqh and I,Y I U = c $ ~ I Dty n U. The set Y of admissible partial valuations is not empty for it contains the empty partial valuation. Y is ordered partially by the relation of extension as defined in the proof of 1.5.6. Any non-empty totally

240

19.3.

SELECTED TOPICS

ordered subset Y' = {yp} of Y possesses an upper bound y' in Y. For if we define the domain of definition of y' by D' = D y ' = U { D y a } , B

and the value of y' for any argument A within D' as the joint value of all ya E Y which are defined for A then we have ya < y' for all ya E Y . Moreover, yl' is admissible. Indeed, let U be any finite subset of S and let J EA. Since U' = U n D' is finite there exists a W A E Y such that U c D y l . Since yn is admissible there exists Y E J such that U' c DpV and W A I U' = +v I U'. But D+v c D' and W A I U' = y' I U' and so U' c D' and y' I U' = pVI U' showing that y' is admissible. By Zorn's lemma, Y contains a maximal element, yo. We claim that yo is a total valuation of S. Supposing the contrary, let A E S - Dyo. Define the partial valuation y1 of S by y 1 = yo on Dy0 and y1 (A) = 0. Then y1 cannot be admissible since ylo is maximal and there exists a finite subset V1 of Ssuch that for some element J1of A the conditions V1 c D+,. and y1 1 V1 = +,. I B y 1 n V1 are not satisfied for any Y E J1. V1 includes A, otherwise y1 I Dy1 n V1 = yo I D yo n V1, Since yo is admissible we may then find a v E J1, such that V1 c Dyo C D y l and +,, I Dyo n n V1 = qv I Dy1 n V1 = 1/11 I V1 and this contradicts our assumption on V1. Similarly, defining the partial valuation y2 by yz = yo on Dyo and y z ( A ) = 1, we infer from the maximum property of yo that there exists a subset VZ of S, A E VZ and an element JZof A such that the conditions VZ c DbV and ylz I V2 = q5,. I Dyl2 n VZ are not satisfied for any v c Jz.Now A is a net, so there exists an element J3 of A such that J1 n JZ J3. And yo is admissible so there exists a Y e 53 such that V1 u VZ c D+v and yo I VI u VZ = 4,. I D ~ nO(VIu VZ). Suppose first that +,.(A) = 0. In this case y1 I V I = #,. I Dy1 n VI, i.e. y1 coincides with +V on the intersection of the domain of y1 with V1 and so ty1 is admissible, contrary to construction. Suppose on the other hand that +,.(A) = 1, in which case a similar argument shows that (YZ is admissible. We see therefore that the assumption that S - Dylo is not empty leads to a contradiction in all cases and we conclude that the valuation yo is total. This completes the proof. Observe that 1.5.6. is obtained from 9.3.2. for A = {I}. Suppose now that A is an ultrujilter (also called a maximal ideal or a maximal dual ideal) in the algebra of subsets of I. That is to say, A satisfies the following conditions. 9.3.3. A is a non-empty set of non-empty subset of Z; if J1, JZE A then J1 n J z E A ; a n d i f J c ZtheneitherJEAorI- J E A .

9.3.1

DIRECT PRODUCTS AND ULTRAPRODUCTS

241

An immediate consequence of 9.3.3. is that if J EA and J' 3 J is another subset of Z then J' E A . For either J' or Z - J' belongs to A by one of the rules of 9.3.3., and if I - J' belonged to A then the empty set which is equal to the intersection of J and Z - J' also would belong to A . This is impossible and shows that J' E A . 9.3.4. THEORFX. Let Z = { v } be an index set for the set @ = {c#~} of total valuations of the set S. Let A be an ultrafilter on I. Then there exists a unique total valuation y of S such that for every finite subset U of S and for every J c A there exists a v E J such that y I U = +v I U.

PROOF.Let yo be a total valuation such as exists according to the conclusion of Theorem 9.3.2. Then it is easy to check that y = yo satisfies also the conclusion of 9.3.4. To show that v/ as described in the conclusion of 9.3.4. is unique, we observe that for any A E S we have v / ( A ) = 0 or cy(A) = 1 according as the set of v such that +,,(A) = 0 does or does not belong to A . Indeed, let J be the set of all v E Z such that +,(A) = 0 and suppose that J E A . Take U = { A } then by the assumption of the theorem we have for some v E J, v/ I U = +,, I U,i.e. y ( A ) = +,(A) = 0. On the other hand, if J does not belong to A then Z - J belongs to A according to one of the conditions of 9.3.3., and we may conclude that in that case (A) = +,.(A) = 1 for some v E I - J. This shows that y is defined uniqueIy by @ and A and completes the proof of 9.3.4. Now let T = { M y }be a non-empty set of similar structures with index set Z and let A be an ultraillter which is defined on Z. We introduce a structure M' = (nMV),,in the following way. The individuals of M' V

are the functions f which are defined on Z such that f ( v ) E M y for all v E Z. For any individual a which is contained in all M y we identify the constant function f ( v ) a with a, as in the definition of the direct product, M * . For any function (XI, . . .,X n ) which is contained in the structures Mv we define +(xi, .. .,Xn) in M' by

+(fi,

- - -,fn) =

d(fi(V1,

- -*,fn(v))

again as in the definition of the ordinary direct product. For any relation R(x1, .. .,x,) which is contained in the My, and for any n-uple of individuals of M', fi, ...,f n let N be the set of v such that R (fi (v), . ,., fn(v); holds in My. Then we define that R(f1, .. .,fn) holds in M' if and only if N belongs to A . The structure M' obtained in this way is said to be an ultraproduct, more particularly the ultraproduct of {Mv} with respect to A .

242

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9.3.5. THEOREM. Let X be a sentence which is defined and holds in all elements of a set of similar structures { M y }and let A be an ultrafilter on the index set I = { Y } of { M y } . Then X holds also in the ultraproduct M’ = ( I I M V ) ” . V

PROOF.Clearly, it is sufficient to prove the theorem for the case that Xis in prenex normal form, To exemplify the procedure, let us suppose that

9.3.6. X = ( 3 x ) ( t l y ) ( 3 z ) ~ u ) ( V v ) O w )Q ( x , y , z, u, u, w, a, b, c ) where the matrix Q does not contain any quantifiers and does not contain any constants other than a, b, c. Let X’be the Skolem or open form sentence which corresponds to X,i.e. for 9.3.6., 9.3.7.

X’

=

Q ( h Y , w(v), u, u, X(Y,u, v), a, by c )

In order to show that X holds in M‘, we have to find a constant function

4, and functions t,u ( y ) and x ( y , u, v) which are defined on the individuals of M’ such that for arbitrary y

9.3.8.

= g,

u = h, u = k in M’ the sentence

Q(4,g, V ( ( Y ) ,h , k, x (g, 4 k), a, b, c)

holds in M’. Note that a, b, c, in 9.3.8. stand for the constant functions f ( v ) G a , f ( v ) fb , f ( v ) c, respectively. We know that the sentence X holds in M y for all Y E I. Thus, for every Y E I, we can find an individual +v in M y and functions w v ( y ) and xy(y, u, u) defined on the individuals of Mv and taking values in M ysuch that for any individuals p , q, r in Mv, the sentence

943.9- Q(& P, w v ( P ) , 4, r, X V ( P ,4,r), a, b y

C)

holds in Mp.Accordingly we define the Skolem functions 4, w, x in M’ by

i

for all

Y E

I

We note that for any set of argument values in M‘, the right hand sides of these equations do indeed define the functional values of $I, ly,x as individuals of M’. We are going to show that with these definitions, 9.3.8. holds in M’ for arbitrary g , h, k in M‘. Now for given g , h, k the truth value of 9.3.8. (“holds” or “does not hold”) is determined entirely by the truth values of the atomic sentences which occur in it. It follows that in order to prove that 9.3.8. hoIds in

9.3.1

DIRECT PRODUCTS AND ULTRAPRODUCTS

243

M’ we only have to establish the existence of a v E Z such that the truth values of the atomic sentences of 9.3.8. coincide with the truth values of the corresponding sentences in 9.3.9. In this connection, the atomic sentence Y(”)= R(a1, . .,0%)in 9.3.4. which corresponds to an atomic sentence Y = R ( f 1 , ...,fn) in 9.3.8. is, by definition, the wff which is obtained from Y by replacing f l y .. .,f n by their functional values in Mv, ai =A(.), v = i, . . ., n. Let Y I , . . ., Ym be the atomic sentences which occur in 9.3.8. in some arbitrary but definite order, and let Y‘Y,’, . ., Y ( 2 be the corresponding atomic formulae in 9.3.9., for any v E Z. By the definition of the truth value of any Yi in M‘, the set JZ of Y such that the truth value of Y y ) in Mv coincides with the truth value of Yg in M’, is contained in A , Ji E A. Let J = JI n J2 n ... n Jm, then J E A and J is not empty. For any v E J the truth values of Y $ ) , i = 1, .. .,m coincide with the truth values of YI,i = 1, . . .,m. This proves 9.3.5. If all Mv coincide with a specific atomic structure M , then the result of the ultraproduct operation, for given Z and A is said to be an ultrapower of Mand is denoted by M i . It follows from 9.3.5. that any sentence X which is defined and holds in M holds also in M i . Thus, M i is an elementary extension of Mywhere it is understood that the individuals of M are identified with the constant functions of M i . Suppose in particular that M is the system of natural numbers endowed with the relations E, S, and P (equality, addition and multiplication). Suppose that I also is the system of natural numbers and that A is an ultrafilter which includes the complements of all finite subsets of I. We are going to show that in this case M i is a proper extension of M. The elements of M are represented in M j by the constant functions f n (v) n, n = 0, 1,2, ... Consider the function f (v) = v, which also belongs to M i . We claim that fn < f i n M i for all natural numbers n. Indeed, for any natural number n, the set N of all v such thatfn ( v ) 2 v, i.e. n 2 Y, is finite and so its complement Z - N belongs to A . This shows that fn
.

