- Email: [email protected]
Appendix I The Axiomatic Method
APPENDIX I
THE AXIOMATIC METHOD
The general nature of this method is usually described as follows. Instead of assertions about abstract properties of speczjic objects and concepts (such as space, material point, probability, etc.), one considers statements of the following form: given any collection of objects (whose nature is not otherwise specified) and given any set of relations between these objects, if the relations satisfy certain logical conditions (called axioms) then they also satisfy certain other logical conditions (called theorems of the given axiomatic theory). In different branches of ordinary mathematics a small number of particular axiomatic systems have been isolated and studied. Thereby a good deal of mathematics has been built up in a systematic and comprehensible way. But one has not been interested in arbitrary axiomatic systems or even in general classes of axiomatic systems. Thus the experience of “ordinary” mathematics provides no reason for supposing that there are useful results about general classes of axiomatic systems which would contribute to the effective use of the axiomatic method. We shall now give some applications of a study of general classes of axiomatic systems, mainly - though not exclusively - of axioms expressed in first order predicate logic, a notion which is defined precisely in Chapters 1 and 2. Broadly speaking, this language can be characterized by saying that its formulas express properties of relations defined on a domain E and that in the definitions of these properties the quantifiers range only over the elements of E and not, say, over the subsets of E. For example, the fact that a relation is an order reIation can be expressed by a first order formula but not the fact that it is a well-ordering. Or again, the fact that a structure is a group (that is to say, that the relation a * b = c satisfies the group axioms) can be expressed by a first order formula. Similarly the property of being a commutative group is of first order. However the fact that a group can be ordered is not expressed by a first order formula since this is the property that “there is an ordering of E
155
THE AXIOMATIC METHOD
compatible with the group structure” or, in other words, “there is a subset of E 2 such that ...”. Nevertheless the property of being an orderable group is equivalent to a certain infinite set of first order conditions. Finally, the properties of being a group having a finite number of generators and of being a countable group are not equivalent to any set, even infinite, of first order conditions. Thus because of the exclusion of higher order quantifiers the class of axiomatic systems for which these general results hold does not include all of mathematics. One can make up for this, at least partially, by use of infinite systems of axioms. By considering structures which satisfy an infinite set of conditions a whole class of problems can be covered which are formulated in higher order terms but which can be reduced to problems about infinite sets of first order conditions. Examples of this are given in Chapters 1-3, mainly in Exercises. The most useful results, all connected with one another, are these: 1. The Finiteness Theorem. This says that if a first order formula A holds in all those structures which satisfy a set d of first order formulas, then there is some finite subset d l of &’ which implies A . 2. The Method of Constants (Chapter 3, Exercise 2). This generalizes the well-known algebraic principle for introduc~ngtranscendental elements (where a structure containing an element 5 satisfying p,(t)#O for all n is derived from a structure which contains, for each n, a &, satisfying p,(t,)#O for all i,
+
156
THE AXIOMATIC METHOD
conditions cannot be replaced by any finite set. We remark in passing that there is an interesting theory which relates the usual algebraic properties of certain classes of structures to the syntactic form of the axioms defining these classes. Thus the group axioms are all equational, the axioms for a field contain Boolean combinations of equations (conditional equations) such as x =0 v x * x - =e ; the axioms for a real closed field all take the form of a string of universal quantifiers followed by a string of existential quantifiersfollowed by a Boolean combination of equations. This theory enables us to answer such questions as ‘Why can we express the fact that a field can be ordered by a set of inequations but not by a set of equations?’. The answer is that if a set of equations is satisfied by a given structure it is satisfied by each homomorphic image of an arbitrary substructure of this structure. Thus if a set of equations is satisfied by the field of rational numbers it is also satisfied by the field of integers modulo 2, but the field of rationals can be ordered while that of the integers modulo 2 cannot. A very elementary example of this theory is given in Exercise 8 of Chapter 3 which provides a useful condition for an axiomatic theory to possess a free model. For recent developments see ABRAHAM ROBINSON, Introduction to Model Theory and to the Metamathematics of Algebra (North-Holland Publ. Co., Amsterdam, 1963). (Chapter 6 of the present book explains the methods used in this theory.) In Chapter 4 there is a more specific use of the notion of a first order formula. This use enables us to exploit the full force of certain particular constructions. For example, the algebraic theory of resultants leads to an equational condition on the coefficients of two polynomials which is necessary and sufficientfor them to have a common root. But this same construct~onprovides much more, namely, an analogous set of conditions for an arbitrary formula in the theory of algebraically closed fields! A similar but more interesting case is that of real closed fields. A long time ago Sturm showed that a polynomial vanishes in a closed segment [a, b] if and only if certain polynomial inequalities (the terms of which are rational combinations of a, b and the coefficients of the given polynomial) are satisfied. Artin and Schreier showed that this result depends only on the axioms for a reat closed field. Once we have the notion of a first order formula it is natural to try to extend this result to aZZ first order formulas of the theory of fields. This problem was mentioned in passing by Herbrand and completely settled by Tarski who proved that each first order formula of this theory is equivalent to a Boolean combination of equa-
