em constructions for a class of generalized quantifiers

Arch. Math. Logic (1992) 31:355-371 Archive for

Mathematical Logic

�9 Springer-Verlag 1992

EM constructions for a class of generalized quantifiers

Martin Otto

Institut fiir Mathematische Logik, Universit/it Freiburg, W-7800 Freiburg, Federal Republic of Germany

Received July 17, 1990

Summary. We consider a class of Lindstr6m extensions of first-order logic which are susceptible to a natural Skolemization procedure. In these logics Ehrenfeucht Mostowski (EM) functors for theories with arbitrarily large models can be obtained under suitable restrictions. Characteristic dependencies between algebraic properties of the quantifiers and the maximal domains of EM functors are investigated.

Results are applied to Magidor Malitz logic, L(Q<~ showing e.g. its Hanf number to be equal to ~l~o(Nt) in the countably compact case. Using results of Baumgartner, the maximal number of isomorphism types of linearly ordered models of regular cardinality is shown to be achieved for theories that admit an EM functor on a typically restricted domain.

Introduction and preliminaries

The classic theorem of Ehrenfeucht and Mostowski [Ehr-Mos 56] for first-order theories with infinite models offers a method to construct models that are generated by embedded chains of indiscernibles of any desired order type. The uniformity of this procedure is best understood in terms of a functorial formulation as presented in [Hodg 84]. Let us adopt the following terminology: For a language S and some subclass ~ C (9, (9 the class of linear orderings,

1. a map F : oY---*str(S) : = {AIA an S-structure} * is called an S-functor on W iff (a) for I e ~g" the S-structure F(I) is generated by I C F(I), and (b) the atomic type of an increasing n-tuple ~ in I, I e f , n e o), in F(I) is

independent of I and the particular choice of ~.

2. With respect to a given logic L and theory T C L[S] an S-functor F for $3 S is an Ehrenfeucht Mostowski functor (EM functor)

* We use boldface letters like A, B, I for structures, A, B, I for their domains

356 M. Otto

(a) for Tiff F(I) IS ~ T for all ! ~ ~'r (b) for L iff condition l(b) above holds when extended to the complete types

in the language L[-S]. This condition will also be referred to as the "adequacy of the functor F for the logic L".

An S-functor is functorial in the sense that any ordermorphic embedding of one element of ~ff into another uniquely extends to an isomorphic embedding of the corresponding images under F. For an EM functor for L these extended embeddings are L-elementary. In particular the functorial aspect gives rise to automorphism theorems, since the group of ordermorphisms of I~ X is thus embedded into the group of automorphisms of F(I).

The uniformity of the model construction achieved through EM functors for a given theory, and the extent to which these models are governed by the embedded chains of indiscernibles, account for the interest in EM functors, for extended logics, too.

Furthermore, suitable EM constructions may serve as substitutes e.g. for compactness properties in questions related to L6wenheim-Skolem theorems or estimates for the Hanf number of a logic.

One may also hope, in certain cases, to reduce the search for non-isomorphic or mutually non-embeddable models to the construction of sufficiently diverse families of linear orderings, from which corresponding families of models can be gained by applying an EM functor suitably chosen. This latter idea gives rise to profound applications of EM constructions in stability theory.

On the other hand, the extent to which a given logic satisfies EM-like theorems (and corresponding automorphism theorems) may be regarded as a model- theoretic criterion in its own right, for the expressive power of a logic. Relevant investigations into EM constructions for extended logics concern L(Q1) [Eb 71], [Flum 72a], and logics with standard part (or related to presentable classes in the sense of [Mak 85]), to be found implicitly in [Flum 72b] or [Ba-Ku 71]. Similar to the last mentioned references, emphasis here is on EM results for theories with arbitrarily large models.

Obviously an S-functor F (on a domain ~{" containing infinite linear orderings) is completely characterized by the atomic S-types r,(}) realized in F(I) by increasing n-tuples from I, I e ~ , n ea~. We call a sequence (r,0})),~ of types coherent iff for n < m < co and ordermorphic map o-:n--* m we have r,((x~(i))i~,) C r,,(}). With this notion it is obvious that any coherent sequence of atomic S-types induces an S-functor on the whole of •. In particular any S-structure A into which an infinite linear ordering I is embedded as a chain ofindiscernibles (with respect to atomic formulae) induces an S-functor F on @. This F is uniquely determined by the condition that for any increasing ~ in J e (g the atomic S-type of ~ in F(J) should agree with that of any increasing/3 ~ I in A, or

(*)

In order to produce an EM functor for a given logic and theory rather than just an S-functor, this procedure has to be applied in suitably prepared circumstances; and this is where Skolemization becomes relevant.