.

244

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I9.4.

It might be thought that the construction of ultrapowers can add but little to our understanding of a particular theory which is formulated in the Lower predicate calculus, since by its very nature the result of the operation has exactly the same elementary properties as the original structure. However, it turns out that by a judicious combination of these properties with other features of the structure whose description is beyond the power of expression of the language L one can obtain far-reaching results. Thus, the theory of ultra-products provides interesting alternative derivations of a number of the results proved earlier in this book. Ultraproducts are of some significance also in the theory of Non-standard analysis, which will be described in the next section.

9.4. Non-Standard Analysis. Let Ro be the set of real numbers. Let K be the set of all sentences that are formulated in the language L of the Lower predicate calculus in terms of (object symbols for) all elements of RO and in terms of (relative symbols for) all relations of positive order that are definable in Ro. Let KObe the set of sentences of K that hold in Ro. Although we do not include any function symbols in the vocabulary of KO,the set KO includes sentences which may be interpreted as statements about functions in Ro in a way used repeatedly in earlier chapters. Thus, if the real-valued functionf(x) possesses the set A c Ro as domain of definition, then there exists a two-place relation F(x, y ) in the vocabulary of KO such that P(u, b) holds in RO if and only if a E A and b = f ( u ) . If E(x,y)is the relation of equality in Ro, and T ( x ) is the one-place relation which defines A (i.e. T(u)holds if and only if a E A ) then the fact that F ( x , y) denotes a function with domain of definition A is expressed by the sentence : (W (VY) r) = W)] A [ (V'X) [ W )= [ (3Y) (V-4[% v) A [ W Y 4 = J W Y z ) ] ] ] ] ] This sentence is an element of KO.Note that F(x, y ) corresponds to f ( x ) with the specijied domain of definition. Again, if S(x, y, z ) and P(x,y , z) are the relations of addition and multiplication in Ro and H ( a , b) holds if and only if b = ea, where ez is the exponential function then the following is a sentence of KO.

[w,

9.4.1. (Vx) (Vy) (Vz)(Vu) (Vv) (Vw) [ H ( x , u) A H ( y , v) A H ( z , w) A S(x, Y , 4 = P(u,u, w)]

Indeed, 9.4.1. represents the functional equation for the exponential function.

9.4.1

245

NON-STANDARD ANALYSIS

Another important relation which is defined in Ro and hence, belongs to the vocabulary of KO,is the one-place relation N ( x ) (say) which holds for any a E RO if and only if a is a natural number, a = 0, 1,2, 3, . . . All models of KOare elementary extensions of Ro. A proper extension of RO which is a model of KO will be called a non-standard Model of Analysis. There exist non-standard models of analysis. To see this, let a be an individual which is not contained in ROand let G be the set of all sentences E(a, by) where by varies over all elements of Ro. Consider the set KO u G. If this set were contradictory then the set KO u G' would be contradictory for some finite subset G' of G. G' cannot be empty since KO possesses a model, Ro, and hence is consistent. Let G'= {- E(a,bl), E(a,b2), ..., E(a,bn)), n 2 1. N

-

-

N

Then KO u {- E(a, bl) A dictory. It follows that

E(a, bz) A

KO C. E(a, bl) A E(a, b2) A

.. . A

-

E(a, bn)) is contra-

. . . A E(a, bn)

But this entails that 9.4.2.

KO C. (VX) [E(x,bi) V E ( x ,b2) V

... V E ( x ,ba)]

The interpretation of 9.4.2. is that all elements are equal to one of bl, b2, . . ., bn and this is clearly not true in any model of KO.We conclude that KO u G is consistent and possesses a model, R*. R* is a proper elementary extension of Ro since it contains the element a which is different, in R*, from all elements of Ro. Thus, R* is a non-standard model of analysis. We can also obtain a non-standard model of analysis as an ultraproduct of Ro. Let the index set, I, be the set of natural numbers and suppose that the ultrafilter, A, contains the complements of the finite subsets of I. Then Roi is an elementary extension of Roy by 9.3.5. To see that Roi is a proper extension of RO consider the element f of R o which ~ is given by f (n) = n. Thus, f takes the functional value n, a natural number regarded as an element of Ro for the argument value n, where n is now an element of I. By the definition of the structure Roi, N c f ) holds in that structure. We show as in 9.3. that f cannot be equal to any natural number 0, 1,2, 3, ... On the other hand, f cannot be equal in R o to ~ any element a of Ro which is not a natural number in Ro, for in that case, N(a) holds N

246

[9.4.

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in Ro and hence in Roi. We conclude that f is different, in Roi, from all elements of Ro and hence, that Roi is a proper elementaryextension of Ro, Roi is a non-standard model of analysis. We notice that Roi is of cardinal cNo= c, where c is the cardinal of the continuum. The existence of a non-standard model of analysis with this cardinal does not follow from previous construction by means of the extended completeness theorem since the cardinal of KOis 2C. Roi, regarded as a field, occurs also in the theory of rings of continuous functions, where it belongs to the class of hyper-realfields. Let R* be any non-standard model of analysis. Since R* is an ordered field and a proper extension of Ro, it must be non-Archimedean. It follows that the two subsets of Ro, M Oand M I , which are defined below are not empty. 9.4.3. MO- the set of all a E R* such that I a M I - the set of all a E R* such that a

I < r for some r E Ro;

I I < r for all positive r E Ro.

The elements of Mo will be said to be5nite. The elements of M I will be said to be infinitesimal. The elements of ROwill be called standard. They form a subset of Mo. Complementarily, the elements of R* - MO are infinite, and the elements of R* - RO are non-standard. MO is a ring and M I is an ideal in Mo. More particularly, it is not difficultto verify that M I is a maximal prime ideal in Mo. It follows that the quotient ring Mo/Ml is a field. Mo/MI contains a subfield which is isomorphic to Ro since different elements of Ro are different modulo M I . On the other hand, every element of MOdetermines a Dedekind cut in Ro and is congruent modulo M I to the element of Ro which determines the same cut. It follows that Mo/Ml is isomorphic to Ro. Earlier we introduced N ( x ) as the one-place relation which determines the set of natural numbers (NO, say) in Ro. The same relation determines a set N * in R*. N* is a strong non-standard model of arithmetic. Indeed, N* is a proper extension of No since the natural number are co-final with Ro, and this is a property which can be formulated within KOand accordingly holds also for N* within R*. And all properties of NOwith respect to addition, multiplication, and equality, which can be formulated within L, belong to KO and accordingly, hold also for N * . We are going to show that within this framework, we can develop a calculus which involves infinitesimal (infinitely small) and infinitely large quantities. This will enable us to restate many of the familiar results of classical Function theory in a language of infinitesimals such as was

9.4.1

247

NON-STANDARD ANALYSIS

envisaged, vaguely, by Leibniz. Our approach yields a proper extension of classical Analysis. Thus the standard properties of specific functions (e.g. the exponential function, the Bessel functions) in the field of real numbers still hold in non-standard analysis. But beyond this, we may now enrich our vocabulary by the use of new concepts, such as the property of belonging to M I , which have no counterpart in ordinary Analysis. From now on, we shall refer to functions, sets, relations, etc. which are defined already in Ro as standard (functions, sets, etc.) The interval a < x < b is standard if a and b are standard numbers (elements of Ro) while the interval E < x < b, b standard and E infinitesimal, is nonstandard (except for E = 0). The function lnx is a standard function which is defined in the standard interval 0 < x < cm,i.e. for all positive x in R*. In order to handle our tools effectively, we have to be able to decide in each particular case whether or not a statement can be formulated within the language L ; at the same time we have to try and reduce the amount of formula writing which is necessary if we wish to express all statement for which this can be done (and which occur in the argument) explicitly within L. Accordingly, we shall make use of accepted expressions such as If ( x ) - a I < E as parts of our formal sentences and we shall leave it to the reader to verify that the expression which we shall use can indeed be translated into L. We select the theory of infinite sequences as our first topic for consideration from the point of view of non-standard Analysis. Let {sn} be a standard sequence. Thus, {sn} is a function of n which is defined in the first instance for all standard natural numbers N and taking values in Ro.Passage to R* extends the definition of {sn} automatically to all n EN*,i.e. also to the non-standard natural number. A non-standard natural number will be called also an infinite natural number (or, an infinitely large positive integer). 9.4.4. THEOREM. The standard sequence {sn> is bounded if and only if

so is finite for all infinite natural numbers o.

PROOF.If {sn} is bounded then there exists a positive standard number p such that 9.4.5.

(W [w= [ I 3s I

PI]

where we recall that the relation N ( x ) defines the natural numbers.