’
157
THE AXIOMATIC METHOD
tions and inequalities. In particular, a formula without free variables is either true in all real closed fields or false in them all. Thus although it is obvious that not all real closed fields are isomorphic they are nevertheless all equivalent with respect to first order formulas which are built up from polynomial equations and inequalities. A proof of Artin's Theorem on the representation of non-negative forms as sums of squares of rational functions follows almost immediately from this result. This is done as an exercise in Chapter 4. If d is a set of axioms all of whose models are equivalent with respect to first order formulas expressible in the language of d,then &' is said to be complete. MORLEY (Categoricity in Power, Trans. Amer. Math. Soc. 114 (1965) 514-538) has recently constructed a remarkable theory of the models of complete sets. This theory is closely parallel to Cantor's theory of closed subsets of the real line. The closed subsets which we most naturally think of are all very special. If they are not themselves perfect their first or second derivatives are perfect (possibly empty). However, for each countable ordinal CI there is a closed set whose a-th derivative is not perfect. In a similar way the ordinals which, in Morley's theory, correspond to the complete sets of axioms which have turned up in other branches of mathematics are all finite, although for each countable ordinal CI there is a (countable) complete set whose corresponding ordinal is CI. It follows from the Finiteness Theorem that each set of first order axioms which has an infinite model has models of different infinite cardinals (which are therefore not isomorphic). ~istorically,the first - and the best known - systems of axioms, for example, Peano's axioms for arithmetic and Dedekind's for the continuum, were introduced to characterize uniquely certain infinite structures. If we look at these systems more closely we see that their intended interpretation does not take into account all the general models, but only some of them. In other words it is not only the meaning of the logical symbols that is laid down, but also that of certain other symbols. In particular, in certain classical systems of axioms "set variables" occur and the models considered are those in which these variables range over the set of all sub-sets (of the set which we earlier denoted by E). Languages which contain such set variables are called higher order languages and the particular models just described are called p r i n ~ ~models, al where a language is said to be of order n if it contains variables over 'p'(E)for each i t n , with ' p o ( E ) = E ,@"' (E)= @ fpP'(E)], 'p denoting the power set operation. The axiom systems of Peano and
'
I58
THE AXIOMATIC METHOD
Dedekind are of second order. Some isolated results, for example, the reduction of validity of order n (n finite, n>2) to second order validity, can be found in Chapter 7, but most of the general results about first order systems cannot be extended to the higher order case. We define a n intermediate class of models, the is, by requiring that the value of one of the unary relation symbols be the set of natural numbers and the value of one of the binary relation symbols be the successor relation. We constantly meet classes of such structures in everyday mathematics, for example, vector spaces over the field of rationals; in contrast, the class of vector spaces over an arbitrary (not fixed) field is just the class of all models (without any restriction) of a set of first order axioms. In Chapter 7 some results about general models are extended to o-models; only, we often have to require that the sets of axioms be at most countably infinite. Much more on the subject of o-models (and, more generally, of models defined by infinitely long formulas) can be found in the references cited in the summary of Chapter 6. The “negative” results about non-categoricity (with respect to first order axioms) do have a “positive” side, namely, the existence of nonprincipal models (which in Exercise 3 of Chapter 2 are also called nonstandard models). Quite recently these models have been used to create Non-standard Analysis. This recent work differs from other attempts at doing Analysis on a non-Archimedean field K by bringing in the set of “integers of K” (which satisfy the axioms of arithmetic considered). The existence of non-principal models implies the existence of non-Archimedean fields which contain such (non-Archimedean) “integers” as well as non-Archimedean “real numbers” (for example, in a Taylor series a,x”, the variable n ranges over all the integers of K and not just over the standard ones). This genuine Infinitesimal Analysis is expounded in ABRAHAM ROBINSON, Non-standard Analysis (North-Holland Publ. Co., Amsterdam, 1966). The applications described so far are applications in the strict sense of the word in that the methods given in the main text enable us to answer questions which are explicitly formulated in ordinary mathematical language. It remains to consider what, in the long run, is the most fruitful rljle of new ideas, namely, the possibility of formulating questions that we have in mind but which we cannot express precisely in ordinary mathematical language (besides possible applications to less common branches of mathematics). In this connection probably the most striking example