* Here [ ] denotes generated substructures

EM constructions for a class of generalized quantifiers 357

In fact, for the classic result of Ehrenfeucht and Mostowski for Lo,~,, the language S of a given theory with infinite models is extended to an S closed under the adjunction of Skolem functions for existential formulae; the theory itself is then assumed to contain the corresponding Skolem theory, Skth[S,]. Compactness together with the standard Ramsey argument produces a model containing an infinite chain ofindiscernibles. The Skolem properties of that model ensure that (*) forces the induced functor to be adequate for Lo, o, as well as for the original theory.

Correspondingly, the two main issues for our study are the following: �9 The construction of coherent sequences of atomic types in a more general

setting not involving compactness arguments. (The restriction to theories with arbitrarily large models will reduce this to an application of the theorem of Erd6s and Rado.)

�9 The formulation of a suitable Skolemization procedure for an extended logic; this must ensure the adequacy of a functor that is induced by an appropriate coherent sequence of types.

Before actually turning to a class of extended logics that allows to deal with the second issue, let us sketch the combinatorial core which settles the first one.

Lemma 1. Let ~ be a class of S-structures closed under isomorphisms and containing arbitrarily large structures and t I an infinite ordinal. Then there is a coherent sequence of atomic S-types (r,(~)),~, and an accompanying sequence (B.).~, of structures in ~ such that for n ~ co:

�9 t/CB~; �9 For ~.~* in q and any atomic S-formula 49(~c) we have B. ~ q~[-~]/ff ~b(~)~r..

I f 2 is the cardinality of the set of atomic S-types realized in structures that are represented in ~ , it is sufficient to suppose that the cardinality of structures in ~ is not bounded beneath :l~+(Ir/I).**

Sketch of proof. This is a standard argument based on the generalization of Ramsey's theorem by Erd6s and Rado (cf. [Ch-K 90]). By induction we choose r.(~) and simultaneously pick suitable ( B . ) ~ , o ~ . The crucial condition in the process is:

For r,(~) and any ~< "~z+(Itll) there is some structure B in ~ containing a as a subset such that all increasing n-tuples ~ in a realize r,(~).

In passing from n to n + 1, the theorem of Erd6s and Rado is invoked to choose subsequences of (n + 1)-indiscernibles from the given sequences a of the n th step. For reasons of cofinality there is a suitable continuation of our sequence of types among the characteristic types of these (n + 1)-chains. []

We should like to point out that these sequences of coherent types and accompanying structures induce an S-functor F on (9 which is characterized by the following property:

For n E co, I E (9, ~ and c} increasing in I and t/, resp.: n 11 n n

([q] , q ) - ( [ a ] ,a). (1)

In particular, all the F(I) only realize those atomic types that are realized in almost �9 all of the B.. An immediate consequence would be a complete analogue of the theorem of Ehrenfeucht and Mostowski for unbounded presentable classes (in the sense of [Mak 85]) and, in fact, for logics with standard part.

* ".~" is shorthand for "strictly increasing" ** We refer to the ::l-hierarchy starting from no(2)=2

358 M. Otto

Local quantifiers

Our conventions regarding Lindstr6m quantifiers are as follows: Let r always be a relational language�9 For R ~ z let nR denote the arity of R.

A Lindstr6m quantifier of type z is given in terms of a class of z-structures K which is closed under isomorphisms. The corresponding quantifier Q~ binds a

�9 - - - - - - 1 / / I �9 �9

z-family of formulae ~b(z)= ((oR(z, x))R ~, where we take this notation to imply that �9 �9 _ 1 1 1 1 . . . .

the variables m z and x are distinct. The result of Q~-quantificatmn then is

with free variables as indicated. The semantics of that formula is given through the following stipulation:

n R

~b~ [a, a ] } R ~ ) e K . A ~ p [ ~ ] iff ( A , { a e A I A ~ - "~

Notice that this convention does not enforce the relativization property for Lindstr6m extensions. All the considerations that follow, however, are left invariant by the usual procedure of gaining relativization through the adjunction of an extra unary predicate (as outlined e.g. in [Eb 85]).

