Axioms for abstract model theory

ANNALS OF MATHEMATICAL LOGIC 7 (1974) 221-265. © North-Holland Publishing Company

AXIOMS FOR ABSTRACT MODEL THEORY t K. Jon BARWISE Department of Mathematics, Universityof ICqsconstn,Madison, Wise., 53706, U.S.A. Mathematical Inrtirute, Oxford, England Reeeive,d 25 May 1974

0, Preface

During the spring of ! 972 I gave some lectures on abstract model theory in the logic seminar in Madison, Wisconsin. The lecture notes from this semivar have been widely circulated in Europe and their unpublished nature has been the source of many complaints. With some misgivings, then, I have decided to publish these notes with only minor changes~ . The most important task of abstract model theory is probably the discovery and investigation of mathematically useful extensions of L,o,o, the usual first order predicate calculus. This task is only treated indirectly in these notes, which accounts for the misgivings mentioned above. Our aim is a modest one: to investiga,e the role elementary syntactic tran,~rmations play in abstract model theory. Experience suggests that a good understanding of this will aid in the study of more delicate matters. This paper is divided into two parts. Part I presents the basic axioms (in Section 1.7) and treats certain preliminary matters. Part II explores four ways these axioms can be u ~ d to good advantage in abstract model theory.

t The te~eareh for this paper was partially supported by grant NSF GP-27633. 2 The revisions consist primarily of rearrangement, the elimination of some tedious w,aterial f~om Section 1.9, and the inclusion of Section II.4. The material in this section was di3tfibuted by means of a handwriRen memo in the wiuter of '71 - ' 7 2 but was not typed as part of the seminar notes,

222

K,Z Barwise, Axioms.for abstract model theory

Part 1. THE AXIOMS § 1. What is abstract model theory? A tremendous amount o f effort has been spent on the study o f logics which strengthen the lower predicate calculus in one way or another, Some o f the logics have proven .t'ruitful, other were destined for obscurity from the start. What are the considerations which give rise to a u ~ f u l , natural logic? How does one achieve the necessary balance between strength and manageability? These are two rather imprecise questions one can investigate by means o f ~:bstract model theory. More precise questions follow. We first describe seven well-known logics. These examples will allow us to isolate some o f the esse~ltial features o f a logic, as well as state some precise questions abstract model theory attempts to answer. ~2~,o - t h e ust, a l l o w e r p r e d i c a t e c a t c u h t s

Syntax: To each set L o f relation, function and constant symbols, there is a set o f formulas o f L,o,,, . They are obtained by introducing variables o0, o 1. . . . , forming terms and atomic formulas as usual and closing under the logical operations o f ^ , v, 7, 3, V. We take = to be a logical symbol, not an undefined relation symbol. Semantics: An L-structure ~l-= (M, . . . ) assigns a meaning to the symbols in the "alphabet" L. One then extends this to give a meaning to sentences o f L ~ o by inductively defining the relation '~ ~ ~a. For example, ~(~^

~,)

iff

~F-]~

iff

not(~Vg),

~ ~x ~ x )

iff

~ a n d ~ , ,

there is an a E M such that (~l, a) ~ ~(c).

~2n - s e c o n d o r d e r logic"

Syntax: We add variables X o, X 1 . . . . and a new logical symbol E. New atomic formulas t ~ X i are allowed. A formation rule is added to those o f L,o,o. If~o is a formula, so are Y X i ~ and :_IXt ~ . Semantics: ~ is interpreted by set membership.

K.J. Barwise, Axioms for abstract model theory

223

~t= YX ~ means that for all X c M. ( ~ , X) t=~o; ~1~ ~ X ¢ means that for some X c M, (9.~, X) ~ ~0. ~2w _ w e a k s e c o n d order logic

Syntax: Exactly like that o f L a, Semantics: ~ is interpreted by set membership. ,lr~~ YX ¢ means that for all finite X c M, ('3.~, X ) ~ ¢; ~ V ~ X ¢ means that for some finite X c M, (~.~LAT)~ ~p. E ( O 1 ) - logic with the quantifier "there exist u n c o u n t a b l y m a n y "

Syntax: Add a new symbol 0 to L~o~o by meaus of the formation rule: if q~ is a formula so is (lxi ~0. Semantics: ~ ~ O x ~o(x) means that there are an uncountable number o f a E M such that (%'q, a) F ¢(c). ( 0 o ) ..... logic wiyh thc quantifier "there exist infinitely m a n y "

Syntax: Exactly like that o f L(O l ). Semantics: 9~2~0 x ~ ( x ; means that there are an infinite number o f a E M such that ( ~ , a ) ~ ¢1.c). E=to - logic with #zfinite conjugations a n d dis]unctions

Syntax: Add the new formation rule to L~,w : if cI, is a set o f formulas then A ¢ and V ~ are formulas. Semantics: ~ ~ A ~ means that ~ ~ ~0 for all ~pEcI,; ~ ~ V ¢ mea~s tha~ ,.~ ~ ~o for some ~o~ ~. Ew z ~ - logic with countable conjugations and dis]unctions

Syntax: Like that o f L~,~ except that ep is required to be countable in the new formation nile. Semantics: Exactly like that o f L=,o. Tile list o f examples is potentially infinite. These examples suffice for isolating some o f the crucial features o f the notion o f a logic. First we see that a !ogle is not just a set o f formulas but rather a mapping which, in a mfiform way, assigns to any set L o f symbols a certain set o f formulas. This mapping is part o f the syntax o f the logic. Secondly, a logic associates a semantics with the syntax, that is, it ~ells o f a sentence ~0o f the syntax whether or not ~o is true ii~ a given structure.

224

K.Z Barwise, Axioms for abatract model theory

The examples above show that radically different logics often have exactly the same syntax. There are certain syntactic transform~ltions 3n sentences that seem inherent in the idea o f a logic. For example, one should be able to replace a binary relation symbol R ~y a new symbol R' in any formula. Similarly, one should be able to relativize a sentence ~ to a unary relation symbol U obtaining a sentence ~,tu~ which asserts about a substructure with domain U what ~psays of the whole structure. We must find a way of building such syntactic transformations into the defmition of a logic if we are to have a reasonably smooth theory. A common feature of the above logics which pla3,s very little role is the use of free variables. There is no essential difference between a free variable and an uninterpreted constant symbol, that is, a constant symbol o f some bigger language. We can thus reduce the number of primitives in ~;ur theory by doing without free variables in the beginning, treating them as a defined concept later o~a. To summarize, a 19gic is going to be an operation which assigns to each set L of symbols a syntax and a semantics such that: (1) elementary syntactical operations (l~ike relativizing and renaming symbols) are performable, (2) isomorphic dtructures satisfy the same sentences. A simple way of making all this turn out smoothly is to use the notion offunctor from category theor3~. We conclude this section by answering the question posed in the title of the section. As often happens, the answer is a seqt ence of further questions. These questions, at least, have p~rtiai answers and the answers are what abstract model theory is all about. Q1. What is there that is basic and fundamental about L,,~, ? We can turn Q 1 around to get one of the most important questions of the subject, already mentioned above. Q2. What considerations give rise to natural extensions of L~,~. extensions that characterize important notions not expressible in L~, o , but extensions which still have a manageable model theory? Q3. It is often asserted that Lw and L(O 0) are essentially the same. How can we make such statements precise?

K.£ Barwise, Axioms/'or abstract model theory

225

Q4. Anyone who has worked with extensions of L~ ¢o knows that some results are entirely "soft" in that they use only very general properties of the logic, properties that carry over to a large number of other logics. Shouldn't such results be part of an axiomatic treatment of

logics? Q5. L(Q 1 ) is nicely behaved in sc~me ways but fails to satisfy some natural closure conditions. Is there a good extension of L(O 1 ) that does satisfy these conditions? The list is by no means complete but it should give the reader a feeling for the answer to our question: What is abstract model theory? 2. A review of terminology and notation from category theory The clearest v, ay of expressing the axioms we are after is to use the notion of functor. This notion is most natural in the contect of category theory so we review some definitions here, largely to fix notation and terminology. A category e consists of: (i) a collection of objects; (ii) for all objects X. Y a set horn(X. Y) of morphisms with domain X and range Y; if~ E hom(X, Y) we ,,,rite a : X-* Y or X ~ Y; and (iii) for all morphisms ~,/3 of the form

x Y, a morphism ~ ~ horn(X, Z). These objects and morphisms satisfy the following two axioms: Associativitv: If X ~ Y, Y ~ Z and Z & W, then 7 ( ~ ) = (~//3)a. Identity: For each object Y there is a morphism Iy : Y ~ Y such that i f X ~ Y, then 1ya = a and if Y ~ Z, then 131y =/3. The morphism 1). is uniquely determined. It is called the identity morphism on Y. Let e and q) be categories. A (covariant) functor * from e to q) consists o f (i) a function which assigns to each object X of e an object X* of ,~,

226

K.£ Barwise, Axioms for abstract model theory

(it), a function which assigns to each morphism X ~ Y o f e a morphism X* ~ Y* o f ~ such that:

(Ix)* = lx , ,

(t~)* =/3"~*

f o r all objects X and morphisms e, ~ o f (3. The assertion that a : X ~ Y,/3 : Y -* Z and ~, = Ba can be expressed by asserting that the diagram

X2, y

commutes. The crucial functor axiom can be expressed by asserting that if the above commutes, so does X * ~ Y*

~Z* We can use diagrams to define some special types o f morphisms. For example, a morphlsm X ~ Y is a retraction if there is a morphism Y ~ X such that

rLx commutes. Since a/3 = 1 r,/3 is called a right hzverse for a. On the o t h e r hand, X ~ Y is a coretraction if there is a Y ~ X such that Ot

X-* Y

l\~e x ~X

commutes, and, since ~ = I x ,/3 is called a left im,erse for ~. I f X ~ Y is a retraction and a coretraction, i.e., if it has both a right inverse/3 and a left inverse ~', then ~ is an isomorphism. Since

/3' =/3' 1 r = Y('~/3) = (/3'~)/3 = 1x/3 --/3, the morphism ~ =/3' is called the inverse ofo~ and is denoted by ~-1 Note that a - 1 ~x - i x , a ~ - 1 _.- I r" and ( ~ - t ) - t = ~.

K3, Barv~qse,Axioms for abstract model theory

227

2.1. l.emma. Functors preserve retracts, coretracts and hence isomorphisms This lemma~ as well as those below, is trivial and can be found in the first few pages of almost any book on category theory. A m o n o m o r p h i s m is a naorplusm X -~ such that for all/3,/3' if

z l xJ,~

¢' J, X~

O(

Y

commutes, then ~ -=/3'. (In most categories this is the same a's ~ being one-to-one.) An epimorphism is a morphism X -~ Y such that for all/3,/3' if O~

X-~ Y

Y;z commutes, then t3 =/3'. (In most examples this is the stme as a being onto.)

2.2. Lemma. (i) Every retraction is an epimorphism. (ii) Every coretraction is a monomorphism. Functors do not always preserve monomorphisms and epimorphisms. Similarly, a morphism can b c a monomorphism and an epimorphism without being an isomorphism.