THE AXIOMATIC METHOD
159
is the theory of uniformly definable sets, explained in Chapter 6, which is illustrated by the following simple questions. Consider the commutative fields of characteristic zero; they all contain a sub-field isomorphic to the field of rationals. So we can ask: 1. Which first order formulas A ( x ) define the same set of rationals in all these fields, i.e. are satisfied by the same rationals in each of these fields? 2. Which first order formulas A ( x ) , satisfied only by rationals, define the same set of rationals in all these fields? 3. Which sets of rationals can be defined in this way? 4. Which sets of rationals can be defined in each commutative field of characteristic zero by a first order formula which may depend on the field? Complete answers to these questions follow as corollaries to quite general theorems about arbitrary sets of axioms. Questions 3 and 4 are equivalent; this provides a new and powerful uniformity condition. The answer to question 2 is that they are (some of the) first order formulas which define finite sets only. In other words we cannot hope todistinguish the rationals by one and the same first order formula in all fields. In fact there is a commutative field of characteristic zero in which the rationals cannot be distinguished by any first order condition (or as an algebraist would put it, they are not algebraically definable). One need only reflect for a moment to see that these questions are only interesting if arbitrary first order formulas A are considered and not just equations or Boolean combinations of equations, Obviously this is another reason why the above questions have never been dealt with in the literature of “ordinary” mathematics. This work on definable sets in general models also extends to the wmodels described above. It provides an example of an application of model theory to two other branches of logic not dealt with in this book, namely, the theory of recursive sets and that of hyperarithmetic sets. This application is based on the following facts. On the one hand the basic notions of recursion theory are those of finite set (of natural numbers) and of recursive set; on the other hand, the sets which are uniformly definable in the usual axiomatic systems for arithmetic are just the finite sets, if definability is taken in the sense of 2 above, or if it is taken in the sense of 1 above, just the recursive sets. Thus we can generalize recursion theory in two directions, either by replacing the usual axioms for arithmetic by other axioms or by replacing the class of general models by some other class of models such as the w-models mentioned above.
THE AXIOMATIC METHOD
The general nature of this method is usually described as follows. Instead of assertions about abstract properties of speczjic objects and concepts (such as space, material point, probability, etc.), one considers statements of the following form: given any collection of objects (whose nature is not otherwise specified) and given any set of relations between these objects, if the relations satisfy certain logical conditions (called axioms) then they also satisfy certain other logical conditions (called theorems of the given axiomatic theory). In different branches of ordinary mathematics a small number of particular axiomatic systems have been isolated and studied. Thereby a good deal of mathematics has been built up in a systematic and comprehensible way. But one has not been interested in arbitrary axiomatic systems or even in general classes of axiomatic systems. Thus the experience of “ordinary” mathematics provides no reason for supposing that there are useful results about general classes of axiomatic systems which would contribute to the effective use of the axiomatic method. We shall now give some applications of a study of general classes of axiomatic systems, mainly - though not exclusively - of axioms expressed in first order predicate logic, a notion which is defined precisely in Chapters 1 and 2. Broadly speaking, this language can be characterized by saying that its formulas express properties of relations defined on a domain E and that in the definitions of these properties the quantifiers range only over the elements of E and not, say, over the subsets of E. For example, the fact that a relation is an order reIation can be expressed by a first order formula but not the fact that it is a well-ordering. Or again, the fact that a structure is a group (that is to say, that the relation a * b = c satisfies the group axioms) can be expressed by a first order formula. Similarly the property of being a commutative group is of first order. However the fact that a group can be ordered is not expressed by a first order formula since this is the property that “there is an ordering of E
155
THE AXIOMATIC METHOD