We shall call a Lindstr6m quantifier Q~ (as well as its defining class K) x-local for a cardinal x iff there is a subclass Ko CK such that:

�9 Ko C str<_ ~(z); �9 K = {A ~str(z)]A ~ A o for some A o e Ko}. Here str 5~(z) denotes the class of'c-structures of cardinality < tc. Notice that the

existential quantifier of first-order logic is l-local. Local quantifiers are existential in the sense that there is an obvious purely existential Leo oo definition for any local class. This existential nature leaves room for a characteristic asymmetry between the quantifier and its dual, which will become apparent later.

To illustrate the definition, we give some examples, specifying K through the subclasses K o in the sense of the definition.

Examples�9 1. z= {U}, U unary,

Ko={(A, Ua)IIAI=Ic, UA=A} , tr o ,

induces the ~c-interpretation of the cardinality quantifier Q;

2. z = {R.}, R. n-ary,

K o = {(A, R~A) [ [A[ = x, R A 3 ~A~"}, x > No,

induces the 1c-interpretation of the Magidor Malitz quantifier Qn; ~A}" denotes the set { ~ A ] a i ~ a j for i< j<n} .

3. z = {R}, R binary, (a) Ko= {(A, RA)I(A, RA)~--(og, <)*}; (b) K o = {(A, Ra)[IAI < ~, (A, R A) ~ (9, not the union of a countable chain of well-

orderings}, x_>_ N i; (c) K o = {(A, RA) I[A[ = X, (A, R A) ~ (9 and there is no embedding of (~:, <) into

(A, RA)}, K >= NO;

4. z = {R}, R ternary,

Ko--{(A, RA)J IAI--x, R A the graph of a binary function on A of finite range}, x > No.


For later use, let us look at the model classes of the formula

~p :=" <linearly orders the universe" ^ -7 Qr(xoxl;xo < x 0

in the interpretations given by the examples 3(a)-3(c). These are: (a) The class of well-orderings. (b) The class of linear orderings for which any subordering of cardinality up to

x can be obtained as the limit of an ~o-chain of well-orderings. We shall denote this class by •o(X), it will become important later.

(c) The class ~(x) of those linear orderings for which any subordering of cardinality x admits an ordermorphic embedding of (x, <). This class will also be used later.

Local quantifiers are singled out here because of their natural Skolemization properties:

Lemma 2. Let QK be x-local of type z and L: = Lo~o,(Qr). L then admits the following Skolemization:

For any language S there exists an extension $3 S, together with an L[S]-theory A[S], such that for a specified unary predicate in S, D say, the following conditions hold:

1. Any sufficiently large S-structure can be extended to a model of A[S] whose D-part is of cardinality < x;

2. From A~A[S] , BCA, DaCB it follows that B-<LA.

This can be achieved with IS] =max(No, Izl, ISI).

Sketch of proof. Fix K o C K as in the definition of x-locality. The idea is to ensure the membership of any definable z-structure in K by parametrizing a K o representative that witnesses this membership (much in the same sense that the usual Skolemization procedure for L~o, parametrizes witnesses for existential claims).

Let the relational language tr consist of the following:

W, D unary, U binary, and R* (nR+ 1)-ary for R in z.

A language $3 S shall be called Skolem closed for QK in L iff it contains two function symbols, f~, m-ary, and gz, (m+l)-ary, for any L[S-]-formula

m n i l r n r l R /" Z(z)=QIc( x ; ~R( z, X ))R~, m~a~.

The Skolem theory for Qr in L[S], QK-Skth(L[S]), is taken to be the following collection of L[S]-sentences:

�9 VxVy(Uxy~Dy), ("" �9 Vw e W QK x ; R * w ~ ^ A Uwxi'~R~, i<nR /

and for Z(~) as above:

�9 V~(Z('~)~--~Wfx'~), �9 u I Ufx~_ isomorphically embeds*

R*fz~'_ into {~lq~R(~,"~)}"), for any R in z.

* Here gx~_ denotes the obvious truncation of gx; the same convention applies to predicates

360 M. Otto

Notice, that in regarding the existential quantifier as l-local we have also introduced a slightly redundant version of a Skolemization for existential expressions in L[S].