2.3, l.emma. (i) I r a is a retraction and a m o n o m o r p h i s m , then a is an isomorphism. ( i i ) / r e is a coretraction and an epimorphism, then ~ is an isomorphisnL

In the category of languages, to be defined in Section 4, there are monomorphisms which are not coretractions, but every epimorphism is a retraction. By the above lemma, every morphism which is a monomorphism and an epimorphism in the category o f languages is an isomorphism. A category e 0 is a subcategory of a category ~ if every object of E0 ~s an object o f e , and ff every morphism X ~ Y m e 0 is also a

228

K,J. Barwise, Axioms.for abstract model theory

morphism in e . e 0 is a full subcategory o f e if, in addition, for all objects X, Y in e 0 , h o m e (X, Y) = h o m e (X, I 0 . o

3. Languages and structures In order to simplify the notion of relativization we introduce :i distinguished unary relation symbol ¥ to be held fixed throughout our discussion. (In this context V is not the universal quantifier but denotes the domain of the universal quantifier. It should cause no confusion.) A language is a set L of relation, fimction and constant symbols with ¥ E L. Each relation symbol end fimction symbol has a fixed finite number of argument places. A constant symbol is a function symbol with 0 arguments. Since V is an element o f every language, we often ignore it in displaying a language. For example, if we write L = {R, ]: ?}, we mean L = {V, R, f, c}. The tril,ial language is the language {V}. Two languages L, K are disioint if L o K is trivial. A partial structure f o r a language L is a function $l with domain _q L such that, writing S ~ f o r m ( S ) and M for V~ (= ~ f ¢ ) ) , the following hold: (0) V~ d o m ( . ~ ) , (i) if R ~ L is an n-ary relatio~ symbol, then R ~ c M n , (ii) i f f ~ L is an n-ary function symbol, t h e n f ~ is a partial functior, from M n to M, (iii) i f c ~ L is a constant symbol and c ~ is defined, then c ~ ~ M. The set M is called the universe o f ~ , A partial structure ~[~ for L is called a stntcture f o r L, or L-structure, i f f ~ is total, for each f ~ L, and c ~ is defined, for each c E L. We use standard notation whenever possible. For example, if L is the language {R, g, c} we usually display an L-structure ~//by ~l = (bt, R, g, c) or even ~ = ( M . . . . ). By the cardinality of ~ we mean the cardinality of its universe. Every language L has associated with it a set o f terms, defined inductively as usual: if c is a constant symbol in L, then c is a term o f L; if f is an n-ary function symbol of L and t 1 . . . t n are terms o f L, then

K.J. Bara,ise, Axioms for abstract modcl theory

229

t n ) is a term o f L. (If L has no constant symbols, then o f course L has no terms.) Let L be a language and let ~ be an L-structure. Every term t o f L denotes an element t ~ o f the universe o f ~ . For constant symbol c, c ~ is already defined. For more complicated terms,

f(t I ...

t.)

....

,

An e x p a n s i o n o f a language L is any language K with L c__K. An Lstructure ~ is a r e d u c t o f a K-structure 9~, and ~ is an expansion o f ~ , if ~ is 92 r L, the restriction o f 92 to the language L. If we are given a language L and some new symbols, say S, g, we write L(S, g) for the language L t2 {S. g) and a structure for L(S. g) might be written as 92 = ( ~ , S, g) where ~I is an L-structure and S = S 9~,g = g'~. 4. The category of languages We want to impose a natural category stru,~'ture on tile collection o f all languages, What are the natural morphisms? Intuitively. a morphism is simply a set o f instructiors o f the following kind: replace R ( x 1 . . . x n ) b y R ' ( x l . . . x n , t l . . . t k ), replacef(x 1. . . x n ) b y f ' ( x l . . . x n , t ~ . . . :;..), replace c by g ( t 1 . . . t k ) and relativize all quantifiers to the set o f x such that V ( x , t 1 . . . t k ) . In other words, it is just an interpretation. 4.1. Definition. Let k .>_ 0 be an integer. Let L, K be languages. A (~t e r m ) i n t e r p r e t a t i o n r, o f L into K consists o f k not necessarily distinct terms (t~, . . . . t~) o f K and a function a mapping L into K satisfying the following: (i) I f R ~ L is an n-at3, relation symbol, then R~( = t~(R)) is an (n + k)-ary relation symbol. (ii) l f f ~ L is an n-ary function symbol, then f ~ is an (n + k)-ary function symbol (iii) I f offx) = V, t h e n x = V. As a special case o f (i) we have ¥~ is a ( 1 + k)-ary relation symbol, often denoted by U or V. Since constant symbols are O-ary fimction symbols, a special case o f (ii) asserts that if c ~ I,, then a(c) is a k-ary function symbol. Also note that if ¥ is in the range o f a, thev k = 0 (by combining (iii) and (i)). There is an obvious way to compose interpretations. Suppose, for example, that a : L ~ K is a unary interpretation with term c and : K ~, J is a binary interpretation with terms t, t'. Then ~ : L --.-J is

230

K.Z Barwise, Axim~t~for abstract model theory

a ternary interpretation with terms g(t, t'), t, t', where g is/3(c). The general definition is entirely analogous. For any language L let 1L : L -~ L be the identity function. Thus 1L is a 0-term interpretation. 4.2. Definition. The category e ** o f all languages has, as objects, all languages. Given languages L, K, hom(L, K) consists o f all interpretations of L into K. Composition and the identity morphisms are defined as above. The reader should verify that C,, is indeed a category, that is, that it satisfies the axioms for a category. Let L, K be languages and let h o m k ( L , K) be the set o f k-tetrn interpretations o f L into K so that h o m ( L , K ) = 0 _ < LI k<

hom~ (L, K)

The elements of hOmk(L, K) are also called k-morphisrns. Note that the composition o f a k-morphism and ~n l-morphism is a (k + 1)-morphism. A 0-term interpretation a : L -, K such that Va = V is called a simple morphism. If L ~ K, t h e n the inclusion map is an example o f a simple morphism, In general, a simple morphism is an interpretation where the universe of quantification is not redefined. 4.3. I.emma. Let ~ : L -~ K be a morphism in the category o f all lang~tages. (i) a is a m o n o m o r p h i s m i f f a iT one-one as a function mapping L into K. (ii) ot is an epimorphism i f f ~ is a retraction i f f ~ is onto K. (iii) tz is an isomorphism i f f ~ is one-one and onto l i f e is both an epimorphism and a m o n o m o r p h i s m . Proof. (i) It is clear that if c~ is one-one, then a is a m o n o m o r p h i s m . Suppose that a is not one-one; for example, suppose that R, S are distinct binary relation symbols such that R a = S ~ . Let J = (R). Let /3 be the identity map o f J into L and let/3' : J -~ L be the simple morphism taking R to S. Then a~ = a/3' but B $ ~ . (ii) By L e m m a 2.2, retractions are always epimorphisms. It is clear that if a is o n t o K, then a is a retraction. An argument similar to that for (i) shows that if t~ is not o n t o , then it cannot be an epimorphism. (iii) This follows from (ii) and Lemma 2.3(i).

K,Z Bar~se, Axioms for abstract model theory

231

Simple monomorphisms are called embeddiugs. Thus an embedding : L -* K amounts to nothing more than replacing symbols of L by symbols of K (of the same arity) in a one-one fashion. § $. Operations on structures induced by morphisms on languages Let L -~ K be a k-morphism and let ~ = ( M , , . . , a t . . . a k) be a typical K-structure, where a i is the interpretation of the term t ~ , i.e., a i = ~ ( t ~ ) . We would like to "pull ~ back" to get a structure 9X-~ for the language L. If a is simple (and hence k = 0) then the definition is simple: ~.'q/-a = ~ * 0t. That is, the universe of ~ - ~ is M and the interpretation of a symbols ~ L is just the interpretation o f s ~ in ~ . I f a is not simple, the definition is not so simple. The universe of ~ - a is, in general, the set N = ( a ~ M : (a, al. . , a k ) ~ U ~ , U = a ( V ) ) , the interpretation of an n-ary relation symbol R ~ L is given by

(bl.,.bn)~R~-~

b 1...b hEN, ( b l . . . b , , , a l . . . a k ) E (R ~ ) ~ , and the interpretation f ~ -'~ of an n-ary function s y m b o l f ~ L is the restriction o f f a ~ ...ak to N n , wherefaa." .ak is defined by fa~...~:(ba'"bn

iff

) = ( f " ) ~ ( h i . . . b n , a ~ . . . a k) •

Since the set N may or may not be closed unde," the function fa a , the most we can say, in general, is that ~ - a is a partial structure for L. tf ~ - a is an L-structure then ~ is said to be t~-invertible. "

-

l

'

"

k

5.1. Example. Suppose L 9 K and that a : L ~ K is the inclusion map. Then every K-structure ~ is ~-invertible and ~ - ~ is just the reduct of to L,

5.2. Example. Let L be a language, V a unary, relation symbol not in L and let K -- L(If). Let a : L -* K be the interpretation which sends V'to V but leaves the other symbols of L unchanged. A typical K-structure has the form (!tJL V), where ~ is an L-structure. t h e stil~cture ( ~ , V) is a-invertible iff the set V is closed under the van')us functions of ~ , in which case ~ - a is just the usual relativization of ~ to the set V, !~-~' = df ~ ( V ) .

232

K.J. Barwise, Axioms for abstract model theory

A slightly ;ess trivial example comes from the formation of the indexed union of a set of L-structures. 5.3. Example. Let L = {R}, where R is binary and let K = (R'. E c), where R ' is 3-ary, V is binary and c is a constant symbol. Let a : L ~ K be the l-term interpretation (1 -morphism) which sends ¥ to V and R to R'. The term of~ is just c. To ~ee the point of such an interpretation, let ( ~ t : i ~ I} be an indexed set of structures for L, ~ i = (M~, Rt). We can code this set of models into a single model 91l = (M, If. R'~, the ind e x e d union of the ~ i , by defining: M = I u i U j Mi , ~ V

if i ~ I a n d a E M

(v, b, i) E R '

i,

if i E I and (a, b ) E R i .

Then we can recapture each ~ i since i)

=

for each i ~ / . (Here i is the interpretation of the constant symbol c in the K-structure ( ~ , i).) The formation of indexed unions for more complicated languages L is carried out in the same way. These three examples are quite simple but they are essential for many arguments in model theory. In all of the above examples the interpretations were monomorphisms. 3 An example of an interpretation that is not one-one comes up in the proof that the interpolation theorem implies Beth's definability theorem. 5.4. Example. Let R, R' be n-ary relation symbols not in L and let ~z : L(R, R') ~ L(R) be the 0-morphism which sends both R and R' to R and ;eaves the other symbols of L unchanged. Every L(R)-structure ( ~ , R) is ~-invertible and R)

=

R, R ) .