compatible with the group structure” or, in other words, “there is a subset of E 2 such that ...”. Nevertheless the property of being an orderable group is equivalent to a certain infinite set of first order conditions. Finally, the properties of being a group having a finite number of generators and of being a countable group are not equivalent to any set, even infinite, of first order conditions. Thus because of the exclusion of higher order quantifiers the class of axiomatic systems for which these general results hold does not include all of mathematics. One can make up for this, at least partially, by use of infinite systems of axioms. By considering structures which satisfy an infinite set of conditions a whole class of problems can be covered which are formulated in higher order terms but which can be reduced to problems about infinite sets of first order conditions. Examples of this are given in Chapters 1-3, mainly in Exercises. The most useful results, all connected with one another, are these: 1. The Finiteness Theorem. This says that if a first order formula A holds in all those structures which satisfy a set d of first order formulas, then there is some finite subset d l of &’ which implies A . 2. The Method of Constants (Chapter 3, Exercise 2). This generalizes the well-known algebraic principle for introduc~ngtranscendental elements (where a structure containing an element 5 satisfying p,(t)#O for all n is derived from a structure which contains, for each n, a &, satisfying p,(t,)#O for all i,
+
156
THE AXIOMATIC METHOD
conditions cannot be replaced by any finite set. We remark in passing that there is an interesting theory which relates the usual algebraic properties of certain classes of structures to the syntactic form of the axioms defining these classes. Thus the group axioms are all equational, the axioms for a field contain Boolean combinations of equations (conditional equations) such as x =0 v x * x - =e ; the axioms for a real closed field all take the form of a string of universal quantifiers followed by a string of existential quantifiersfollowed by a Boolean combination of equations. This theory enables us to answer such questions as ‘Why can we express the fact that a field can be ordered by a set of inequations but not by a set of equations?’. The answer is that if a set of equations is satisfied by a given structure it is satisfied by each homomorphic image of an arbitrary substructure of this structure. Thus if a set of equations is satisfied by the field of rational numbers it is also satisfied by the field of integers modulo 2, but the field of rationals can be ordered while that of the integers modulo 2 cannot. A very elementary example of this theory is given in Exercise 8 of Chapter 3 which provides a useful condition for an axiomatic theory to possess a free model. For recent developments see ABRAHAM ROBINSON, Introduction to Model Theory and to the Metamathematics of Algebra (North-Holland Publ. Co., Amsterdam, 1963). (Chapter 6 of the present book explains the methods used in this theory.) In Chapter 4 there is a more specific use of the notion of a first order formula. This use enables us to exploit the full force of certain particular constructions. For example, the algebraic theory of resultants leads to an equational condition on the coefficients of two polynomials which is necessary and sufficientfor them to have a common root. But this same construct~onprovides much more, namely, an analogous set of conditions for an arbitrary formula in the theory of algebraically closed fields! A similar but more interesting case is that of real closed fields. A long time ago Sturm showed that a polynomial vanishes in a closed segment [a, b] if and only if certain polynomial inequalities (the terms of which are rational combinations of a, b and the coefficients of the given polynomial) are satisfied. Artin and Schreier showed that this result depends only on the axioms for a reat closed field. Once we have the notion of a first order formula it is natural to try to extend this result to aZZ first order formulas of the theory of fields. This problem was mentioned in passing by Herbrand and completely settled by Tarski who proved that each first order formula of this theory is equivalent to a Boolean combination of equa-
’
157
THE AXIOMATIC METHOD
tions and inequalities. In particular, a formula without free variables is either true in all real closed fields or false in them all. Thus although it is obvious that not all real closed fields are isomorphic they are nevertheless all equivalent with respect to first order formulas which are built up from polynomial equations and inequalities. A proof of Artin's Theorem on the representation of non-negative forms as sums of squares of rational functions follows almost immediately from this result. This is done as an exercise in Chapter 4. If d is a set of axioms all of whose models are equivalent with respect to first order formulas expressible in the language of d,then &' is said to be complete. MORLEY (Categoricity in Power, Trans. Amer. Math. Soc. 114 (1965) 514-538) has recently constructed a remarkable theory of the models of complete sets. This theory is closely parallel to Cantor's theory of closed subsets of the real line. The closed subsets which we most naturally think of are all very special. If they are not themselves perfect their first or second derivatives are perfect (possibly empty). However, for each countable ordinal CI there is a closed set whose a-th derivative is not perfect. In a similar way the ordinals which, in Morley's theory, correspond to the complete sets of axioms which have turned up in other branches of mathematics are all finite, although for each countable ordinal CI there is a (countable) complete set whose corresponding ordinal is CI. It follows from the Finiteness Theorem that each set of first order axioms which has an infinite model has models of different infinite cardinals (which are therefore not isomorphic). ~istorically,the first - and the best known - systems of axioms, for example, Peano's axioms for arithmetic and Dedekind's for the continuum, were introduced to characterize uniquely certain infinite structures. If we look at these systems more closely we see that their intended interpretation does not take into account all the general models, but only some of them. In other words it is not only the meaning of the logical symbols that is laid down, but also that of certain other symbols. In particular, in certain classical systems of axioms "set variables" occur and the models considered are those in which these variables range over the set of all sub-sets (of the set which we earlier denoted by E). Languages which contain such set variables are called higher order languages and the particular models just described are called p r i n ~ ~models, al where a language is said to be of order n if it contains variables over 'p'(E)for each i t n , with ' p o ( E ) = E ,@"' (E)= @ fpP'(E)], 'p denoting the power set operation. The axiom systems of Peano and