248

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9.4.5. can be formulated as a sentence of KOand, accordingly, holds also in R. It follows that 1 sm I < p also for all infinite positive integers w. Conversely, if so is finite for all infinite natural numbers o then it is finite for all natural numbers w. Accordingly, if p is any infinite positive number (i.e. a positive number which belongs to R* - Mo) then (Vx) “ ( x ) 3 [ I sz I < p ] ] holds in R* and so 9.4.6.

= [ I sz I

(3Y)(VX) “(x)

Yl]

holds in R*. But 9.4.6. can be formulated as a sentence of K and so it must hold in Ro. (For if the negation of 9.4.6. hold in Ro then it would be expressible as a sentence of KOand hence would hold also in R*). Thus, there exists a real number yo such that (Vx) “(x) =) [ I sz Q p o l ] holds in Ro. In other words, {sn} is bounded.

I

9.4.7.

THEOREM. The standard number

sequence

{Sn},

lim

n+

Sn

s is the limit of the standard = s if and only if s - so is infinitesimal for all

m

infinite positive integers o. PROOF. Suppose that lim sn = s. Let E be any standard positive number. n+m

Then there exists a standard positive integer v such that the statement 9.4.8.

Vx) “(x) A [x > V]

3

[ I - sx

I < &I]

holds in Ro. 9.4.8. can be formulated as a sentence of KOand so it holds also in R*.Now any infinite natural number o is greater than v and so, by 9.4.8., I s - so I < E. Since this is true for all standard positive numbers, I s - so I must be infinitesimal, and the same must be true of s - so. Conversely, suppose that Sn and s are standard, and that s - sm is infinitesimal for all infinite positive integers o.Let E be any standard positive number, and let v be a particular infinite positive integer. Since s - so is infinitesimal for all infinite positive integers the statement 9.4.9.

(Vx)[N(x) A

[X

> V]

2

[ I s - sz I

< el]

holds in R*.This statement cannot be formulated within Kand it contains the individual v. However, by 9.4.9., 9.4.10- ( 3 ~ “(Y) )

A (Vx) [ N ( x )A

[X

> Y]

3

[ I s - sx

I < &I]]

holds in R* and 9.4.10. can be formulated within K . We conclude as before that 9.4.10. holds in Ro, there exists a (standard) positive integer YO such that

9.4.1

NON-STANDARD ANALYSIS

249

A [X > YO] 2 [ I s - ~z I < &I] holds in Ro. This shows that lim sn = s according to the standard (VX) “(x)

n+co

definition and completes the proof of 9.4.7. For any a and b in R*,we shall write a 2: b if a - b is infinitesimal and we shall say in this case that b is infinitely close to a. In this terminology, 9.4.7. states that lim sn = s for standard isn> and s if and only if

n+ m smis infinitely close to s for all infinite natural numbers w. The relation

N

is a relation of equivalence.

9.4.11. THEOREM. The standard number s is a limit point of the standard sequence {sn} if and only if sw is infinitely close to s for some infinite natural number w.

PROOF.Suppose the standard number s is a limit point of the standard sequence {sn}. Then the following statement holds in Ro 9.4.12. ( V X ) ( V ~ ) [ [ X > O ] A N ( . Y ) ~ ( ( ~AZ[z>.Y]A[Is-s~I<x]]] )[N(Z)

This statement can be reformulated within KO and, accordingly, holds also in R*. Let E be positive infinitesimal and let be an infinite natural number. Then, by 9.4.12., the statement (3.4 “(z) A rz > ttl A [ I s - sz I < El] holds in R*. It follows that there exists an element w of R* such that N(w) holds in R* - i.e. w is a natural number -, and w > q holds in R* i.e. o is an infinite natural number such that I s - sw I < E holds in R*. Thus s N so, the condition of 9.4.11. is necessary. Conversely, suppose that for standard isn} and s and for some infinite natural number w, so is infinitely close to s. Let E be a standard positive number, otherwise arbitrary, and let 17 be any standard natural number. Then the following holds in R*

> 01 A N ( y ) 3 (32) “(2) A [Z> .Y]A [ I s - sz 1 < E ] ] . Indeed, the implicate of 9.4.13. is (32) “(2) A [z > q] A [ I s - sz 1 < E]] and this holds in R* since N ( o ) A [w > q] A [ I s - so I < E ] ] holds in

9.4.13.

[E

R*.Thus, 9.4.13. holds in R* and hence, by the same argument as before, holds in Ro. This shows that s is a limit point of {sn}. Corresponding to Cauchy’s general convergence test, we have

9.4.14. THEOREM. A standard sequence {ss) converges (possesses a limit) if and only if SA 21 sc for all infinite positive integers Iz and p.

250

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[9.4.

9.4.14. may be proved, after the manner of 9.4.7., by relying on Cauchy's test, which states that the standard sequence {sn) converges if and only if lim (sn - sm) = 0 . %+a0

rn-m

However, we may also adopt another and in some ways more interesting procedure. That is, we may try to use R* in order to eliminate, as far as possible, the traditional &-technique.Of course, this idea can only serve as a general guiding principle since all the traditional methods are still applicable, at least to all standard concepts. Thus, we shall now prove 9.4.14. using the condition that sa,'Y s for all infinite o as the dejinition of the concept of the limit. It follows immediately from this definition that the condition of 9.4.14. is necessary. Indeed, let I and p be any two infinite positive integers. Then $1 N s and s, 'Y s by assumption, and so SA II s, since II is a relation of equivalence. Suppose now that the condition of 9.4.14. is satisfied for the standard sequence {sn}. We claim first of all that SA is finite for some infinite natural number 1.Indeed, if this were not the case then {sn} would be unbounded, by 9.4.4. Accordingly, it would be true in Ro and hence in R* that for any positive v there exist (Y and j3 greater than v such that 1 sa - sg I > 1. Taking v infinite, we find that sa is not infinitely close to sg for infinite a and /3 and this contradicts the condition of 9.4.14. It follows that SA is finite for some infinite natural number A, SA E Mo.As stated above, every finite number a E MO is infinitely close to a unique element b of Ro. We call b the standardpart of a, and we write b = st (a). With this notation, let s = s t ( s ~ ) .We claim that s is the limit of {sn}. Indeed s 1: SA since s = st @A), and SL II s, for all other infinite natural numbers p, by assumption. Hence, s 2: s, for all infinite natural numbers p, s is the limit of {sn}, 9.4.14. is proved. Next, we shall give a non-standard proof of the theorem of BolzanoWeierstrass, 9.4.15. THEOREM. Every bounded standard infinite sequence {sn} possesses a (standard) limit point s.

PROOF.For bounded {sn} and any infinite natural w, sw is finite, by 9.4.4., and possesses a standard part s. By 9.4.11., s is a limit point of Isn>.This proves 9.4.15. The following alternative proof is instructive. Choose a standard a such that - a < sn < a for all standard positive integers n (and hence,

9.4.1

25 1

NON-STANDARD ANALYSIS

also for all infinite positive integers). Let m be any standard positive integer and divide the interval - a < x < a into 2m sub-intervals A k of equal length alm, so that x E A k if

(v

- 1) a < x

<

(k

- 1) a, k =

1,2,

.. ., 2 m .

Since the number of intervals A k is finite, there exists a k such that A k contains Sn for an infinite number of indices n. Thus, the following statement holds in Ro. 9.4.16. “For every positive integer w, there exists a positive integer

x,x < w, such that for every positive integer y there exists a positive integer z, z > y , such that

It is not difficult to check that 9.4.16. can be formulated as a sentence of KO and hence, that it holds in R*. Accordingly, if we choose w as an infinite positive integer m then there exists a positive integer k < m such that if y = v is any infinite positive integer, then there exists a positive integer z = p which is greater than Y and hence also infinite, such that

(E

-

1) a <

sg

<

(;

- 1)

a.

Now lets be the standard part of

(;-+

Since

is infinitesimal, s is also the standard part of

( $ 4 ) . and of s,. Thus, sg N s, s is a limit point of {sn}. This provides a second proof for 9.4.15.

252

[9.5.

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The proof of Cauchy’s general convergence test is usually carried out with the help of 9.4.15. However, in non-standard Analysis, 9.4.15.can be circumvented by proving first 9.4.14.and by showing afterwards that 9.4.14. is equivalent to the general convergence test, as indicated above. The theorem that the limit of the sum of two standard sequences is the sum of the limits of the two sequences follows immediately from the fact that A41 is an additive group. Now suppose lim sn = s, lim tn = t, .00 +

%+W

for standard { s ~ } ,s, (t.}, t. Then sa,11 s, ta, 21 t for all infinite natural numbers o and so s,,,to, 111 st since A41 is an ideal in Mo. This proves that lim sntn = st. -00

9.5. Non-standard Theory of Functions of a Real Variable. Let a be a standard real number and letf(x) be a standard function which is defined in an interval b < x < u, b standard. We consider the behavior of the functionf(x) as x tends to a from below. 9.5.1. THEOREM. The standard number I is the limit off(x) as x tends to a from below, lim f ( x ) = I if and only iff(a - q) N I for all infinitesimal ta positive q.