Syntactically, cz is the instruction to erase all the primes from R' in the formulas of L(R, R ' ) 3 All of the z~.,sultsin Part II are pzoved by using monomozphisn~ It seems dear, though, that the other types of morphisms ate just as natural and should be bm~t in to the deflation

of "zone'.

l~.J. Barwise, Axioms for abstract model t~:eoo,

233

5,5. Lemma. Let L ~ K be a morphism (i) I f L cmitains no function or constant symbols then every K-s~ructure is a-invertible. (fi) I f L contains some function or constant symbol then a is simple i f f every K-stntcture ~l~ is ~-invertible. (iii) t~ is a monomorphism iff ereo, L-structure ~.~ is 9~-~ fbr some K-structure ~. All parts of this lemma are trivial. §6. Subcategories of languages Not every logic is defined on the category ~** of all languages. Consider, for example, admissible fragments ~2A of ~o.~, where A i'~,the admissible set. The only languages L for which LA makes sense are those L which are (coded as) A-recursive subsets of A. The only raorphisms (interlrretations) which behave properly are those a : L ~ K given by an A-recursive function ~. We wish to define a notion of category of languages rich enougl~ to carry out the syntactic manipulations one uses but restricted enough to allow the above example. We aEo want the following examples to be categories of languages. 6.1. Example. Let ~: be any infinite cardinal. The category e K is the full subcategory of e~ whose objects are those languages of power less than t~. 6.2. Definition (teatative). A subcategory e of e® is a category o f languages if iz satisfies the following conditions: (i) If L and K are objects in e, so are L n K and L u K. (ii) If L and K are objects of e and L ~ K, then the inclusion map is a morphism o f C. (iii) If L is a finite language, then L is isomorphic to some L' in e . Moreover, for any K in e there is an L' in (3 isomorphic to L but disjoint from K. (iv) If L and K are in e and if ~ : L -~ K is a 0-term interpretation which is the identity except on a finite number of elements, then o~is a morphism of ~. (v) If L is in E, then there is a constant c not in L and a language K in e with c ~ K and a 1-term interpretation a : L -~ K with term c which :is one-one and a morphism o f C.

234

K.J. Barwise, Axion~ for abstract model theorj,

By using (i) and (iii) we may take any L in e and "add a new binary R " to form L(R) in ~. By (ii), the inclusion of L in L(R) is a morphism of e; similarly for relation symbols of any arity and for function and constant symbols. We have used the word "tentative" in Definition 6.2 because there are probably other natural closure conditions which will be needed for other work. We have simply listed those needed in this paper. A really good definition of this notion will take more experience to formulate. It may well require a solution to problem P6 (see Section 11.5): § 7. What is a logic? In this section we give the axloms for logics referred to in the title of the paper. The axioms were arrived at, after trial and error, by keeping in mind (i) the evident common features of concrete logics like those discussed in Section !, (ii) the experience gained by actually working with some of these logics, (iii) the desire to strike a balance between generality on the one hand, and naturalness and manageability on the other. This last point needs emphasizing. There is no single result that needs all the conditions imposed on a logic in this section, but style has to count for something, so we try to resist the temptation toward maximum generality at the expense of intuition. The axioms use the notion of functor. We could eliminate all mention of functors and categories by building those definitions into the axioms. This would have the advantage of not repelling those readers hostile to category theory. It would be slightly dishonest, though, since it is exactly the functor that lies at the heart of tile syntax. As we observed in Section 1, a logic ~2" consists of a syntax and a semantics which fit together nicely. Tile s vntctx" o f ~.* is a functor • on some category e of languages to the category of classes. (In most examples L* is always a set.) The elements of the various L*, for L in C, are called ~2*-sentences. The functor • satisfies the following axiom. Occurrence Axiom. For every £*-sentence ~o there is a smallest (under c_) language L, in e such that ~0~ L*. If i : L, c K is an inclusion morphism, so is i* : L~* C_ K * .

K,J, Bar~qse, Axioms for abstract mo&,l theory

235

The elements of L¢ are said to occur in C so that L¢ is the set of symbols which occur in C. The axiom asserts that every sentence C has a smallest set of symbols which occur in it, and that C ~ K* whenever every symbol occurring in C is an element of K, The semat~tics o f ~* is a relation ~ such that if .q]~PC, then N? is an L-structure ?or some L in C, and so ~ L*. It satisfies the following axiom: Isomorphism Axiom. If ~ ~'-C and ~ ~- g~, then 9? DO. The relation ~ DC is read .~Jlis a model o f c . Given a morphism L 2; K, the syntax • gives us a function a* : L* ~ K*. I f c ~ L*, we write ,p~ for a*(C) The syntax and the semantics of ~2" fit together according to the final axiom. Translation A~iom, For every ~d*-sentence C, every morphism L ¢ ~ K and every K-s~a'uctvre ~ , ~'~ F : ~

iff

~ is o~-invertible and ~ - a ~C.

This last axiom is named a: it is because we think o f c a = o~*(C) as the translatior, of ¢ into the language K by means of the interpretation a. 7.1. Example. Let C be an ~d*-sentence and let L~, be the set of symbols occurring in C. Let V be a unary relation symbol not occurring in C and let a : L ~ L(V) be the relativization defined in rZ,xample 5.2. We use the more stanc~ard notation C(v) for C '~ (= a*(C)). For this morphism, the translation axiom asserts that an L(V)-structure ( ~ , V) P C (v) if and only if the set V is closed under the functions of ~ and ~ ( v ) ~C. We call C(v) the relativization oJ'c to V. In order to get a feeling for tile content of these axioms, we explore some of their simplest consequences. We fix a logic ~2" on a category e of languages for tile rest,of this section. 7.2. Proposition, Tile functor • preserves all hwlusion morphisms from e. Proof. Let L ~ K be an inclusion morphism and let C ~ L*. We need to show that a*(C) = C. Let L 0 be the set of all symbols occurring in C and let i 0 : L 0 c_ L and i 1 : ~ c. K be the inclusion morphisms. S i n c e , io L0-', L

K.J. Barwise, Axioms for abstract model theory

236

commutes, so does L~L* i~++x, ++,*

K*

But i~ is the inclusion morphism L~ c_ L* and similarly for i ~ , by the Occurrence Axiom. Thus a*(¢) = ~. 7.3. Proposition. I et ~ ~ L* t3 K* and let ~ and ~ be L - a n d K-structures which agree on the symbols occurring in ~. Then ~ ~ ~ i f f ffl ~ ~. Proof. Let ~ : L~, c_. L and/~ : L~o c__K be the inclusion maps. Then ~ is e-invertible, ~ - a = ~ t L~, 9/ is i3-invertible and ~ - a = 9/P L~, and 9~ I" L¢ = 9/1' L~, by hypothesis. By the translation ~xiom ~ ~ ~0iff

7.4. 1.emma. Assume that ~ is a fuU subcategory o f ~,~ . Let ~ : L -* K be an interpretation which maps L onto K. Then every sentence ~k ~ K* is ~ :for some ~ ~ L*. Proof. Since B is full, every epimorphism in ~ is a retraction and the functor • preserves retractions. Thus e* : L* ~ K* is a retraction and hence an epimorphism in the category of classes. Let ~o, ~ ~ L ~ We say that ~, ~ are logically equivalent and write ~0~* ~, if ~ and ~ have the same L-structures as models. By Proposition 7.3 this is independent of L as long as ~, ~ ~ L*. %5. Proposition. Let L ~" K be a morphism. (i) f f ~ ~ L* and ~ is an e-invertible K-stmcn+re, then ~ ~ ~ if]+ ~J~-~ ~ ~o. (ii) f f ~ is a monomorphism but ~o~ = ~/a f o r some ~, ~k E L*, then ~oand ~ are logically equivalent, ~o~ ~. Proof. (i) Let ~ : L~ + K be the restriction of e to L~,. Given an a-invertible ~ , ~I~is also fl-invertible so ~ F ~ iff ~ c ~ ~ , But ~ - ~ = ~ - ~ f L~, so by Proposition 7.3, ~ ¢ ~ i f f ~ -a ~ . B u t L C_L

al: °

K,Z Barwise, Axio.n for abstract model theory

237

commutes, so L* c L* ~,x ~* K* commutes, so ~ = 3"(~) = e * ( ¢ ) = ~ a Thus, ~)~~ iff ~ - ~ ~ o . (ii) Since a is a mor~omorphism, every L-structure ~ is ff/-~ for some K-structure ~ . Thus, ~.3~~ ~o iff ~ P- ~0a iff ~ ~= if, using (i). Its not hard to construct examples to show that Proposition 7.5 (ii) cannot be improved to cenclude ¢ = ~b. (Define L * to be Lto~o except that any sentence containing constant symbols is identified with all other sentences o f L* equivalent to it and containing the same symbols. A l-monomorphism which introduces a constant symbol into a sentence that had none wilt collapse some equivelent sentences.) §8. Atomic sentences Every language L gives rise to a (possibly e m p t y ) set o f atomic sentences, If t 1 , t 2 are terms o f L, then the expression (t 1 = t 2) is an atomic sentence of L. If R ~ L is an n-ary relation symbol and t t . . . t n are terms o f L, then R ( t I . . . t n) is an atomic sentence o f L. Let ~d* be a logic on a category e o f languages. We say that ~d* contains all atomic sentences if for each language L of e the following conditions are satisfied: (i) if (t I = t 2) i:, an atomic sentence o f L, then it is an element of L* and, for each L-structure $~,, ~ ( t I = t 2)

iff

tl~=t~

;

(ii) i f R ( t I . . . t n) is an atomic sentence o f L, then it is an element o f L* and, for each L-structure $1, ~R(tl...t

n)

iff

(t~ . . . .

,tn~)~R ~ .

We usually assume that ~2" contains all atomic sentences. 8.1. I.emma. Suppose ~,~*contains aH atomic jbrmulas and let ~obe an atomic sentence o f L * , f o r some L in e~ The symbols which occur in ~o, in the sense o f the occurrence axiom, are V and the symbols which occur in ~ in the usual setLse. 4 4 Note thai " ~ " is being treated as a logical symbol and that it does not occur as an element

of lanSuagcs L

238

K.Z Barwise, Axioms for abstract model theory

Proof. The symbol V occurs in all sentences since it is an element o f every language. Suppose, to take a typical example, that ~ is R ( c 1 , f ( c 2 )). We want to prove that L~, = K, where K = (V,R, f c I , c2}. Since ~o is a sentence o f K, ¢ ~ K* by the above definition. Thus K D_ L~,. Now suppose some symbol of K, say c I , is not in L~. But L-structures do not even assign an interpretation to c ! so the equivalence (ii) must fail since it implies that all o f R s~ and t ~ . . . . t~ are defined.