'
I58
THE AXIOMATIC METHOD
Dedekind are of second order. Some isolated results, for example, the reduction of validity of order n (n finite, n>2) to second order validity, can be found in Chapter 7, but most of the general results about first order systems cannot be extended to the higher order case. We define a n intermediate class of models, the is, by requiring that the value of one of the unary relation symbols be the set of natural numbers and the value of one of the binary relation symbols be the successor relation. We constantly meet classes of such structures in everyday mathematics, for example, vector spaces over the field of rationals; in contrast, the class of vector spaces over an arbitrary (not fixed) field is just the class of all models (without any restriction) of a set of first order axioms. In Chapter 7 some results about general models are extended to o-models; only, we often have to require that the sets of axioms be at most countably infinite. Much more on the subject of o-models (and, more generally, of models defined by infinitely long formulas) can be found in the references cited in the summary of Chapter 6. The “negative” results about non-categoricity (with respect to first order axioms) do have a “positive” side, namely, the existence of nonprincipal models (which in Exercise 3 of Chapter 2 are also called nonstandard models). Quite recently these models have been used to create Non-standard Analysis. This recent work differs from other attempts at doing Analysis on a non-Archimedean field K by bringing in the set of “integers of K” (which satisfy the axioms of arithmetic considered). The existence of non-principal models implies the existence of non-Archimedean fields which contain such (non-Archimedean) “integers” as well as non-Archimedean “real numbers” (for example, in a Taylor series a,x”, the variable n ranges over all the integers of K and not just over the standard ones). This genuine Infinitesimal Analysis is expounded in ABRAHAM ROBINSON, Non-standard Analysis (North-Holland Publ. Co., Amsterdam, 1966). The applications described so far are applications in the strict sense of the word in that the methods given in the main text enable us to answer questions which are explicitly formulated in ordinary mathematical language. It remains to consider what, in the long run, is the most fruitful rljle of new ideas, namely, the possibility of formulating questions that we have in mind but which we cannot express precisely in ordinary mathematical language (besides possible applications to less common branches of mathematics). In this connection probably the most striking example
THE AXIOMATIC METHOD
159
is the theory of uniformly definable sets, explained in Chapter 6, which is illustrated by the following simple questions. Consider the commutative fields of characteristic zero; they all contain a sub-field isomorphic to the field of rationals. So we can ask: 1. Which first order formulas A ( x ) define the same set of rationals in all these fields, i.e. are satisfied by the same rationals in each of these fields? 2. Which first order formulas A ( x ) , satisfied only by rationals, define the same set of rationals in all these fields? 3. Which sets of rationals can be defined in this way? 4. Which sets of rationals can be defined in each commutative field of characteristic zero by a first order formula which may depend on the field? Complete answers to these questions follow as corollaries to quite general theorems about arbitrary sets of axioms. Questions 3 and 4 are equivalent; this provides a new and powerful uniformity condition. The answer to question 2 is that they are (some of the) first order formulas which define finite sets only. In other words we cannot hope todistinguish the rationals by one and the same first order formula in all fields. In fact there is a commutative field of characteristic zero in which the rationals cannot be distinguished by any first order condition (or as an algebraist would put it, they are not algebraically definable). One need only reflect for a moment to see that these questions are only interesting if arbitrary first order formulas A are considered and not just equations or Boolean combinations of equations, Obviously this is another reason why the above questions have never been dealt with in the literature of “ordinary” mathematics. This work on definable sets in general models also extends to the wmodels described above. It provides an example of an application of model theory to two other branches of logic not dealt with in this book, namely, the theory of recursive sets and that of hyperarithmetic sets. This application is based on the following facts. On the one hand the basic notions of recursion theory are those of finite set (of natural numbers) and of recursive set; on the other hand, the sets which are uniformly definable in the usual axiomatic systems for arithmetic are just the finite sets, if definability is taken in the sense of 2 above, or if it is taken in the sense of 1 above, just the recursive sets. Thus we can generalize recursion theory in two directions, either by replacing the usual axioms for arithmetic by other axioms or by replacing the class of general models by some other class of models such as the w-models mentioned above.