W.l.o.g. we now assume the given S to be disjoint from o- and take S to be the minimal extension of Su~ which is Skolem closed for both QK and 3 in L. It is easily seen that A[S] := 3-Skth(L[S])uQK-Skth(L[S]) satisfies the claim of the lemma:

The relevant expansion of S-structures of cardinality above both K and IK o mod ~- J rests on an embedding of a system of representatives for K o modulo -~ in the shape of

(Ux-,(R*x-)R~)x~w. []

From Example 3(a) we infer that locality alone is not sufficient to grant a reasonably general EM theorem. Hence we further specify K as well as QK to be called

1. co-robust iff the complement of K is closed under limits of countable chains, and

2. strictly K-local iff K0 C K witnessing the locality of K can be chosen so as to have

�9 Ko ( str=~(z);* �9 For A e Ko and any A' C A of cardinality K there is a substructure A" C A,

A"~ K with finite A"\A'.

Remarks. �9 For strictly K-local K, K infinite, co-robustness is equivalent to CfK > CO.

�9 K-local quantifiers are K+-securable in the sense of [Mak 75], cf. [Mun 85]. But, whereas K-locality is a stronger condition than ~ +-inductivity, co-robustness is weaker than co-inductivity (or continuity in [Tharp 743).

�9 If Qr is local and co-robust, then L = L,oo~(Qr) has the Tarski property for countable chains.

�9 The status of our examples is as follows: - Examples 1 and 2 lead to strictly K-local quantifiers, - K of 3(a) is strictly No-local (hence not co-robust), - K is co-robust but not strictly K-local for any uncountable K in Example 3(b);

that this class is not strictly Nl-lOcal will become clear from our EM results and the fact that there are linear orderings in ~F(N 1), of cardinality N 1, yet not presentable as countable unions of well-orderings, cf. [Bau 76].

- K in 3(c) is strictly K-local, so is K in 4.

co-robust local quantifiers provide the setting for the following EM theorem:

Theorem 3. Let z be relational and of bounded arity ( i f infinite); let K be ~r and co-robust, ~c uncountable; L: = L~,~(Q~). Then any L-theory T with arbitrarily large models admits an EM functor for L and T on •o(K) (as defined above).

Proof W.l.o.g. TcL[S] for some S with S= S, TDAES], in the sense of Lemma 2; for the specified predicate D of Lemma 2, T has arbitrarily large models whose D-part is of cardinality K.

Hence, for a set of new constant symbols C : = {d~ ] ~ < K), the sentence

~ := A T/x Vx(Dx~\ ~<,~V x=d~)

* str=~(z) is the class o f z-structures o f cardinality tr


admits arbitrarily large models. In applying Lemma 1 to the class of models of 6 we obtain a coherent sequence of atomic S u C-types (r,(~)). ~ ~ and an accompanying sequence of SuC-structures (B,),~o, such that for n~o~:

�9 B.~6; �9 K + C B . ; �9 For increasing ~ in ~+ and any atomic Su C-formula q~(~) we have B, ~ q~ [~]

iff ~b(~) ~ r,.

Let F be the induced SwC-functor on (9. Note that a priori F is not adequate for L. In fact we shall see later that it cannot be adequate for L on all of (9 in general. But,

I r n l B n ~lx as a consequence of Eq. (1) of the introduction, tk~d , ~) can be isomorphically n B m ~ embedded into ([~] , ~) for m > n. Owing to the Skolem properties of B,,, this

embedding is L[SuC]-elementary. Hence

f,(~):={4~(~)eL[SwC][B,~c~[~] for ~/~ in lc +}

provides a coherent sequence of complete L[SwC]-types. By Skolemization, Eq. (1) itself extends to

F(I) L~>" ([~]F(,),~) , B -~ ([~] % ~) "<r B~ (2)

for any I ~ (9 and increasing tuples ~ in I and ~ in tr +, m > n. We now fix I e ~o(~), and reduce the proof to the following

Claim. Let A=F(I). For n~o~, ~ increasing in I and Z(~)~L[SwC] we have:

A ~ Z [~}] iff Z ~ rn0}) �9 (3)

We shall prove this claim by induction on formulae. First observe that, if (3) has been established for all ,p which can be obtained from a fixed X' by substitution of terms, then for any ~ in I the following substructure relation bolds:

In the induction with respect to the rank of z we shall therefore be able to assume (4) for all formulae X' of lower rank than Z.