§9. Syntactic operations on logics In this section we present a workable definition o f finitary syntactic operations which includes the usual finitrary propositional operations, all known quantifiers, as well as other less c o m m o n examples. Indeed, some o f the examples are so bizarre as to suggest that our definition is too weak to be o f m u c h use. We are far less sure o f its "correctness" than we are o f the axioms in section 7. Still, it serves our purpose here by giving us a way o f introducing propositional connectives and quantifiers into logics. 9.1. Definition. Let ~2" be a logic on e . An (m, n)-ary s y n t a c t i c o p e r a t i o n o n ~* is a mapping r which assigns to each L in e a function rL : DO'L) ~ L*, with domain D(r L) c L m X (L*) n , such that for any embedding t~ : L --, K the following hold: (i) for all x ~ L m X (L*) n , x ~ D(r L) iff a(x) ~ D(r K ) , where t~(x) is evaluated coordinate by coordinate: (ii) the following commutes: D(r L) ~ D(r K) rL,[ ~rK L*

-)"

9.2. Lemma. L e t s 1 . . .

K* s m ~ L ca K, ~oI . . . ~0n ~ L* ca K* a n d l e t x = ( s l . . . s n ,

~... ~). (i) x ~ D(r L ) i f f x E D(r K ), (ii) i f x ~ DO'L) t h e n r L ( x ) = r r ( x ) , a n d t h e s y m b o l s o c c u r r i n g in rL(X) are amo~tg s I . . . s m a n d ttle s y m b o l s o c c u r r i n g in ~Pl " • • ~°n. Proof. Let L 0 be the smallest language containing s 1 . . . s m , and tile symbols occurring in any o f sp1 . . . en- Apply the above to the inclusion morphisms a : L 0 c_ L, # : L 0 c__ K.

K.Z Barwis¢, Axioms for abstract model theory

239

This lemma usually allow:; us to write D(T) for D(T L) and r(x) for rL(X) without danger o f c o t fusion. The simplest examples o f syntactic operations are propositional operations; in fact, they are so simple that they are quite misleading, so we begin our examination o f syntactic operations with the notion o f quantifier, 9.3. Definition. Let q be an ira, n)-ar3, syntactic operation on ~d* and let 1 <_ k -< m. We call q a k-ary quantifier (of type (re7, n)) if for all x = (s 1 , . . s m , ¢I- • • ¢n) ~ D(q), the symbols s I . . . s k do not occur in the sentence ~ = q(x). If, ~n addition, x E D(q) implies that each o f the symbols s ~ . , . s k is a coi~stant symbol, then q is a h--ary quantifier

on indivkh~als. 9.4. Example. Add to the syntax of ~2ww the formulation rule: i f ~ is a formula and s is a symbol c f L then ~ s ~ is a formula. Define ~l ~ 3s ,# iff 92 t= ~ for son:e expansion 92 = ( ~ sg~ ) o f ~L This is a 1-ary quantifier of type (1,1), It is not a quantifier on individuals, sil, ce D(q L) = L × L*. This extension o f ~2wo~ is ~me version of second order logic. 9,5. Example. Add to ~2ww tile formation rule: l f c is a constant symbol, < a binary relatioa symbol o f L and ~p a formula then ~1arb. lrge (<) X ~p (x)

is a formula. Extend the semantics by defining ~IN 3 arb'trge<<)x ¢(x)

iff

~NVy3x(y
This is a 1-ary qunatifier q on individuals o f type (2,1). D(q L) = {(c. < , ¢): c a constant symbol o f L, < a binary relation symbol of L, ¢ ~ L* ) .

9.6. Example. Lindstrom [8] gave us a simple way to introduce new quantifiers on individuals. Let q( be a class o f structures closed under isomorphism for some finite language. Suppose, to simplify thee definition, that the structures in q( have the form ~ -- , where U is unary and R is binary. We may use c~ to define a new 3-ary quantifier a ~ o f type (3,2) by using the following definition o f satisfaction:

240

(1)

K.£ Bar~dse, Axioms for abstract model theory

'~

lick XlX2X 3 [~(Xt); ~'(X2,X3)]

if and only if the structure (/t4, U, R) is in q(, where M is the universe o f U= {a~M: ( ~ , a ) P ~o(ct )} R = {(a, b): ( ~ , a , b ) ~ qKc2,c3)}. We list some special cases of this below. (a) The existential quantifier corresponds to llcK when 9( is the class of all structures of the form (M, U), where U is a nonempO, subset of M. (b) The quantifier "there exist uncountably many" is licK when 9( is the class of all structures (M, U), where U is an uncountable subset of M, (c) The Henkin quantifier. Let 9(o be the class of all structures of the form (M, R, f, g), where R is 4-ary.f, g unary, which satisfy the axiom

v x vz R(x, f(x), z, g(z)) . Let 9( be the class of all reducts of structures in 9(0 to the language (R}. The quantifier 09( x, y, z, w ~ x , y, z, w) is usually written as

Vx 3y ~(x, y; z, w). Vz 3w

$1 is a model o f this sentence if for every x there is a y and for every z there is a w depending only on z such that ~(x, y;z, w). This is an example of a 4-ary quantifier on individuals. It cannot be expressed within ~ 2 ~ . A moments reflection on the Henkin quantifier shows that there is a serious deficiency in the above method o f introducing new quantifiers. Surely if one permits the Henkin quantifier in a logic, one also wants to be able to express, in the same logic,

VxEU 3yEU Yz~U

3w~u~(X'Y;Z'W)"

In other words, one wants to be able to relativize the new quantifier, s s One might also want to express

Vx~X 3y~" Vz~Z Bw~W~(x'y;z' w) but this is not as essential for the p u l p o ~ s of this paper. It does stt~est an even more general definition of a quantifier Q|9( ] ~ased on a class ~ of structures, though.

K.J. Barwise, Axioms for abstract model theory

241

This suggests a more general definition c f O cK which we write O(cK ). Let cK be a class of structures as before of the form ~).~ = (M, U, R >, where U ~ unary and R is binary. The quantifier O( c~ ) is a 4-ary quantifier on individuals o f type (4,3) with satisfaction defined by: (2)

9"r~t= O(°tC) XoXIX2A'3 [0(N0): ~P(xI); t.0(N2'N3)]

if and only if the structure
Proof. This is just a special case o f the definition o f syntactic operation since a is an embedding. In situations like that described in this proposition we o f t e n write q x 1 . . . x k ~o(x I . . . x k) for q ( c I . . . c k , ~p(cI . . . ck)) to emphasize the

242

K.J. Barwise, Axioms for abstract model theory

f a c t that the constants c 1 . . . c k no longer occur in q ( c ~ . . , c k , ¢ ( c l . . . ctc)). We n o w turn from quantifiers to the simpler notion o f propositional operation. An n-ary propositional operation on ~* is a total (0, n)-ary logical operation with the property that the truth o f ~ ~ r(~pl... ~o,~) depends solely on the " t r u t h values" o f ~ ~ ~! . . . . . ~ ~ ~on ; not on ~ or the ~o1 . . . ~on . Before making this precise let us give some examples o f ( 0 , n)ary logical operations which are trot propositional operations. 9.9. Example. (i) Define ~ ~ r(~0) iff ~1P ~o and 'JR is uncountable. (ii) Define 2R ~ r(~o, ~k) iff every relation symbol occurring in ~o occurs in ~k. (iii) Define ~l ~ r(~o) iff 92 ~ ~o for some 92 c_ ~ . Let t, f be the " t r u t h values" ! and 0. An n-ary tnlth f u n c t i o n is a function p : {t, f}n -~ {t, f}. A (0, n)-ary logical operation r is determitred by tile truth f i m c t i o n p if for every L, D(r L ) = (L*)n and for every L-structure ~ ~er(cp 1...¢,z)

iff

p(e I . . . . .

%)=t,

where e i = t if ~ ~ ~oi, e i = f if ~ ~ ~oi. An n-ary propositional operataon is a (0, n)-ary logical operation determined by some truth function. A binary propositional operation ^ is a conjunction for ~2" if it is determined by the following truth function: p^ (el, e 2) = t

iff

e I = e 2 = t,

in which case we write (~p ^ ~k) instead o f A(¢, ~k). A unary propositional operation -I is a negation for ~d* if it is determined by the following truth function: p_l(e)=t

iff

e=f.

The theory o f propositional operations is not very rich. 9.10. Lemma. Every truth f i m c t i o n can be built up f r o m the following tWO:

pT(e) = t

iff

p^(e:,e 2)=t

e=f , iff

e 1 =e 2 =t.

Proof. Well k n o w n to undergraduate students o f logic.

K.J. Barwise, Axioms.for ~'bstract model theory

243

Hence if a logic ~2" contains conju~ation and negation, then for every truth function p, there is a propositional operation on ~2* determined byp. 9.1 1. Proposition. Let r be a propositional operation on ~2" and let L ~ K. For any ~ t ' ' • ~°n ~ L* h't ~ = r(¢ 1 . . ~o .n ),. 0 =. r(~p~ . ~0n~), and let ~ be an ~-invertible K-stnwtttre. Then ~ ~ ~b~ i f f ~ ~ O. Proof. Let e i = t i f f ~ ~ ¢~ and let d i = t iff 93~-~ ~ ~Pi. By proposition 7.5 (i), e i = d i for all i, s o p ( e l . . , e n) "-'-p(dl... d n) for the truth function p which determines r. Thus, !lY/P ~ka iff 9.~-,~ ~ ~ i f f p(d 1. . . d n) = t i f f p ( e 1, .. e n) = t i f f ~)~ ~ 0. This result does not assert that ¢)a and 0 are logically equivalent, only that they are equivalent on a-invertible structures. A sentence o f the form "1~ shows that we can't do better since ("l~p)'~ asserts, among other things, that the structure in question is ~-invertible, wlfile "l(~p~ ) can hold simply because the structure is not c~-invertible. § 10. Some prelhninaries or, partial isomorphisms In this section we review some p:eliminaries on back and forth arguments and ~**~. We refer the reader to the survey paper [4] for ally proofs not D~'en here, even when the proof has to be slightly modified to fit in witE the slightly different way we treat atomic formulas here. Let ~ and 92 be partial structures for a language L and let F be a one-one fu~-ction maFping the universe M o f ~ onto the universe N o f ~ . F is an isomorphism o f ~1l and ~r~if it is an isomorphism when the partial functions are treated as relat;ions. For example, if k = (R, g ) , where R is binary and g is anary, i f ~ = (M, R, g) and 92 = (N, R', g'), where g and g' are partial functions, then F is an isomorphism if for all x, y ~ M: R(x, y ) i f f R ' ( F ( x ) , F ( y ) ) , x ~ domain (g) i f f F ( x ) ~ domain ( g ' ) , i f x E domain (g), then g'(F(x)) = F(g(x)). The last two clauses could be e.':pressed by saying (x, y ) E graph(g) iff (F(x), F ( y ) ) E graph(£").