PROOF.Suppose lim f(x)

= 1. Then for every positive

E

there exists a

zta

positive 6 such that

WY) “0 < Y < 61 = If@- Y ) - 2 I < E l ] This sentence can be formulated within KOand, accordingly holds in R*. It follows that I f ( a - v ) - I I < E for all infinitesimal positive q. Since E may be any standard positive number, this shows thatf(a - q) - I is infinitesimal and proves that the condition is necessary. Conversely, suppose thatf(a - q) 2: I for all infinitesimal positive q. Then, for any standard positive E there exists a positive 6 (in fact any positive infinitesimal number) such that for any y, 0 < y < 6 implies ] f ( u- q) - I I < E. Disregarding the bracketed phrase “in fact any positive infinitesimal number”, the statement just made can be formulated in K and is true in R* and hence also in Ro.This shows that the condition is sufficient. A corresponding result holds for the limit from above Let f ( x ) be a standard real function of a real variable defined in a standard interval I, which may be open or closed or half-closed.

9.5.1

NON-STANDARD THEORY OF FUNCTIONS OF A REAL VARIABLE

253

9.5.2. THEOREM. The functionf(x) is continuous at the standard point

+

xo which belongs to the interior of Z if and only iff(xo q) 1:f(xo) for all infinitesimal q. This theorem follows immediately from 9.5.1. We conclude further

9.5.3. THEOREM. The standard functionf(x) is continuous in the standard

open interval Z if and only if f(xo and for all infinitesimal q.

+ q)

N

.f(xo) for all standard xo in Z

The notion of a standard open interval Z possesses a very simple nonstandard characterization as follows. 9.5.4. THEOREM. A standard interval I is open if and only if for any standard x E Z and any infinitesimal y, x y also belongs to Z.

+

PROOF.If Z is open and x is a standard number which belongs to Z then for some standard E > 0, x y E Z for all y such that I y I < E. Hence, in particular, x y E Z for all infinitesimaly showing that the condition is necessary. Conversely if the condition is satisfied, then for any infinitesimal E, the following statement holds in K*.

+

+

9.5.5.

(VY) [ [ I Y

I < E l A J(x) = J(x

+ v)]

*

In this statement, J ( x ) is the relation which defines the interval Z. J ( x ) belongs to the vocabulary of K, by assumption. From 9.5.5. follows 9.5.6- ( 3 4 [I [z > 01 A

[ ( A[ [ I Y I c 21 A JW = J(x

+Y ) ] ] ] ]

and this statement can be formalized within the vocabulary of K and hence, belongs to KOand holds in Ro. But this shows that Z is open, by definition and proves the theorem. Introducing the appropriate modifications for the notion of continuity at the endpoints of the given interval of definition, we obtain for an arbitrary interval, in place of 9.5.3. 9.5.6. THEOREM. The standard functionf(x) is continuous in the standard interval I if and only if f ( x o q) N f ( x 0 ) for all standard xo in Z and for all infinitesimal q such that xo q belongs to I. In order to pass from ordinary continuity to uniform continuity we have to strengthen the condition of 9.5.6., as follows.

+

+

9.5.7. THEOREM. The standard function f(x) is uniformly continuous in the standard interval Z if and only iff(x0 q) N f ( x 0 ) for all .YO in Z and for all infinitesimal q such that xo q belongs to Z.

+

+

254

r9.5.

SELECTED TOPICS

PROOF.Let J ( x ) be the relation which defines the interval Z, as before. Suppose that f ( x ) is uniformly continuous in I and let E be a standard positive number. Then the following statement holds in Ro for some standard 6 > 0. 9.5.8. J(x) A J(x

( W W Y ) [ I Y I < 61 A +Y) = [[ f ( x +Y) - f ( 4 I < 4

*

This statement must be true also for R*.It follows that for all x in 1 and for all infinitesimal 6, I f ( x y ) - f ( x ) I < E provided both x and x y belongs to I. Thus, f ( x y ) - f ( x ) is infinitesimal provided both x and x y belong to Z, showing that the condition is necessary. Conversely, if the condition of 9.5.7. is satisfied, then 9.5.8. holds in R* for arbitrary standard positive E and for arbitrary infinitesimal 6. It follows that the statement

+

+

+

+

( 3 4 [ [z > 01 A [ W X ) ( V Y ) [ [ I Y I c IA = [ If(x + Y ) - f W I < E ] ] ] ]

J(x)AJ(x +Y)

can be formulated within K and holds in R* and hence in Ro.This shows that f ( x ) is uniformly continuous in I. If we regard the conditions of 9.5.4., 9.5.6., 9.5.7., 9.4.24. as definitions and moreover, define a closed interval as an interval whose coniplement is a union of open intervals then the proof of the following well-known theorem is almost immediate. 9.5.9. THEOREM. A standard function f ( x ) which is continuous in a standard closed and bounded interval Z is uniformly continuous in that interval. PROOF.Suppose that f ( x ) is continuous but not uniformly continuous in I. Then there exists an a E Z and an infinitesimal 1 such that a 1 belongs to Z and such thatf(a v ) - f ( a ) is not infinitesimal. Now if Z is closed and bounded, as supposed in 9.5.9., then a is finite and possesses a standard part ao. a0 belongs to Z, by 9.5.4. since the complement of Z is a union of (two) open intervals and a, which is infinitely close to ao, belongs to I. It now follows from ordinary continuity thatf(u) 2: f ( a o ) and f ( a v ) 2: f ( a o ) and so f ( a ) 2: f ' ( a v). Thus, f ( a v ) - f ( a ) is infinitesimal, contrary to assumption. This proves the theorem. Next, we give a non-standard proof of Weierstrass' theorem on zeros of continuous functions.

+

+

+

+

+

9.5.1

255

NON-STANDARD THEORY OF FUNCTIONS OF A REAL VARIABLE

9.5.10. THEOREM. Let a and b be two standard numbers, a < by and let f ( x ) be a standard function which is defined and continuous in the closed interval a < x G b. Suppose that f ( a ) < 0 and f ( b ) > 0. Then there exists a standard number, c, such that f ( c ) = 0 where a < c < b.

PROOF.Let m be any standard positive integer and define

k m

k = 0 , 1,

ak=a+-(b-a),

..., m .

Then ao = a and am = byf (ao) < 0 and f ( a , ) > 0. It follows that for some standard number j , f ( a j ) < 0, f ( a j + l ) 2 0. Thus, the following statement holds in Ro. 9.5.11. For every positive integer m ythere exists a non-negative integer j , j < m such that

and

f(.

+i+l m (b - ) .

> 0.

But 9.5.11. can be formulated as a sentence of KOand so it holds also in R*. Let o be any infinite positive integer, then for some positive integer 1

(+c )

9-5-12. f a

- (b - a )

<0

The numbers

1 a - (b - u) and a 0

(+ 2 ' )

and f a

~

(b - a)

2 0.

1 + A+ -- (b - a) w

both have the same standard part which will be denoted by c. Thenf(c) is a standard number, and

( +: )

9.5.13. f u

-(b - a )

N

(+ T )

f(~)f , u

--

(b - U )

N

f(c)

.

Combining 9.5.12. with 9.5.13., we obtainf(c) < 0 on one hand, and f ( c ) 2 0 on the other hand. Hencef(c) = 0, a < c < b. But c cannot be equal to either a or b since f ( a ) < 0 andf(b) > 0 and so the theorem is proved.

256

[9.5.

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9.5.14. THEOREM. Let f ( x ) be a standard function which is continuous in the closed standard interval a < x Q b. Then there exists a standard point c, c2 < c < b, at which f ( x ) attains its maximum, f ( c ) 2 f(x) for all x in the interval.

PROOF.Introduce

.

m any standard positive integer, k = 0, 1, . .,m, as in the proof of 9.5.10. Then for some k = h, ~ ( u A2) f @ k ) for k = 0, 1, ...,m, since the number of points a k is finite. This fact can be formulated in KOand accordingly is true also in R*. Take m infinite, and let c be the standard part of an appropriate ah. We claim that c satisfies the conclusion of the theorem. Suppose on the contrary thatf(c) < f ( c ‘ ) for some standard c’. Since m is infinite, some a k is infinitely close to c’. Hence, by continuity, f ( c ’ ) II f ( U k ) and at the same timef(c) N f(an). It follows thatf(c) <
interval I it is necessary and sufficient that in I and for all infinite natural numbers w.

f 0 ( x 0 ) II f ( x 0 )

PROOF.The condition is necessary. Suppose that lim n+

00

for all xo

fn(x) = f ( x )

uniformly in I. Let E be any standard positive number. Then there exists a standard natural number v such that I fn ( x ) - f ( x ) 1 < E for all natural numbers n > v and for all x in I. This is a fact which can be formulated within K and holds in ROand hence also in R*. It follows that

9.5.1

NON-STANDARD THEORY OF FUNCTIONS OF A REAL VARIABLE

257

I f m ( x ) - f ( x ) I < E for all infinite natural numbers o and for all xo in I (and not only for all standard xo in I). Since this is true for all standard positive E, we conclude that fo (XO)2: f ( x o ) for all xo in I. The condition is also sufficient. Suppose that it is satisfied, let E be any standard positive number and let v be an infinite natural number. Then it is true in R* that for every natural number y which is greater than v and for every x in I, Ifu(x) - f ( x ) I < E. It follows that the statement

9.5.16. There exists a natural number z such that for every natural number y which is greater than z,

I f u ( 4 -f ( x ) I < E -can be formulated in K and holds in R* and hence, holds also in Ro. And since 9.5.16. holds for arbitrary standard positive E it follows that fn(x) converges tof(x) uniformly in I. This completes the proof of the theorem. Observe that if { fn ( x ) } is a standard sequence of continuous functions defined in a standard interval I, then for infinite o,f m ( x ) is not in general a standard function. fm ( x ) is still continuous according to the standard (Weierstrass) definition of continuity since the statement “For every natural number n, fn(x) is continuous (in the standard sense)’’ - can be formulated as a sentence of KO and hence, holds also for non-standard n. However, for such n,f n ( x ) need not satisfy the nonstandard condition of 9.5.2. For example, if f n ( x ) = (x2 n2)-2, then the condition of 9.5.2. is not satisfied, for infinite n, at the point xo = 0. We may use a combination of standard and non-standard arguments in order to prove the following standard result

+

9.5.17. THEOREM. Suppose that the standard sequence of continuous

functions { f n ( x ) } converges uniformly to the standard function f ( x ) in the interval I. Thenf(x) is continuous in I.

PROOF.Let xo be any standard point in I. In the formulation of our proof, we shall suppose, for simplicity, that xo is an interior point. Given any standard positive number E > 0, we have to prove that the statement 9-5-18. ( 3 4 [ [ z > 01 A [ (W [ [ I Y

1 < Z I = [ If (xo + A -f

(xo)

I<E]]]]

holds in Ro. Since 9.5.18. can be formulated within K, it will be sufficient to show that it holds in R*. Let o be any non-standard natural number. Sincefa ( x ) is continuous in I, in the standard sense, as explained above, the statement

258

19.5.

SELECTED TOPICS

+

9.5.19. ( 3 4 [ rz> 01A [ WY) [ [ I Y I < ZI = [ IfO(X0 Y ) -fw(xo) I < E l ] ] ] holds in R*, where the number z = 6 which exists according to 9.5.19. may be standard or non-standard. Butfw (xo) II f(x0) andf,(xo y ) II = f(xo y ) and so I fW(xo Y ) -fO(xo) 1 < E entails If(x0 y ) - f ( x o ) I < E. This shows that 9.5.18. holds in Rand proves the theorem. Letf(x) be a standard function which is defined in a standard interval I and let xo be a standard number which belongs to the interior of I. Then the following theorem follows immediately from the non-standard condition for a limit, 9.5.1.

+

+

+

+

The standard number a is the derivative of f ( x ) at 9.5.20. THEOREM. the point x = xo if and only if, for all infinitesimal q, # 0, 9.5.21.

?l

Next, we prove Rolle’s theorem. Let the standard functionf(x) be defined and differ9.5.22. THEOREM. entiable in the standard closed interval a < x < b, and suppose that f ( a ) = f ( b ) = 0. Then there exists a point c in the interior of the interval such that f’(c) = 0. PROOF.If f(x) = 0 throughout the interval then for any standard interior point c and for any infinitesimal q, # 0, rl

and s o f ’ ( c ) = 0 by 9.5.21. Suppose thatf(x) # 0 at some point of the interval. More precisely, we may suppose without any essential limitation of the argument that f(x) > 0 at some point of the interval. Then, by 9.5.14. f(x) attains its maximum at some interior point c of the interval. Let q be positive infinitesimal then

+

f ( c V ) G fk),f ( c - r) G f ( c > Letf’(c) = a then, by 9.5.21.

and --TI

9.5.1

NON-STANDARD THEORY OF FUNCTIONS OF A REAL VARIABLE

259

Thus, a is infinitely close to both positive and negative numbers, and since a is standard we conclude that a = 0. This proves the theorem. The mean value theorem may be deduced from 9.5.22. in the usual way. However, the above examples will suffice to prove that the nonstandard Differential calculus can compete in simplicity against the more orthodox approach to the subject. Next, we consider definite integration, to begin with for continuous functions. For convenience only, we shall confine ourselves to the partition of the interval of integration into subintervals of equal length. Let f ( x ) be standard and continuous in the closed interval a < x d b. Let

k -(b-a), m

ak=af

0

< k < rn ,

where rn is a standard positive integer. Then m

Sm =

C

b-a

b-a

= --

-m-f(ak)

C f(ak)

rn k=l h is, for large m, an approximation to the value of the integral J f ( x ) d x. a However, sm is defined also for infinite m and we have 9.5.23. THEOREM.

k=l

b

sm

N

J f ( x ) dx a

for all infinite positive integers m. This follows immediately from 9.4.7. and from the fact that, on the assumptions of the theorem, b

lim Sn = f ( x ) d x

n+ co

a

.

When we come to deal with Riemann integration, it seems an artificial limitation to consider partition of the interval of integration into subintervals of equal length only. Accordingly, we set up a formalism which permits us to include more general partitions. Let I be a fixed standard closed interval, I = [a, b]. Let p be any 1 standard real numbers ao, al, . . ., U k , such that sequence of k

+

a = ao < a1 < a2 <

...
=b

where k may be any positive integer p will be called a partition of I, and the set of all partitions of the interval will be denoted by P a b . It is easy to to verify that P a b has the cardinal of the continuum, 1 Pab I = 2'0. We select an arbitrary but fixed one-to-one correspondence between P a b and

260

[9.5.

SELECTED TOPICS

Royand we denote by r p the real number which corresponds to the partition p E P a b in this correspondence, p t-)rp. Then the following relations belong to the vocabulary of K. F (x,y), a relation which holds if and only if x is a real number, and y is the length of the partition which corresponds to x (i.e. the positive integer k); G(x,y), a relation which holds if and only if x is a real number and y is the maximum length of a sub-interval, (az,ai+l) that occurs in the partition corresponding to x. Thus, y is a positive real number; N(x,y, z), a relation which holds if and only if x is a real number, and y is a natural number, and z is the yth element of the partition which corresponds to x (beginning with the 0th element), z = a3/. The relations F(x, y ) and G (x,y) determine functions y = 4 (x) and y = y (x) which are defined for all real numbers. Similarly, N(x, y , z) determines a function z = v (x, y ) which is defined for all real numbers x and for all natural numbers y such that y < 4 (x). Let y =f(x) be a fixed standard function which is defined and bounded on I. For any two numbers XI and x2 in I, XI < x2, we define two functions z = m (XI,x2) and z = p(x1, x2) as the least upper bound and greatest lower bound of the values off(x) in the closed interval [XI,x2] respectively and for any number r in Ro, we put k

9.5.24.

J(r) =

and

C

t=1

(as - at-1) m

(at-1,

ad

k

9.5.25.

j(r) =

1 (at - ar-1) p h - i , a d t=1

where p = (UO, a, .. .,ak) is the partition which corresponds to the real number r. The relations and functions introduced above exist in R* as well as in Ro.We shall call the non-standard real number r admissible (in the present context only) if the maximum length of a sub-interval in the partition p that corresponds to r, i.e. y (r), is infinitesimal. We note that in this case 4 (Ithe ), number of intervals in the partition, is by necessity infinite. Let

a

f ( x ) dx and

b

Ja

-

f(x) dx be the upper and lower Darboux

integrals off(x) over the interval I, respectively. We prove by a technique with which the reader is by now familiar,

9.6.1

NON-STANDARD ANALYSIS OF FUNCTIONS OF SEVERAL VARIABLES

9.5.26. THEOREM. J(r)

admissible r in R*.

1(

fb a

261

b

f(x) dx and j ( r ) 21 fa f(x) dx, for every

Hence,

b

9.5.27. THEOREM. The Riemann integral f f ( x ) dx of the function f (x), a

which is standard and bounded in the interval [a, b], exists if and only if J ( r ) N j ( r ) for any (and hence, every) admissible r. It will be seen that the choice of the real numbers r which label the partitions p was to a very large extent arbitrary. Moreover, we may modify our procedure, e.g. by considering all partitions of finite intervals simultaneously, or by considering the integrals of a family of functions simultaneously. In view of this flexibilityone might suspect that on passing to non-standard r, the quantities J ( r ) and j ( r ) , which were defined originally as sums of products, lose most of the properties which one would tend to associate with such expressions intuitively. Fortunately, this is not the case. Indeed, it is still true that many of the important properties of these expressions can be formulated within KO.We may use this fact, similarly as before, in order to provide alternative proofs for the standard theorems of Riemann integration. To give only simple example, suppose that the standard functions f(x) and g ( x ) are bounded and Riemann integrable in the interval [a, b]. Let Jf(r),if@), Jg(r), jIl(r), j f + I(r), l jf+Il (r) be the functions defined by 9.5.24. and 9.5.25. for f ( x ) , g ( x ) and f ( x ) g(x) respectively. In any sub-interval of [a, b], min ming(x) Q min ( f ( x ) g(x)) Q max (f(x) g(x)) < max f(x) f ( x > max g(x). Reference to 9.5.24. and 9.5.25. now shows immediately that for all standard r, h ( r ) i I l (r) Q jf +p. (4 < Jf+B (r) G 4 (4 JIl (r)

+

+

+

+

+

+

But j f (r)

cy

+

Jf (r) and jIl(r)

1 :JB(r)

ir(r) +j f 7 (r) =! j f + 8 (4

This shows thatf(x) b

J a

(f(x>

-

by assumption, and so

= Jf+B

(4 2! J f (4 + JIl 0-1 -

+ g(x) is Riemann integrable and

+ g ( 4 d x =aSf<.>d x + Ja g ( 4 d x b

b

*

9.6. Non-standard Analysis of Functions of Several Variables. Let R: be

the n-dimensional Euclidean space over the field of real numbers Ro. The

262

[9.6.