244

K.J. Barwise, Axioms for abstract model theory

Let ~ be a structure for a language L. For any subset M0 o f the universe of ~/, there is a partial substructure of ~2 with univer~ M0 , obtained by just restricting the functions and relations t,a M0 . Let ~ and 9/be structures for the language L. A partial morphism of to 9/is just an isomorphism F : ~ 0 ~9/0 for some partial substructures o f ~ and 9/, respectively. 10.1. Def'mition. Let I be a set of partial isomorphisms from ~ to 9/. We say that I has the back and forth property and write I : ~ff/~-p 9/if for every F E I and a E M (resp b E ~ there is a G E I with F c G and a ~ domain (G) (resp. b E range (G)). We write ~ ---p 9/, and say that ~ / a n d 9/are partially isomorphic, if there is an I : ~ ~p 9/. This relation - p is an equivalence relation which will prove important in the following result, which Grandy has pointed out is due to Bertrand Russell, not Cantor. 10.2. Theorem. Let ~ and 9 / b e countable (or countably generated) similar structures. Then ~3l ~_ 9 / i f f ~ ~-p 9/. In fact, i l l : ~ ~- p 9 / a n d F 0 ~ I, then there is an F : ~ ~- 9t with F 0 ~ F. The following result is due to Carol Karp. 10.3. Theorem. Given structures ~l~ and 9/ for the lang,age L, the following are equivalent: (i) ~ - ~ , ~ 9/ (ii) ~ ~ 9/, (iii) There is an I : ~ ~p 9/where each F ~ I has domabl a substructure o f ~ and, hence, range a substructure o f g/. The usual method of assigning quantifier ranks to atomic formulas is not compatible with our use o f partial substructures and is too crude for some of the results in Section 11.3, in particular LindstriSm's Theorem (3.4). For example, the usual m e t h o d assigns quantifier rank 1 to the formula vx (vn<,~ s ( s ( . . . ( o ) . . . )) = x ) . n

However, if we think o f S as a relation symbol and rewrite the above as

Vx (V. < ~, S y l . . "Yn-~ (S(0) =Yl ^ S ( Y l ) =Y2 ^ . " ^ S ( : " , - I ) - x ) ) ,

K.J. Barwise, Axioms.tbr abstract mode' theory

245

it turns o u t to have quantifier rank ¢0 + 1. We wish to define quantifier rank o f atomic formulas so that the two expressions both have quantifier rank co + 1. F o r this we need to define the notion o f s u b t e r m o f a term. If t is a constant symbol, then t is a subterm c f itself. The subterms o f f ( t I . . . ~, ) consist o f this term itself and any subterm o f any of the ti. 10.4, Definition. The quantifier ra~zk o f a sentence of~o is defined inductively as follows. (i) If ~o is a atomic formula il~ which no function symbol ( o f arity > 1) occurs, then qr(~p) = O. (ii) Suppose ~0 is an atomic sentence o f the form (t I = t 2) in which function symbols occur. Let n be the number o f terms which are subterms o f t 1 or t 2 . Then qr(¢) -- n - 1. (iii) Suppose ¢ is an atomic sentence of the form R ( t I . . . tn). Let n be the number o f subterms o f any o f t 1 . . . t n . Then q r ( ~ ) = n. (iv) For more complicated formulas, qr is defined b y qr(-I~) = qr (~o), q r ( ¥ x ~ x ) ) = q r ( ] x ~(x)) = qr(~(c)) + 1 , q r ( A ~ ) = q r ( V ~ ) = sup(qr(~o): ~ ~ ~ ) . Given an ordinal ~ we let L~,~ denote the subset o f L~,~ o f sentences of quantifier r~.nk <_ a. L~w is closed under A, V, -1 but not under 3 and V unless c. is a i!mit ordinal. I f ~ >_ co, then L~,~ contains all atomic sentences. The following refinement o f Theorem 10.3 is also due to Karp. 10.5. Theorem. Given structures 'JR, 71~f o r the language L and an ordinal a, the following are equiva!ent: (i)Eq -~ ~d~ (ii) There z ~ sequence

ZoZl, where, for each ~ < c~, I~ is a nonempty set o f partial isomorphism between partial substructures o f ~ , ~It suct,, that if[3 ÷ 1 <_ a and F E !~÷ 1, then f o r each a ~ M (resp. b ~ iV) there is a G ~ I~ with F c__ G and a domain(G) (resp. b ~ range(G)). Let L be a language. We wish to define, for each ordinal ¢~, each L-structure ~ , and each sequence a = a I . . . a n ~ M a sentence

K.J. Barwise, A x i o m s f o r abstract m o d e l theory

246

o ~ ,a) ~ L(c~... cn)**,o. We do this by induction on

c~:

o °( ~ , a ) = A (V: ~ an atomic or negated atomic sentence of L(c~... cn) , qr(V) =O, and ( ~ , a ! a ~ ~} For any t~, -,~'(~~+,~a ) is the conjunction of Ot

o(~

,a)'

^

a~M

1

Vx.+l ~VM

o

(~,al...an,

a)~Xn+l]'

°~,~l ...o,,,,0 [ e"+! ~, W', + 1 !

For limit ordinal ~ > 0,• o h( ~ , a )

is the conjunction ~
t ra( ~ , a ) "

10.6. I.emma. For each ~ , a 1 . . . a n and a, (i)q K o ~ , a)) = a, (ii) (• ~ , a 1. o . a n ) ~ trS~ '~ , a). , (iii) (ffl> ot then o ~ , a ) implies o ~ , , ) , (iv)/.f (91, h i . . . bn) ~ o ~ , a ) then for every a ~ M there is a b ~ N s u c h that(91, bl. . • b n ) b ) ~ o~,a~...an ' a ) ) (v) if(91, bl . b n ) ~ 0 t(x~+,l a ) t. n- e n f o r e v e r y b ~ N t h e r e i s a n a ~ M such that (91, b 1 . . . bn, b) ~ o ~ , a ~ . "'an'a ). .

.

10.7. Theorem. Given L ( c l . . . c n ), structures ~ , ~, 9t ~ o~ (tf

~ - ~ , ~.

The cardinality I~olof a formula ~0of ~2~0~ is, by definition, tile cardinality of the set sub(~0) of subformulas of ~. The set sub0p) is defined by sub(~o) = (~o} if~0 is atonfic, sub(~) = {~}U sub(V) i f ~ i s Y x V, 3x V or-IV, sub(9)={9}O U sub(V) i f ~ i s A ¢ o r V O . We define ~d~ to be the sublogic of E**~ with i~ol< ~: for all ~, for all cardinals tc _> to. Let :10(A) = ~, 11c,+1(;~) = 2 a~tX)and aa(h ) = supa<# ~lc,('A)for limit/L Let a , = as(O). In reading the next lem1,~a note that, by our definition,

to= ~ =~0

"

K.Z Barwise, Axioms ]br abstract model thcorp

247

10.8. Lemma. L e t ~ = :1~ a n d iLl < ~:. For each a < ~ there are less than x sentences o f the f o r m o(~ ~ , a)' as ~3~ ranges over all L-structures, a n d each o f these s e n t e n c e s / n L~,~ . Proof. First examine the p r o o f for ~: = co. The result can be seen by a routine induction on a < co. (It was for this lemma that we defined qr as we did on atomic formulas so that if L is finite, then o ° ~ a) ~ L~,~ ) For ~ > ~ let k = ILl + ~0 and prove, by induction on a < t~ that there are _< :la + 1 (X) such o(~_~, a) and each o f them has -< '~,~+ 1 CA) subformulas. The following, while not used later, does give us the crucmi hint needed for the prcofs o f Section II.3. 10.9. Corollary. L e t ~ = ~1~ , iLl < ~: a n d let ~ ~ E~®w f o r s o m e a < ~. Then ~o is logically equivalent to a s e n t e n c e o f ~2~ o f the f o r m V o~ t , i~= _t

where Iit < e.

Proof. ~ is logically equivalent to V { o ~ : ~ ~ ~o), by various results above, and there are < x such sentences o ~ . 10.10. Corollary. I f L s finite, then ~ =~2o~ 9t implies ~ --E~,o 92 f o r all L-structures ~ a n d 9L Proof. If ~ ~ ~w 9~, then there is an a < co such that ~ ~ ~d~w 92.

248

K.Z Barwise, Axioms for abstract model theory

Part 11. SOME ABSTRACT MODEL THEORY {} 1. Up from mythology One of the abstract model t h e o r y ' s lesser functions is to turn mythology into theory. We illustrate this function in the present section. The results serve as useful warm-up exercises in the use of the machinery set up in Part I. We assume throughout this section that E* is a logic on a category C of languages, that ~* contains all atomic formulas and is closed trader conjunction, negation and the existential quantifier. All languages are assumed to be members of e. The simplest bit o f mythology that is not completely trivial is the fact that the interpolation theorem implies Beth's definability theorem. The reader should verify for himself that the standard proof goes through with our axioms. If ~0(R) ~ L(R)* is tile sentence implicitly defining R, one uses first the morphism ¢t :L(R)--~ L ( R , R ' , c) which is the identity on L and takes R to R'; c is a new constant symbol. Both ~o(R) and ~0(R') (=~*(~o(R))) are in L(R, R', c)* and hence so is the logically valid sentence (~(R) ^ R(c)) -~ (~(R') -~ R'(c)) . Interpolation gives a sentence ~(c) ~ L(c)* implied by the left-hand side and implying the right-hand side of the above. If we now apply the morphism /$ : L(R, R', c) -~ L(R, c) which takes both R and R' to R we obt~,ia the logical validity o f ~ R ) ~ JR(c) ~, ~(c)]. Applying the universal quantifier gives the desired conclusion: ~o(R)-~ Yx [ R ( x ) ~ ~ ( x ) ] . More complicated results involve more complicated interpretations. 1.1. Definition. (i) Structures ~I/and ~Rare similar if they are structures for the same language L, i.e., if d o m ( ~ ) = dom(gD = L. LetCg be a class of similar structures, say for the language L. We say thatCg is ~,*definable if there is a ~o~ L* such that

K.J. Barw,:~e,Axioms for abstract model theory

249

We say that ~ is ~] in )2* :f it is the class of relativized reducts of some ~2*-definable class, i.e., if there is a 0-morphism L ~ K which is the identity except for the possibility that a(¥) ~ V, and an t2*-definable classg(' of K-structures such that every'il.R ~ 9(' is a-invertible and

If there is a ~11 class cKl of L-structures such that ~ ~ c~ i f f ~ ~ 9( 1 , forallL-structures ~l, then cg is II[ in ~.*. I f ~ is both Z[ and rl], then 9( is A[ in E*. (ii) A structure ~ is definable, E~, II[ or A[ in ~2" if the class 9( = (~: • similar to ~ and ~ -~ ~ } is definable, ~1, II11 or A I . (iii) A set or class A of ordinals is definable, ~ , stntctures 9(= ( ~ : ~

etc., if the clas~ of

( a , < ) for some a ~ A}

is definable, Z~, etc,, in ~d*. In logics where one does not have the downward L 6 w e n h e i m - S k o l e m theorem, ~[ classes play a more important role than PC classes (classes of reducts, rather than relativized reducts). Modulo finit, structures, the two notions are equivalent for ~2~,~. 1,2. Theorem. The following are equivalent: (i) The collection o f finite sets ~s Y,~ definable in ~2", (ii) (ca, < ) is Y,~ defin.able in ~2" . (iii) There is a class A o f ordinals, Y,] in ~2", which contains a~i infinite ordinal. Proof. Tile implicatioa (ii) = (iii) is trivial. We first show that the following consequence (i') of (i) implies (ii) and then show (iii) =, (i). So assume (i') There is a classq( of finite sets, E[ in k!*, such that no n < ca is a bound to IMI for M E oK. Let ¢ be an ~2" sentence in which the unary U occurs such t h a t M ~ 9( iff M = U~ for some ~ ~ ¢. For each n < ca there is an 92n ~ ~ with IU ~n t >- n. We may (by the Isomorphism Axiom) assume that each

250

K.J. Barwise, Axioms [o., abstract model theory

U ~n is actually an initial segment o f the integers. Let 9~,~ be the indexed union o f the 9/n , n < co, and let < be the natural ordering on co, a subset universe o f ~tto. Let a, d be a l - m o n o m o r p h i s m , L~ -~ K and let K 1 = K(W, < ) where W is unary and < is binary. Write ¢ (d) f o r ~ , U' = a (U). The K 1 -structure (ff/o~, to, < ) is a model o f the conjunction of the following K~-sentences: "< is an infinite linear ordering o f tie' Yx [It(x) --* 3 y (~o'(y) ^ ~ (z < x - - * / f ( z , y ) ) ) ] . (To see that (9~o, to, < ) is a model of the second sentence, given x ~ to let y be large enough so that x ~ ~ly .~ Now let (if/, W, < ) be any model of ~k. We need to see that
(W ~, < ~ ) :~ ( ~ , < ) , Let ~ ~ L¢ (U)* be the conjunction o f ~ and

v x (U(x) -~ W(x)), Yx vy [U(x) ^ y < x -~ U ( y ) ] , Yx [ U ( x ) ^ ~ y ( y < x ) - * ~ y ( y < x A V z q ( y < z < x ) ) ] . Then M is finite i f f M = U ~ for some 9~ ~ ~o. Of the exanmles in Section 1 (to, < ) is definable in all but ~ ~2(Q1 ), where it is not Z 1 definable.

and

1.3. Lemma. (w, < ) / s II~ definable in ~2". Proof. The second order Peano axioms show that (co, < ) is II~ definable in ~,,.,~ ,.hence in ~d*. An ordinal a is ~*-accessible i f a ~ A for some class A of ordinals which is ~ in £*. Thus, one consequence o f T h e o r e m 1.2 is that

K,J, Batwi~c, Axioms lbr abstract model theory

251

(to, <) is ~ in E* iff to is E*-accessible. For general a this is not true, as the reader can see by considering the logic obtained by adding the quantifier Otwo) to ~2,~,,,, where WOis the class of all well-ordered structures (M, <). Every ordinal a is ~2*-accessible whereas only certain countable ordinals (a, <) are ~ in ~2". For ~2,o,o the definable and the accessible ordinals coinqide: they are both just the finite ordinals. 1,4. Theorem. Let A be a class o f ordinals El definable in ~2". The class A' o f ordinals <_~for some t~ ~ A is ~. ~ in ~*. I r A is a set and = sup{~: a ~ A ) , the~ ~ is ~2*-accessibIe. Proof. The proof of the first assertion is especially simple. Let ~ be the ~ definition of A, let If, < ' be the new unary and binary relation symbols and let ~k ~ L,,(U', <')* be the col~iunction of

vx (tf(x~ -~ U(x)), "<' is the reduction o f < to U' " This sentence ~ obviously is a E~ definition of A'. Now assume A is bounded and ~ is its supremum We may assume/3 = A by the first assertion. The proof that (iS, <) is accessible is very nmch like the proof of (i) =* (ii) in Theorem 1.2. Let t~ : L~ -~ K be a/-monornarphism with constant d, write ¢'(d) for ¢~, ~ for a(<). Let 0 be the conjunction of the following sentences of K(W, ~ ). where W is unary and -~ i~. binary: " ~ is a linear ordering of I¢"

Vx [W(x) -* ~ y ~ ( y ) ^ Vz [z -~ x -~ R ( z . x, y)])]. We leave it to the reader to check that lbr every ~ ~ 0, ( I4t~ , ~-~ is. well-ordered, and that for some ~ ~ 0,
~ .

By 1.4, the class of ~2*-accessible ordinals forms an initial segment of the ordinals, proper or non-proper. 1.5. Theorem. The class o f ~2*-accessible ordinals is closed under ordinal addition, multiplication and exponentiation. In ]'act, it is closed under all primitive recursive functions on the ordinals.

K.J. Barwise, Axioms for abstract model theory

252

Proof. The proof of [5, Theorem 1.91 goes through in this general case, once you recall that there is a finite T c__ZF such that the standard part of any model of T is an admissible set. ~2" is bounded (bounded by [3) if every E] classA of ordinals is bounded (resp. bounded by/3). All of the logics in Section 0 are bounded with the exception of ~2n. Note that for every ordinal t~, (a. < ) is definable in ~d~.~. Thus ~2**u is bounded, but is not bounded by any [3. 1.6. Definition. If ~d* is bounded by some [3, then wo(~2") is the least such [3. The ordinal w o ( E * ) will play an important role in Section 4. The Hanfnumber of a logic ~d* is the least cardinal ~ such that if a sentence ~ of ~2" has a model power >_ ~:, then it has arbitrarily large models, provided such at¢ exists. 1.7. Theor,: m (HanD. Assume e is a full subcategory o f e**. I f there is a cardinal ; such that IL* ~I< - k for all ~.*-sentences ~, then the Hanf number o f 9.* exists. Proof. This is one of the least constructive proofs in mathematics. It uses the axiom of replacement of ZF in an essential way. Define the following equivalence relation on ~2*-sentences: ~p--. ~

iff

there is an isomorphism L

L¢ such that ~ = ~

Each of the following is routine given the hypothesis cf tile theorem: (1) For all ¢, iL~0t -< X. (2) There are < ;~o nonisomorphic languages L~ in ~. (3) There are ~ X% equivalence classes ~p/~ for 9,*-sentence ~. Line (2) uses (1) and the "fullness" hypothesis. Line (3) uses (2) and the hypothesis that tL*~t < X, for each ~. For each such equivalence class ~p/ --, let t~(¢) be the least cardinal such that ~phas no model of power >- ~:(¢), if such a cardinal exists, x(~o) = 0 otherwise. The cardinal K(~o) clearly depends only on ~ / ~ , not on ~p. Let ~: = sup{~:(~): all equivalence classes ,p ] -,}. By the axiom replacement, ~: is a cardinal. If a sentence ~pof 9,* has a model of power > ~:, then ~:(~p) = 0 so ~p has ~lrbitrarily large models.

.

K.Z Barv.q,e, Axioms for abstract model theory

253

All o f the logics in Section 1 satisfy Theorem 1.7 except ~,0~o, which does not have a Hanf number. The sequence of beth cardinals is d.efined by ~0 = 0, ~~+ ~ = 2 a , "~x = sup~ < x :l~ if k is a limit ordinal (as in Section I. 10). 1.8. Theorem. Assume that the t h m f mtmber e~ o f ~2* exists. (i) ~ = ~ for some limit ordinal Is. (ii) l f (~, <) is ~.*-definabte, then [3 < Is, ]br any ordinal [3. (iii) f f the class o f well ordering i~ definable in ~*, then K = I~. (iv) I r a sentence ~oo f ~* has a model o f power >_ 3,for each ~ < ~, then ~ohas arbitrarily large models. Proof. To prove (i) we need to show that i f k < ~:, then 2 a < ~:. Let ~0be a sentence o f E* which has a m o d e l o f power ~ k but not arbitrarily large models. Let K = L~(U. E) where U and E are unary and bi:aary relation symbols. Let ® be the conjunction of: ,¢(U) , Vx Vy [E(x, y) -~ U : x ) ] , Vx Vv [vz (E(z, x) ,- E(z, y)) -; x = y ] . Now, for any model 3 = (N, U. E . . . . ) of 0, INI _<2 iUI and ~(v) b so O does not have arbitrarily large models. To show that/9 has a model 9~ = (N, E, U , . . . ) o f power ~ 2 n let F : ~ --- ~ ' be given by F(a) = (a), let N = P(M), let U = M; and define (x, y ) ~ E iff. = (a) for s o m e a E y. For symbols s E L define ~(s) = ~ F (s). Then ~2(U) = ~ ' ~ ~ so ~ ,= ~, and 9t has power 2 IMI _>. 2 x . This proves (i). Next we prove (ii) and leave the similar (iii) to the reader. Let ~obe an ~2*-sentence in which only (¥ and) < occt~rs such that UIL < ) ~ ~o iff ()If. < ) _~ (/L <). Relativize ~0 to a unary U and let E L , ( U , / 3 be the conjunction o f

Vx Vy [Vz (E(z. x) o E(z, y)) ~ x = y ] ,

Vx U(.f(x)), v.,: v y rE(y, x) -, f ( y ) < . f t x ) l . The sentence ~0 has a model o f power :le but none larger.

K.J. Barwise,Axioms foe abstract model theory

254

Now, to prove (iv), we use the following simple fact: if(M, < ) is a linear orderin~ with every x ~ M having fewer than ~: predecessors, then M has p o w e r at most ~:. Let us suppose that there is a sentence ~o such that for each ;k < ~ there is a model ~ , = (M x . . . . ) o f ¢ o f power > but that ~odoes not have arbitrarily large models. Then each ~l~ has power < r by the definition o f ~. B.~ the Isomorphism Axiom we may assume M~ is an ordinal < ~. Let ~ : L, ~ K be a 1-monomorphism with constant d, write ~o'(d) for ~p", and let K 1 = K ( < ) , where < is a binary relation symbol. Let ~ be the conjunction of: " < is a linear ordering o f the universe", Yx 3 y [ ~ ' ( y ) ^ " t h e < predecessors o f x are a subset o f

{z: V(z, y)) "], where V= a ( ¥ ) . Let 9~ = (N . . . . ) be the indexed union o f the ~ x , < ~, so that N = to. I f < is the natural ordering o f ~¢, then ( ~ , < ) is a model o f ~k of powei ~:; hence ~ has arbitrary large models. Let ( ~ , < ) be some model o f ~ o f power > ~:. Then some element a0 ~ ~ has >_ predecessors. Pick a b ~ ~ such that (~g~,b) ~ ~p'(d) and the < predecessors o f a 0 are contained in {a ~ ~ : (a, b) ~ V ~ } . Then (2"q, b) - a is a model of~p of power >_ ~, a contradiction.

§ 2. Logics with the Karp p:operty We continue to assume t h a t ~* is a logic on e that ~2" contains all atomic formulas, and that ~d* is closed under conjunction, negation, and existential quantification. We assume in addition, in this and the next section, that only a finite n u m b e r o f symbols occur in an ~*-sentence ~o; i.e., that L~ is finite. Let L be a language in e and let $1,92 be L-structures, We say that and if/are ~2*-equivalent, and write =*92 if 91/and 92 are models o f tl:= ~.me sentences ~ G L*. 2.1. Definition. The logic E* has the ,V,arp property if for all L in e and all L-structures ~ , 92, if ~ ~p ~ , then ~ ~* ~ . We said in Section I. 10 that ~2,~ has the Karp property. There are

K,J, Barwise,Axioms for abstract model theory

255

much stronger logics which also have the Karp property. For example, Keisler showed that the logic ;~(6J) has the Karp property. Barwise [2] shows that every "absolute" logic has the Karp property. In Section 3 we will use the Karp property to characterize certain sublogics of ,,i The next result simws that a great many logics have the Karp property. ~2* has the Lowenheim property if every ~2*-sentence ~ which has a model has one of power _< S 0 . 2.2. Theorem. I f ~* has the Lowenheim property, then it also has the Karp property. Proof. Suppose that ~* does not have the Karp property and let L be a language in ~3 with two structures ~!~0 and ~1 such that ~)~0 - p ~J-~: but such that for some ,p ~ L*, ,3.~0 ~ ~0,

~1~: ~ "q~.