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distance between two points P = (XI, . . .,X n ) and P' = (xi, . . ., x i ) of Ro is defined by PP' = 1/((x1(XZ - x;)~ ... (xn- x 2 ) . Let R*n be the corresponding space over R*, with the same definition for distance, R*n 2 R:. A point P of R*" is standard if all its coordinates are standard, i.e. if it belongs to R:. Let S be a set of points in 3.: S is given by a relation Q(x1, ..., X n ) in Kin the sense that any point (al, ...,a,) of R: belongs to S if and only if Q (al, . . .,an) holds in Ro. The same relation determines a set S* in R*n. Such sets will be called standard while all other subsets of R*n are non-standard in agreement with our previous terminology. The injnitesimal neighborhood of a point P in R*n is defined as the set of all points P' such that PP' is infinitesimal.

+

+ +

9.6.1. THEOREM. A standard subset S of n-dimensional Euclidean space is open if and only if for any standard point P in S,the infinitesimalneighborhood of P belongs to s. PROOF.Suppose that S is open and let P be a standard point of S. Then there exists a standard positive number 6 such that all points P' for which PP' < 6, belong to S. It follows, in particular that all points in the infinitesimal neighborhood of P belong to S. Conversely, suppose that the infinitesimal neighborhood of a standard point P of S belongs to S. Then for any infinitesimal positive 6, the set of points P' such that PP' < 6, belongs entirely to S. Accordingly, the statement

9.6.2. There exists a positive z such that any point P' for which PP' < 6 belongs to S holds in R. But 9.6.2. can be formulated as a sentence of K and so it holds also in Ro. This shows that P is an interior point of S in the standard sense. And since P is an arbitrary standard point of S it follows that S is open. 9.6.3. THEOREM. If the standard S is closed and bounded in the standard sense, and if the point P belongs to S, then the point st (P)whose coordinates are the standard parts of the coordinates of P,also belongs to S. PROOF.Since S is bounded, the coordinates of P are all finite and so st(P) exists. Let S' be the complement of S in R*n. If S' contains st (P) then it must contain also P,which belongs to the infinitesimal neighborhood of st (P).This is contrary to assumption and proves the theorem.

9.6.1

NON-STANDARD ANALYSIS OF FUNCTIONS OF SEVERAL VARIABLES

In the sequel we shall, sometimes, writef(P) instead off(a1, where P = (a,...,an).

263

...,Gn),

.. ., xn) be a standard function which is defined in the standard set S. In order that f(x1, ...,xn) be continuous in S, it is necessary and sufficient that for any two points P and P' in S such that P is standard and the distance between P and P' is infinitesimal, f(P) N f(P'). In order thatf(x1, . . .,xn) be uniformly continuous in S, it is necessary and sufficient that f (P)= f ( P ' ) for all points P and P' in S such that the distance PP' is infinitesimal. 9.6.4. is a straightforward generalization of 9.5.6. and 9.5.7. and the reader will have no difficulty in proving this theorem. We shall also dispense with the proof of the following theorem which generalizes 9.5.15. 9.6.4. THEOREM. Let f(x1,

9.6.5. THEOREM. In order that the standard sequence of functions { fn (XI, ..,X n ) } converge uniformly in S to the standard function

.

f (x) it is necessary and sufficient thatf, (P)N f (P) for all infinite positive integers o and for all P in S.

By way of an example, we shall give a non-standard proof of Dini's

theorem for the case of functions of several variables.

. .., xn)} be a standard sequence of continuous functions defined on a standard closed and bounded set S, and converging to a standard functionf (XI, . . .,xn) on that set. Suppose that ( f k ( x 1 , .. .,x n ) } is a non-increasing sequence at all points of S, f k ( P ) 2 f m ( P ) for k 2 m. Then the convergence of { f n } tofis uniform in S. 9.6.6. THEOREM. Let {fk(xi,

PROOF.We may suppose f(x1, . . ., x,) = 0 throughout S, if necessary after the preparatory step of considering { f n - f } in place of { f n } . Let P be any point in S, standard or non-standard. By 9.6.5. we have to show that f, (P) is infinitesimal for all infinite positive integers w. At any rate,f,(P) 2 0, sincefn 2 0 for all standard n, and f o ( P ) < < (P) sincefn is non-increasing. Suppose now thatfm(P) is not infinitesimal. Then f,(P) > E for some standard positive E and so f n (P) > E for all finite E. Let PO = st(P), then PO belongs to S by 9.6.3. Also, POP is infinitesimal, and sof n (PO)N f n ( P )for all standard n. Hence,f n (PO) >E and this contradicts the assumption that lim fn(P0) = 0. The proof of n-m 9.6.6. is now complete.

264

[9.6.

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We observe that, unlike the classical proof of Dini's theorem, the above argument does not involve the theorem of Bolzano-Weierstrass. Nothing new is involved in the definition of the partial derivatives. In order to discuss differential notation we introduce the following symbols. For any finite or infinite set of elements of R*,{qy}, we denote by 0 {qy) the set of all finite sums avqv,where the an are finite, and we denote by o {+} the set of all finite sums bvvvwhere the b,. are infinitesimal. If the set {qy} is finite, {qv} = { q ~ ,. . .,qk}, we write also O(q1, . . ., qk) and o (111, . . ., qk) in place of 0 {qy) and o {qv} respectively. Then MO = 0 (l), MI = o (1). The sets 0 {qy} and o {qv} are modules with MO as domain of operators. Accordingly, we shall employ the notation a = bmod 0 { q y } i f a - b is in 0 {qv} and a = b mod o {qv>i f a - b is in o {qv}.

9.6.7. THEOREM. Let PO = ( X O I , . . ., ~ 0 % )be a standard point in ndimensional Euclidean space. Let f(x1, ...,xm) be a standard function which is defined and possesses continuous first derivatives at all points P such that POP< r, where r is a standard positive number given in advance. Let (dxl, . . ., dx,) be an arbitrary set of infinitesimal numbers, and let d f = f ( x o l + dx1, -.., Xon dxn) - f ( ~ 0 1 ., .., XOn). Then

+

9.6.8. d f r

(5) dxl + ... + axl

dxnmodo(dxl, ..., dxn)

where

denotes the partial derivative

taken at the point PO.

+

PROOF.Put g ( t ) = f ( x o i tdxi, . . ., X0n meter. Then df = g(1) - g(0). Also, for 0

+ tdxn) where t is a para-

< t < 1,

9.6.1

NON-STANDARD ANALYSIS OF FUNCTIONS OF SEVERAL VARIABLES

265

where the

are taken for the particular value of t mentioned on the left hand side. Hence, by the mean value theorem.

..

where 0 < 0 < 1. All these equations would certainly be true if dxl, . , dxn were finite, and since they can be expressed within KO,they hold also for infinitesimal dxl, ,dxn. Now

...

):( -(g)o t =B

=br

is infinitesimal, i = 1, . . ., n, since the first derivatives are continuous in the region under consideration. Hence

= bl dX1,

+ . . . + bn dxn

and bidxi . . . bndxn belongs to Mi.This proves 9.6.7. Although 9.6.8. does not yield ordinary equality between the two sides of the congruence, it can be applied without difficulty in a number of cases, e.g. in order to calculate the derivative of a function which is defined implicitly. With corresponding modifications, we may also justify in this manner the use of infinitesimals as they occur in classical Differental Geometry and in Analytical Mechanics. Even though we considered in the first instance only standard functions, we were led in a natural way to consider also functions of a more general type. For example, if ( f i ( x ) } is a standard sequence then f o ( x ) , o infinite, which arises naturally in the discussion of the sequence, is not in general a standard function. To present such functions in a more

266

SELECTED TOPICS

[9.6.

comprehensive setting, let G (x, t) be a standard function of two variables, and let T be any element of R *. Then g(x) = G(x, 7 )

9.6.9.

is said to be a quasi-standardfunction of x. This definition can be extended to the case of several variables XI, ...,X n and several parameters, f1,

. . ., f k .