By I Proposition 7.3, we may assume that L = L, and hence is finite. Let U, W be unary relation symbols E a binary relation symbol, p a 2-ary function symbol ("E" is for equwalence, "p" for pairing). Let qJ ~ L(U. W, E, p)* be the conjunction of~0(v), (-Ko)(w) and sentences expressing the following, where we write (x, y) for p(x, y), ( x l . . . x n + 1) f o r p g x : . . . Xn), Xn+ 1 ): (1)

Vx,y, z, w ((x,)') = (z, w) ~- (x -- z A y = W))

(2)

V x y u [ E ( x , y ) ^ U(u) ~ "~w (W(w) ^ E((x, u), (y, w)))],

(3)

Vxyw [E(x, y) ^ W(w)--~ 3u (U(u) ^ E((x, u), 0', w)))],

(4)R VXl'''Xn

YI'''Yn

[E((Xl'" "Xn)'(Yl"'Yn A

:<.i<_n

))

(U(x~) ~ W(Yi))

^ (R(x I • . . x n) ,~ R(y t . . . y , ) ) ] ,

(5)/- '¢Xl...xn+ l y l . ' - y n + I [E((Xl'"Xn+l ) ' ( y l ' ' ' y n + I ) .-~

A

l
(U(xi) ,-, W(yi) )

^ (f(Xl" " "Xn) = X n + l ' ~ f ( Y l " ' Y n ) =Yn+l)] The last two sentences are required for each n-ary R ~ L and n-ary f ~ L. See in this regard problem P5 in § IL5.)

256

K.J. Barwise, Axioms [or abstract model theory

Since ~ o -~p ~J~l, ~J~0 ~ ~0, ~ 1 ~ "]~0, there is a model ~}of ~, hence one of power < 80. If L ~ L(U, W, E, p) is the identity except for 0~(V) = U and L g L(U, W, E, p) except for t~(V) = W, then ~-'~ and ~t--~ are countable models of~o and "t~0, respectively. But 9~- a ---p ~ - ~ so ~l-~ _~ ~-~ which contradicts the isomorphism axiom. The converse of Theorem 2.2. fails since k~,,,,, has the Karp property but not the L6wenheim property. For logics satisfying the interpolation theorem, however, the two are equivalent. ~* has the interpolation property ~ if any two disjoint classesCg0, cg I , each Z] in 32", can be separated by an ~*-definable class qC ~f0 c_c_c~,

cg I ~ cg = 0 .

2.3. Theorem. Assume that ~d* has the interpolation property. Then ~.* has the Karp property if and only if it has the L6wenheim property. Proof. By Theorem 2.2, we need only show that if ~2' has the Karp property then it has the L6wenheim property. The proof breaks into two cases. Case (i). The ordinal w is not ~] in ~2". We shall prove in t h e o r e m 3.2 that in this case every ~oin ~2" has the same models as some 0 in ~ , ~ , assuming ~d* has the Ka"p property, so the result follows. Case (ii). The ordinal to is ~ in 32". But then (to, <) is A~ in U* by Lemma 1.3, hence definable if ~2" has the interpolation property, say by the sentence o of ~d*. Let 0 be a sentence of ~2" which has a model but, for the sake of argument, no model of power <_ ~0. Let L = L,, u L0, and let ~ 0 = (M: for some L-structure ~ = (M . . . . ), 212 ~ o ) , qCl = {M: for some L-structure ~ =
K,.I Barwisc. Axioms for abstract ,~odel theor.

The above p r o o f is an abstract version o f Malitz' proof that ~2 not have the interpolation property.

257

does

§3. O~aracterizing logics with the Kaq~ property Lindstrom characterized ~2~,~ as the strongest logic satisfying the compactness theorem and the Lbwenheim property. The significance of such a result in the search for useful extensions of 52=,~ is obvious. By pulling Lindstr~m's p r o o f apart at the seams we obtain characterizations o f ~2~,o whenever ~c = ~ , in particular, for ~: inaccessible. (Recall, from our definition o f :1~, that co = :1~o.) 3.1. Definition. Given logics ~* and ~# on the same category e of lalkguages, we say t!~at E # is as strong as 52", and write E # >_ 52*, if for every E*-sentence ~ there is an 52#-sentence ~k such that: (i) every symboi occurring in ~b occurs in ¢, (ii) ~.~ P * ~p iff ~.~~ # ~b for ~dl L~-structures . We say that ~ # isstronger than ~*, and write E * > 52*, if 5J# >- E* but not ~2" ~ ~2". If E* ~ E# and ~2# >_ ~2" then we write E* -=- 52#. We assume t h r o u g h o u t the rest of this section that 52* is a logic satisfying the assun~ptions stated at the beginning of Section 2. Recall the definition of wo( 52*) given in Definition 1,6. 3.2. Theorem. Let ~: = "~ >_>-co. The logic" E~,~ !s as strong as ~2" if and only Jf 52* has the Karp property and wo( t2 ) <- to.

Proof. (3) Lopez-Escobar proved that for any regular cardinal k, wo( 52.x~,) -< ( 2x)÷. (A direct proof can be found in [5], If ~: = aK > co, then every ~ K,~ -sentence ,p is an ~d~,,~-sentence for some regular k < ~¢, so the result follows, l f ~ = co, then the result is well known. ('~) Now suppose 52* ~ E~w and let ~0 be a sentence of ~d* not having the ~ m e models as any sentence of 52K,, . We will show that ~: is 52*-accessible. First, note that for each e < ~ there are L~,-structures ~ , ~l such that ~0~ ~a ~ , ~/~ ~0, but ~ ~ "1~, for otherwise ~0 would have K~O . the same mode'Is as V {o~ : ~q ~ ~ ) , a set~tence of 52,,0 by results in Section I. 10. Let L be t'ae language obtained from L~, by adding the following new symbols: tL Wi, W2 (all u n ~ y ) , < (binary), p (a binary function symbol), E (3-ary). We are going to describe a sentence ~ o f L* such that

K.J. Bar~qse, Axioms for abstract model theory

258

(i) if 9~ ~ ¢,, then ( U ~ , ' < 's~ ) is a well ordering, (ii) for every a < ~: there is a model ~ ~ ~k with ( U ~, < ~) o f order type a. It will follow, by Theorem 1.4, that ~: is E*-accessible. The typical model 9~ of ~9 will be of the form (P,; U, <, W0, 14'~. p, E, . . . ) where:

92(wo) -

~l(w~ ), w h e r e a is the order type o f ( U . < ) .

Expiicitly, ff is the conjunction o f (a)--(g) below.

(a) ~o(Wo),

(b) (-I ¢)~w~), (c) " < is a linear ordering o f U " , (d) "p is a pairing function", (e) " i f E ( b , .x', y) and b' < b, then for every a E Wo there is ab ~ Wl and for every b E Ill1 there is an a E W0 such that E ( b ' , ( x , a ) , (y, b~), (OR if E ( b , (x 1 . . . x n ) , ( Y l " " • Yn )) and U ( h ) , then n

i--IA(l¢~(xi) ^ W! (.v,), and

R ( x I . . . . x , ) o R ( y ! • " • Yn ) •

(g)¢ similar to (f) but for function symbolsf. We have to write (f)n and (g)f for every relation symbol R and function symbo! f of L¢. "[his is where we need to have L,~ finite. We are using the same notation regarding (x I . . . x~) as in the proof of Theorem 1.2. Now clearly (ii) above holds, by the assumption on ¢ and Section 1.10. Suppose 9t is a model o f ~, with (/Z < ) not well ordered. Let X c U be a subset with no least element. Let I be the set of partial isomorphisms F from ff/(wo) to ~n,~ ) such that for some b E X, F is given by x 1 ~)'1 . . . . . Yn ~'Yn, where E ( b t (x 1 . . . x,>. O"l yn)). Then I " 9~('~'o~ -~ 9~(w~) which contradicts ~(wo) ~ ~p, ~tfw~) ~ -t¢, since ~* has the Karp property. By letting x = oo (the class o f all ordinals) in Theorem 3.2 we obtair the following characterization o f E,~,,~. (This is a strengthening o f a result in [2] about absolute logics.) 3.3. Corollary. £ a o W is as s t r o n g as k~* i f a n d o n l y. i f ~2" is b o u n d e d a n d has t h e Kar;~ p r o p e r t y .

K,J. Barwise, Axioms ]br abstract model theory

259

At the other end of the line we can let ~ = co in Theorem 3.2 to obtain the following characterizations of 52,o,o, due to Lindstr6m [8, 9]. By taking our approach to logics, however, we have been able to eliminate the annoying exceptions of the finite models. 3.4. Corollary (Lindstrom). If 52* has the LOwenheim property and either o f the properties stated belou,, then ~* -~ 52~,o . (a) (Upwald L6wenheim-Skolem) Ira sentence o f 52* has an infinite model, then it has a tmco,Jntable model. (b) (Countable compactness) I f T is a countable set o f sentences o f L*, .for some L in ~, and if every finite subset o f T has a model, then T has a model. Proof, Since ~* has the L6wenheim property, it has the Karp property by Theorem 2.2. Svppose that ~2" is stronger than 52,o,~. By Theorem 3.2, co is ~*-accessible. Hence, by Theorem 1.2, (co, <) is E] definable in ~2". This clearly contradicts countable compactness. In conjunction with the k6wenheim property, it also contradicts upward L6wenheimSkolem. For if ¢(L~ <) is the Z I definition of(co, <), then its conjunction with ':f is a one-one function with range ~ U" is a sentence of ~* with only countable models. This is probably as good a place as any to discuss tile difference between Lindstr6m's nolion of logic and that given here. In one sense, the only difference is that our logics permit relativization whereas his do not. In a more fundamental sense, however, the two approaches are quite different. Lindstr6m avoids syntactic considerations altogether since he deals directly with classes of structures, rather that the "sentences" which define them. We find this approach unsatisfying on two grounds. In the first place, it seems contrary to the very spirit of model theory where the primary object o f ~tudy is the retationsh;'p between syntactic objects and the stnwtures they define, Secondly, it fails to make explicit that the closure conditions on the classes o f structures (like the formation of indexed unions and its inverse) arise out o f natural syntactic considerations, considerations which seem implicit in the very idea of a model-theoretic language. The main purpose of these notes is to emphasize the importance of syntactic considerations for abstract model theory.