A standard function is also quasi-standard. The function fo (x) mentioned above is quasi-standard. All functionals or operators which apply to the functions gt(x) = G(x, t) for all standard t belonging to a given standard set T can be extended in a natural and unambiguous way to the function g(x) given by 9.6.9. for some non-standard z which belongs to T. Instead of demonstrating this fact for the general case, we consider a particular example which will be useful later on. Suppose that the functions gt(x) are integrable in some definite sense, e.g. Riemann integrable, in the standard interval [a, b], for all standard t in T,

9.6.10.

b

J a

b

gt(x) dx = [ G(x, t) dx = J ( z ) . a

b

We then define the integral J g (x) dx for non-standard z simply by a

9.6.11.

b

J g(x) dx = J(t). a

This definition preserves the properties of an integral to the extent to which they are definable in the Lower predicate calculus. For example, if g ( x ) is infinitesimal in the closed interval [a, b] then it attains its maximum and its minimum in the interval, which must therefore both be infinitesimal and since

(b - a)minh(x)

b

<J

a

h(x)dx

< (b - a)maxh(x)

for all standard h (x), the same holds for the quasi-standard function g(x). It follows that if g(x) is infinitesimal in the entire interval [a, b], b

then J f(x) dx is infinitesimal. a

The above definition presents itself in a more concrete form if we take as our non-standard model of analysis one of the ultrapowers R o ~ introduced at the beginning of section 9.4. We recall that in this instance,

9.6.1

NON-STANDARD ANALYSIS OF FUNCTIONS OF SEVERAL VARIABLES

261

I denotes the natural numbers and A denotes an ultrafilter in the set of subsets of I, The elements of R o are ~ the functionsf(v) from the standard natural numbers into the standard real numbers, although distinct functionsf(v) may be equal according to the definition of the relation of equality in Roi. Among the elements of Ro,’, the functions which are constant, or which are constant on a subset of I that belongs to A constitute (a field isomorphic to) Ro.The number z = f ( v ) belongs to the standard set T, if the set of natural numbers v for whichf(v) belongs to T is an element of A . Suppose now that the standard function F ( t ) is defined for the elements of T, then for any 7 = f ( v ) which belongs to T there is an element 1of A such that F(z) as a function of Y is defined by F ( f ( v ) ) for all v which belong to 1 (and arbitrarily elsewhere). In particular, if F ( t ) coincides with the integral J ( t ) as given by 9.6.10., then for all v in a set Iz which belongs to A , F ( z ) as a function of v takes b

the value f G (x, f (u )) dx. a

Quasi-standard functions can be used to represent generalized functions such as Dirac’s delta function, but we shall not follow up this possibility here. Instead, we give as our last example a non-standard proof of the existence of a solution for a differential equation of the first order with a given initial value. As it stands, the theorem is formulated for the field of real numbers, Ro. 9.6.12 THEOREM. Suppose that the functionf(x, y ) is continuous in the closed rectangle I x - xo I 6 a, y - yo 1 < b, such that f(x,y) I < M

I

+

I

throughout the rectangle. Then there exists a function (x) with a continuous first derivative in the closed interval [XO - c, xo c], c = min ((I, bM-I), such that (XO) = yo and

+

9.6.13.

for xo - c Q x Q xo

+ c.

+

+

9’( 4 =f(x7 (4)

PROOF.We make the preliminary observation that it is sufficient to prove that there exists a function 4 (x) which is continuous for all x such that xo 6 x 6 xo c and which satisfies the equation

+

9.6.14.

+ (4

+ J f ( t , 9(0)dt X

= Yo

5 0

For in that case, 9.6.13. follows from 9.6.14. by differentiation for the interval [XO,xo c] and a similar consideration for [XO - c, XO]com-

+

268

19.6.

SELECTED rnPICs

+

pletes the argument for the entire interval [XO - c, xo c]. Moreover, for x = XO, 9.6.14. yields (XO) = yo, as required. In order to obtain a function (x) which satisfies the modified condition, divide the interval XO, xo c into m sub-intervalsof equal length by means of the points X k = xo kc/m, k = 0, 1, . ..,m, m any positive integer (in Ro). Define the function +m (x) by

+

+ +

+m(xo) = yo

9.6.15.

+m

(4 = +m (Xd

+f(x4,

+m (XI))

(x

- xz)

for xt < x < x~+I,i = 0, 1, .. ., m - 1. One shows without difficulty that 9.6.15. yields a continuous function in [XO, xo c], such that is constant in the sub-intervals (i.e. the graph of + m ( x ) is a polygonal line). Moreover for all x and x’ in the interval under consideration

+

+,

1 +m (x) - + m (x‘) I < M I x - x’ 1 and in particular I (x) -yo1 < M I x - x ~ I a n d h e n c e , I + , ( x ) I < M c + I y o I < y o + b . Also, given any E > 0 there exists a positive integer p such that for all m>P 9.6.16.

9.6.17.

I +&I

+

-f(x,

+,<XI)

I<

+A(.)

at all points of [xo, xo c] at which the derivative exists. All the properties of +A(x) are obtained by some simple finite algebra involving a number of equations and inequalities. Moreover, from 9.6.17. we obtain by integration

1 +m ( x ) - (yo + J f ( t ,+m (t)) df)1 < E (X - XO) < E c xo < x < c , m > p . 2

9-6-18.

for

20

Up to this point we have followed standard procedure (in both senses of the word standard). We now break away from it by setting m equal to an infinite positive integer o.Then 9.6.18. holds for all standard positive E and so 9.6.19.

+J f(4 X

+m(4

2:

yo

20

+u@)

dt

-

Observe that dm(x) is a quasi-standard function, and so is f (x, +w (x)), and the right hand side of 9.6.19. has a meaning in the sense indicated above. Also, since the inequalities 9.6.16. are satisfied for all finite m, they hold also for m = o.

9.6.1

NON-STANDARD ANALYSIS OF FUNCTIONS OF SEVERAL VARIABLES

269

The function &(x) is continuous since the functions q5m(x) are continuous for finite m.Beyond that, the inequality I +m (x) - cjW(x’) I < < MI x - x‘ I shows that &(x) 21 +m(x’) provided x N x’ for any x and x’ in the interval [XO, xo c]. The function &(x) is bounded by a standard number in the interval [XO, xo c ] , by 9.6.16. Accordingly, we may define a standard function in that interval by putting + ( x ) = st ( &( x)) for all standard x in [XO, xo c]. Notice that although st(+@ (x)) exists also for non-standard x it is not usually equal to (x) at such points. By the definition of +(x), (x) H + W ( x ) for all standard x in the interval. Hence, by the first inequality of 9.6.16.

+

+ +

9.6.20. [XO, xo

+

+

I +(x) - +(x’) I < M I x - x’ 1 + q for + c] where q is some infinitesimal number.

I+

+

I

standard x, x’ in

I

Since both ( x ) - (x’) I and M x - x’ are standard numbers, the q on the right hand side of 9.6.20. must be equal to 0. This shows that + ( x ) is continuous. Next, we claim that r#m (x) N ( x ) for all x in the interval [XO, xo c]. This is true for all standard x, by the definition of C#J (x). For any other x, let x‘ = st (x). Then (x‘) II 4 (x’) as stated just now, and & (x‘) 21 2: & (x) as mentioned above. Also, cj (x’) II 4 (x), by 9.5.6. since 4 (x) is continuous. Combining these three relations we prove our assertion I+ ( x ) for all x in the interval. Moreover f ( x , +m(x) 2: that I$~(X) I 2: f (x, (x)) for all x in [XO, xo c] since f ( x , y ) is continuous in the closed rectangle x - xo I < a, I y - yo I < b, hence uniformly contin(x)) and (x, +(x)) belong to that rectangle, uous, and the points (x, and the distance between them is infinitesimal. It follows that the integral

+

+

+,

+

7

+

I

[ f ( t ,+ w ( t ) ) - f ( t , + ( t ) ) ]dx is infinitesimal for all standard x in

20

+

[XO, xo c] (and also for all non-standard x in the interval, but we shall not require this fact). Thus, bm(x) 2: +(x) for all standard x in the X

interval, and at the same time f f (t, & ( t ) ) dt 20

ing these relations with 9.6.19., we obtain 9.6.21.

2

1:

+

f f (t, (t) dt. Combin20

= Yo + J f ( t , +@I)df. 2

+(XI

20

But the numbers on both sides of 9.6.21. are standard and so we may replace N by the sign of equality. This yields 9.6.14. and proves the theorem.

270

[9.7.

SELECTED TOPICS

Theorem 9.6.12. with the requirement of continuity alone forf(x, y ) is due to Peano. It is a matter of historical interest that in one of this papers on this subject [Peano 18901 Peano employed an early version of Symbolic logic. He justified the use of his notation by the remark that the complete development of the argument in terms of an ordinary language would be excessively complicated. In our argument, we replaced the application of compactness principle, usually the theorem of Ascoli-Arzelh, by certain non-standard considerations. We recall that a similar change took place in the case of Dini's theorem, 9.6.6. This is not to say that our methods eliminate such principles and incidentally the use of the axiom of choice altogether. However, the need for tools of this nature now arises at a different point, in connection with the construction of a non-standard model of Analysis. 9.7. Problems 9.7.1. Let

X

= (Vxl)

- - - (Vxn)( 3 ~ 1-)- - ( 3 ~ k )Q ( X I ,. -,xn, Y I , - - -

9

~ k )

be a sentence formulated in terms of E, S,and P (equality, addition, and multiplication) and without individuals, where Q may contain further quantifiers. Suppose that X holds in the field of complex numbers, F. Prove that there exists a partition of the Cartesian space Fn into a finite number of subsets, D1, . . .,D , and algebraic functions j i 5 (xi, . . ., xn), i = 1, . . .,m,j = 1, . . .,k such that Q (ai, . ..,an,jil ( ~ 1.,. .,an), . .., fik(a, . . . , a ) ) holds in F for all (al, . . . , a n ) in Fn, and for suitable determinations of the values of the &, ( a , . . .,an) (This is a special case of a theorem proved in Lightstone - Robinson 1957a). 9.7.2. Prove that if a structure Mi is elementarily equivalent to a structure M Z(with respect to a given vocabulary) then M I can be embedded in an ultrapower of M Z(T. Frayne). 9.7.3. Show that a non-standard model of analysis cannot be compact. How is it that we were neverthelessable to prove the theorem of Bolzano Weierstrass 9.4.15. by means of non-standard Analysis? 9.7.4. Let (so, s1, $2,

2 be the set of all bounded sequences of complex numbers

.. .) regarded as a linear space. Prove the standard theorem

that there exists a complex linear functional which is defined on all of and which coincides with the limit of (sn) for all convergent

2

9.7.1

PROBLEMS

271

sequences (s,~}. (Hint. Such a functional is provided by the standard part of so where o is a fixed infinite natural number. The example is due to Dana Scott). 9.7.5. Prove the theorem of Ascoli-Arzeli on equicontinuous sequences

of functions.