260

K.£ Barwise, Axioms for abstract model theory

{}4. The operation A 7

A logic satisfies the Sousiin-Kleene property if every class of structures qCwhich is .,x] in ~2" is actually definable in ~2". Obviously every logic with the interpolation property alse has the Souslin-Kleene property. Unlike the properties of logics discus~d earlier, there is an obvious way to round out a logic not having the Souslin-Kleene property to one that does have it, Furthermore, this rounding out sometimes produces logics having the interpolation property. It also gives us a way of answering questions like Q3 (in SectionI.1). Let ~2" be a logic on t3~ which contains all atomic formulas and is closed under conjunction, negation, and existential quantification. We want to define A(~2"), the smallest logic as strong as ~2" which satisfies the Souslin-Kleene property. The class cK of L-s~ructures is going to be definable in A( E *)just in case it is A~ in ~2". What we must check, though, is that there is a sensible way to develop the syntax of A(~2") so that the axioms on logics are satisfied. The most elegant ~ way to do this is to define A ( E * ) to be the logic obtained from ~2~,,, by adjoing each of the quantifiers O('x), whereg( is a class of stn~ctures A~ in ~*. It is fairly obvious, with this definition, that A ( E * ) is as strong as ~*, that it is closed under conjunction, negation and existential quantification, and that it satisfies the Souslin-Kleene property. Details can be found in [ 11 ]. Recall, from Section 1.1, the definitions of weak second order logic, w, and the smallest logic ~2(O 0) with the quantifier "there exist infinitely many". Let HYP be the smallest admissible set containing co and let ~dHyP be the admissible fragment of ~2~,~ given by HYP, except that only formulas with a finite number of symbols are considered. 4.!. Theorem. A ( ~ w) - A( ~2(O0))-= ~ HVP" This result was the first description of A( E *) for some logic ~2" not satisfying the Souslin-Kleene theorenl. It was motivated by, but of course does not answer, a question of Keisler as to whether A( E (01 )) satisfies the interpolation theorem. It has the following corollaries. It is not clear who first considered the operation A. We first learned of it from conversations with Friedman and Keisler in the ~ r i n g of 1971. s OuT original method was by means of equivalence clasps of ordered pairs of ~ntenoes. This definition was suggested by M~ko~,vsky [101.

K.J Barwise, Axioms for abstract model theory

261

4.2. Corollary. The logic\~" A( ~dw ) and A( ~2(O0)) satisfy the interpolation

l.,roperty. Proof. By Barwise [ 1 ] (or. more precisely, by [3]) ~2HYP has the interpolation property. 4.3. Corollary [ 121. ~2w and ~2(O 0) do not satisfy the Souslin-Kleene

property. Proof. Let ~ = (w, <), and let L = {<). The set of all sentences of L(O o) is an element of HYP and satisfaction for L(O 0) is HYP-recursive, so the set of all X c. w definable by a formula ¢(x) of L(O 0) is an element of HYP, by Z 1 -replacement. Thus there is an X E HYP not definable by any ~(x) in L(Oo), but ever3, such X is definable by some ¢(x) in LHyP. Thus L e e P, which is A(L(O 0)) by the theorem, is stronger than L(Q 0 ). Exactly the same r r o o f works for ~ w. We now turn to the proof of the theorem, Proof of Theorem 4.1. The l~gic ~HYP is as strong as ~dw and ~ (Q0) since it is easy to imitate the new quantifi:~rs with infinite disjunctions of finite strings cf quantifiers. Since E HYP has the interpolatior property, it also has the Souslin-Kleene property, s~ it must contain A( ~ w ) and A(~(Q0)). To prove the o:her half of the theorem, we r, ust show that e~ery class % of L-structures which is definable by some sentence ~0 of Lnv P is at least A] in L w and L(O 0). For this, a~l we need to know about these logics is that the ordinal ~ is Y.~ -definable and that they c~ntain $~uw" The first part is immediate by Theorem 1.2. We also need some simple facts about admissible sets with urelements which can be found in our forthcoming book "Admissible Sets and Structures". ( l ) For every a E HYP there is a Z l -formt~la which defines a in every end extension of HYP. (2) The satisfaction relation " ~ ~0" between L-structures ~ and sentences ~oof L=,o is A l -definable in KPU. (3) There is a single sentence ~ in the language L(U, S, E) such that is true in every admissible set with a set of urelements, and the standard pa~t of any model of ~ is an admissible set with a proper set of urelements (i.e., satisfies KPU +). Let ~oE HYP be a sentence of Lny P which we want to prove defines

262

K.J. Barwise, Axioms for abstract model theory

a A] class in, say L(Qo), Let o(o) be the finitary X;1 -formula which defines ~pin the sense of (1). The following are then equivalent, for every L-structure ~ : (a) ~ ~ ~o. (b) There is a structure ~ = (~l;A, E) which is a model of ~ and to is standard in ~, and 3a [o(a) ^ " ~ ~ a ' ] .

(c) For every structure 9~ of the form (9.1kA. E), in which to is standard, ~ is a model of Ya [~0 ^ o(x) -> " ~ a " ] .

The equivalence of (a) and (b) gives a ,v] definition of the models of ~o, in any ~2" in which to is ZI-definable, The equivalence of(a) and (c) gives the Ill definition. There is an obvious relativization of the above results to logics such that (to, <, X) is definable. For example, for any X ~ ~, let Qx be the quantifier Q(%), where eg is the class of structures isomorphic to (to, <, X) By simply modifying the ~.bove proof we get the following. 4.4. Theorem. A( ~dto,o(fix)) --- ~'nxP~x), where HYP(X)is the smallest admissible set containing to and X as elements. This has corollaries analogous to Corollaries 4.2 and 4.3. It also has the following as a corollary. 4.5. Corollary. ~dto.to is tile smallest logic satisfying the Sousiin-.Kh, ene property in which each structure o f the form (to, <, X ) i s de.tinable. Proof. Every sentence of ~dto~to is in HYP(X') for some X c_ to. These last two results have been improved by Makows•y [ 10]. § 5. Some open problems As we mentioned in the preface, the most significant task for abstract model theory, at least in the immediate future, is the discovery and investigation of useful extensions of ~dto,o.Some of the extensions which already exist raise interesting specific open questions. The logic :f2~,to(QI) with the quantifier "'there exist uncountably

K.J. Barwise, Axioms Jbr abstract model theory

263

m a n y " is countably compact and has a simple complete axiomatization, but it ,loes not satisfy the Souslin--Kleene theorem, by an example o f Keisler. The following question was asked by Keisler in the fall o f 1971 and is ,,:till open. PI. Does A( ~dw,o(O 1 )) satisfy the interpolation theorem? A related problem, which might help in the solution of P 1, but which is of interest in its own right, is to prove an analogue o f T h e o r e m 4.1 for &( ~2,,,to(Or)). P2, Find a simple description o f &( ~2,o,o(O1)) analogous to T h e o r e m 4 . 1 Admissible fragments La of L~,w have proven to be use.hd extensions of L. Are they also natural, or can they be strengthened to even more useful logics? One characterization aimed at showing they are natural is contained in [2], What we obtain there is a characterization of the class of all admissible fragments in terms of "absolute" logics. For a single specific admissible set, however, this result is not too useful. P3. For which admissible sets A is there a convincing characterization 9 of P.A'? For some admissible sets we know that LA is weaker than other logics associated with A in a natural way. For example, i f A is ~21 -compact and essentially uncountable (i,e., every countable subset of A is an element of A), then the logic ~2A (O (wo)) is also l/l -compact. P4. Let A be an essentially uncountable l~ 1 -compact admissible set. What is the strongest natural logic associated with A which is still ~; l -compact? The questien is still o p e n in the classical ease of weakly compact cardinals. P4 (Special case). Let ~: be a weakly compact cardinal. What is the strongest logic based on ~: which still satisfies the ~-compactness condition (and some other natural conditions suggested by ~:)? For K = w, we can consider Lindstr6m's theorems as solutions to P4 (Special case). For ~: "> to, one needs a convincing characterization of some logic based on ~. Our final problems are connected more with the axioms on logics given here. Several of the results in Sections 2 and 3 (especially T h e o r e m 3.2) suggest that there is an axiom missing in our formulation o f logics. One should be able to prove these results without the assumption that 9 For A of the form tiYP(X), with X ~ to, Let is the smallest logic satisfying the SouslinKteene property in which the structure (to, <, X) is definable, by 11.4.4.

264

K.J. Barwise,Axioms [or abstract model theory

each sentence of ~d'* has at most a finite number of symbol occurring i n it. P5. Find the missing axiom. In order to solve P5 onc: may well have to solve P6. P6. Find syntactic noti, ms of :d*-finite and ~d*-recursive. Presumably, a reasonable logic will have the property that a language L is ~2*-finite if and only if L is the set of symbols occurring in some ~2*-sentence. Experience with admissible sets suggest that a solution to P6 may be important for questions of completeness and compactness. A solution to P6 may well be needed to solve P2 or, for that matter, any of the other problems. Since the writing of the notes on which this paper is based, a number of interesting papers have appeared. In particular, the reader is referred to the three papers of Feferman and the recent paper of Makowsky referred to in the list of references.

Added in proof A final problem, which we forgot to include, is suggested by any number of examples. Consider, for example, ~,~,, (Q0) and ~to~~o- Any reasonably direct proof that ~toto (C~0) < ~dto.to actually proves more in that it gives a natural transformation (in t~e sense of category theory) of the syntax of ~dto,o(Oo) into the syntax of ~o~. ~- Or consider two different formulations of ~ , one with 7, A, 3 'as primitive, one with 7, V, ¥. Any reasonable proof of equivalence will set up a natt~ral equivalence of the corresponding syntaxes. So define ~* c_C_~2~ if there is a natural transformation T of the functor * to the functor # such that for any ~d*sentence ¢, T(~) is in (L~) # and has the same model as ~. Define ~2" ~ ~2# if T can be chosen as a natural equivalence. P7. How can one extend Theorem 3.2 to conclude that ~2" c__.~2#? How ca.,,, one improve Lindstrtim's Theorem (3.4" *o conclude that

References [ l l K.L Barwise, lnfinitary logic and admissible sets, J. Symbolic Logic 34 (i969)226-252. [2] K.J, Barwise, Absolute logics and L=•, Ann, Math, Logic 4 (1972) 309-340, [3] KJ. Barwise, A preservation theorem for interpretations, Cambridge Summer School in Math. Logic, Lecture Notes in blathematics 337 (Springer, Berlin 1973) 618-621,

K.Z Barwi~¢,Axioms for abstract model theory

265

[4} K,L Barwis¢, Back and forth through lnfinitary logic, in: M. Morley, ed., Studies hn Model Theory, M A A (1973) 5-34. [5] K.L Barwise and H.K, Kunen, Hanf numbers for fragments of L~c ~, IsraelJ. Math. (197 i). |61 S. Feferman, Applications of many-sorted interpolation theorems, to appear in the proceedings of the Tarski Symposium. [7] S. Feferman, Two noles on abstract model theory; [, Properties invariant on the range of definable relations between structures. I1. Langu:~gesfor which the set of valid sentences is semi-invariantl~¢ implicitly definable, Fund. Math., to appear, [8] P. Lindstr~m, First order logic and generalized quantifiers, Theoria 32 (1966) 186-- 195. [9] P. Lindstri~m, On extensions of elementary logic, Theoria 35 (1969) 1-11. [ 101 3.A. Makog~ky, Securable. quantifiers, g-unions and admissible sets, to al~pear in the proceeding of the Bristol 1973 summer school in logic. [11| J.A. Makosky, S. Shelah and J. Stavi, A4ogics and generalized quantifiers, to appear. [12] A. Mostowski, Craig's interpolation theorem in some extended systems of logic in: B. van P,ootsetaltr and J.F. Staal, ¢ds., Logic, Methodology and the Philosophy of Science I11 (North-Holland, Amsterdam, 1968) 87-104.