References. Theorem 9.1.5. is given in Hilbert-Bernays 1934-1939 as a supplement to one of the &-theorems.The theory of section 9.2. is taken from A. Robinson 1958. Tarski’s elimination procedure for real closed field is in Tarski-McKinsey 1948/1951, compare also Seidenberg 1951. Seidenberg’s work on resultants is given in Seidenberg 1956. Skolem’s construction is in Skolem 1934, compare Skolem 1955. The formulation of the ultraproduct notion was given by Lo5 in EoS 1955. More recent work in this field is due to Frayne-Scott-Tarski 1958, Chang-Morel 1958, and in particular to S. B. Kochen 1961 and J. B. Keisler 1961. The approach used here is taken from A. Robinson 1961a. For the theory of rings of continuous functions see Hewitt 1948, Erdos-Gillman-Henriksen 1955, and Gillman-Jerison 1960. For papers on non-standard models of Arithmetic, in addition to Skolem 1934, 1955, see Henkin 1949, GilmoreRobinson 1955, Kemeny 1958, Mendelson 1961, MacDowell-Specker 1961, Scott 1961, A. Robinson 1961, Rieger 1961, Rabin 1961. Possibilities and results in non-standard analysis are described in Robinson 1961b, which contains references to further papers on the subject of a calculus of infinitely small or infioitely large quantifiers. In this connection, the work of Schmieden and Laugwitz, see Schmieden-Laugwitz 1958,Laugwitz1961, 1961a, deserves special mention.

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INDEX OF AUTHORS Ackermann, W., 272 Artin, E., 2, 214, 220, 223, 225, 272 Asser, G., 272 Bar-Hillel, Y., 272 Bernays, P., 22, 87,230,271,272 Beth, E. W., 2, 117, 118, 137, 272 Bing, K., 272 Birkhoff, G., 171, 172 Bruijn, N. G. de, 50, 272 Cantor, G., 90 Carnap, R., 272 Chang, C. C., 87,271,272 Church, A., 272 Craig, W., 116, 118, 137,272 Curry, H. B., 273 Davis, M., 273 Ehrenfeucht, A,, 273 Eichler, M., 50,273 Engeler, E., 50, 273 Erdos, P., 50,271,273 Fefermann, S., 273 Fraenkel, A. A., 273 FraIssC, R., 273 Frame, T. E., 270,271,273 Frohlich, A., 273 Gillman, L., 271,273 Gilmore, P. C., 271,273 Godel, K., 12, 22, 273 Halmos, P. R., 274 Henkin, L., 22, 50,225,271, 274 Henriksen, M., 271,274 Herbrand, J., 51, 52, 87, 274 Hewitt, E., 271,274 Heyting, A., 274 Hilbert, D., 2, 22, 52, 87, 125, 127, 214, 225, 271, 274 Horn, A., 274 Jerison, M., 271, 274 Kaplansky, I., 137, 274 Keisler, H. J., 87, 271, 275 Keineny, J. G., 271, 275

Kleene, S. C., 275 Kochen, S., 271,275 Kolchin, E. R., 214, 275 Kreisel, G., 138, 225, 275 Lang, S., 223, 225,275 Langford, C. H., 90, 111,275 Laugwitz, D., 271, 275 Lefschetz, S., 90 Leibniz, G. W., 247 LCvy, A., 275 Lindenbaum, A., 88, 111 Lightstone, A. H., 270,275 Lorenzen, P., 275 Lo& J., 50, 78, 87, 271, 276 Lowenheim, L., 20,22,255,276 Lyndon, R. C., 87,276 MacDowell, R., 271, 276 McKinsey, J. C.C., 111,271,276 Malcev, A. I., 22, 45, 46, 50, 171, 276 Mendelson, E., 271,276 Montague, R., 276 Morel, A., 271, 276 Mostowski, A,, 87, 276 Miiller, G. H., 277 Neumann, B. H.,50,277 Oberschelp, A., 277 Orey, S., 277 Ostrowski, A., 50,277 Peano, G., 207, 277 Presburger, M., 277 Rabin, M. B., 63, 87,271,277 Rado, R., 22,277 Rasiowa, H., 22, 277 Raudenbush, H. D., 224, 277 Rieger, L., 271, 277 Ritt, J. F., 137, 210, 211, 212, 214, 224, 225, 277 Robinson, A., 22, 50, 87, 101, 111, 137, 138, 171, 191,225,270,271, 278 Robinson, J., 278 Robinson, R. M., 87, 278 Rosser, J. B., 279 Ryll-Nardzewski, C., 279

282

INDEX OF AUTHORS

Schmieden, C., 271 Schreier, O., 214,225 Schroter, K., 279 Scott, D., 50, 271, 279 Seidenberg, A., 133, 137,238,271, 279 Shepherdson, J. C., 279 Shoenfield, J. R., 279 Sierpinski, D., 64, 87, 279 Sikorski, R., 22, 279 Skolem, T., 20, 22, 51, 52, 239, 271, 273, 279 Specker, E., 271,280

Steinitz, E., 90, 11 1, 280 Suszko, R., 78, 87, 280 Tarski, A., 21, 22, 50, 55, 87, 105, 111, 173, 191, 237, 271,280 Vaught, R. L., 55, 87,89, 111,280 Wang, Hao, 280 Zakon, E., 111, 280

SUBJECT INDEX adjunction, 83

admissible (antomorphism), 186 algebraic (predicate), 154, 161 associated (predicates), 152 axiom, 25 Beth’s theorem, 117 block (of quantifiers), 65 bounded (predicate), 149 co-extensive (relations), 23 compactness theorem, 21 complete (set of sentences), 88 completeness theorems, 12 conjunction, infinite, 10 conjunctive, 68 connectives, 3 consistency lemma, 114 consistent, 8 contraction, 33 contradictory, 8 convex, 80, 161 convex (strongly), 83 core model, 83 correspondence, 9 Craig’s lemma, 116 definite (models, sets of sentences), 165 diagram, 24 diagram, negative, 24 diagram, positive, % direct product, 238 disjunction, infinite, 110 disjunctive, 68, 179 divisor, 152, I73 dummy symbol, 3

ideal, differential, 209 ideal, indecomposable, 173 ideal, irreducible, 173 ideal, order, 215 ideal, perfect, 21 1 increasing chain, 35 individual, 9 infinitesimal, 246 invariant, 67 irreducible (predicate), 153 isomorphism, 25 localization, principle of, 21 matrix, 65 model, 10 model-complete, 91, 105 model-completion, 109, 128 model-consistent, 84, 105 modus ponens, 7 monk (polynomial), 30 non-standard analysis, 244 non-standard model of analysis, 245 non-standard model of arithmetic, 243 normal, 24 normal chain, 47 object symbol, 3 obstruction, 75

Herbrand functor, 51 homomorphism, 25

partial structure, 24 perfect, 167 persistent (under extension), 66, 67 persistent under restriction, 66, 67 point, 201 point, compound, 201 point, generic, 203 polynomial (metamathematical), 141 polynomial structure, 142 predicate, 3 prefix, 65 prenex normal form, 8 prepolynomial, 141 prepolynomial structure, 142 pretransitive, 142 prime model, 94

ideal (mefamathematical), 173 ideal, arithmetical, 205

quantifier, 3 quantifier, free, 69

elementary extension, 55 elementarily equivalent, 55 elimination of quantifiers, 231 equality, 23 existential (sentence), 72 free (variable), 4

284

SUBJECT INDEX

quasi-elementary, 47 quasi-standard function, 266 reducible (predicate), 153 related (sets of sentences), 21 relative symbol, 3 restriction, 34 rules of deduction, 6 rules of formation, 3 saturable (predicate), 150 semantic interpretation, 9 sentence, 3 separability (metamathematical), 167 simple extension, 83 Skolem function, 51 Skolem functor, 51, 226 standard, 247 standard part, 250 structure, 9 structure, generic, 194

substitutivity, 23 substructure, 24 syntactical transform, 52 o-persistent, 78 T-system (Tarski system), 21 theorem, 6 transfer principle, 35 transitive, 142 ultrafilter, 240 ultraproduct, 241 universal (sentence), 69 valuation lemma, 13, 239 variety, 21, 192, 202 variety, irreducible, 194 variety, reducible, 194 vocabulary, 31 well-formed formula (d) 4,

Introduction to Model Theory and to the Metamathematics of Algebra (Studies in Logic and the Foundations of Mathematics, 32)