Of the inductive steps just ~- and Qr-quantification are of interest. As the existential step is canonical (due to the underlying existential Skolemization for L), we concentrate on QK-quantification and assume that (3) has been shown for all

n n R n n R formulae of lower QK-rank than Z(z) : = Q,~( x ; CbR(Z, x ))R~"

Let c~ be increasing in I. Suppose that (i) X(~) e t , .

Then, for any increasing ~ in tc +" B. ~ Z [~], whence (with (2)) [-~]a ~ Z [-~]. (4) for the ~bR, and the locality of Q~: give A ~ Z [~], as desired.

(ii) A ~ Z [q]- Let A* be the structure that Z is about:

n n R A*=(A,(RA)R~,) for RA={"~IA~C~R[q, a]}, R in z.

A* e K, so by x-locality there is some J ' C I, ]J'[ <= ~c such that for A' : ~ [j,]a we have A*]A's K. Using the fact that I belongs to :((0(~) we set

A'~={t('~)A[~eJ'm, t an S~C-term}

for a chain of well-orderings (J')m~o) whose union is J'.

362 M. Otto

By co-robustness there is mo and a well-ordering J C I of order type less than ~ + containing ~ such that for Ao : = {t('~~ [ ~'o~ j , t an Su C-term} we have A* [A o e K.

Any ordermorphic embedding Q:J~(tc +, <) canonically extends to an isomorphic embedding

n R n R ^ . , n q .A [Ao--*(Bk,{ b IBk~C~R[O(q), b]}R~),

provided that k is chosen to be sufficiently large: �9 k>2mo settles the injectivity of ~ by means of (2); �9 on the basis of (2), and (4) for the q~R, the condition k > max (n + monR)

ensures compatibility with the relevant predicates. ~

Hence )~(~) is contained in some rk, and so by completeness and coherence it must be in ~,, too. []

In close analogy we also obtain:

Theorem 4. Let z be relational and of bounded arity; let K be strictly x-local for some uncountable and regular ~ ; L: = Lo, o,(QK). Then any L-theory T with arbitrarily large models admits an EM functor for L and r on X(x) (as defined above).

Observe that the stronger hypothesis on K yields a larger domain for the functor.

Sketch of proof. The only alteration required in the proof of the preceding theorem is the way in which Ao is singled out in the second part of the QK-step. In the situation considered there, we now set, for m e co:

Am: = {t(Y)AlY~ I, t an S~C-term}.

By strict ~c-locality, as witnessed by some fixed K o C K, there is some A' CA with A* I A' ~ Ko and A' C A,, o for some too.

mo For suitable sequences (t~)~<~ of SuC-terms and (j (e)),<~ of mo-tuples in I, the set A' can be parametrized without repetitions as

Without impairing the cardinality of A' we may assume that for i< m o the set {j,(~)[~ < x}

is a singleton or of cardinality x, and

in the latter case is ordered of order type x in I (since I ~ YfOc)).

So finally we gain a well-ordering ,1 C I of order type less than ~c. co, containing ~, and such that for some (new) mo ~ co and

Ao:= {t(3~ ] 7~ t an SwC-term}

we have A*IAoeK, as desired. []

Remark. The generalization to whole families of quantifiers is obvious. Also, the Skolemization of Lemma 2 is easily adapted to yield corresponding results for infinitary versions Lu,o(QK ).

Corollary 5. Let Q~; be K-local, co-robust and of a type of bounded arity. Then the logic L= Lu~o(QK)


�9 is bounded with respect to pinning down ordinals (cf [-Eb85] for the terminology),

�9 satisfies an automorphism theorem for theories of an unbounded spectrum: these have models in any sufficiently larg e cardinality 2 with 2 ~ automorphisms,

�9 has Hanf number less than or equal to "~ +(K) for theories in a language S with [L[S][ <Q, where ;~= 2 ~, v = max(~c, [zl, ~).

Remark. Recall that Jgo(tQ=Mod(ip) for a sentence ~p in a x-local, co-robust extension of L~o~o. The same holds for x4r(~c) and a strictly ~- local extension. Since X0(~) and oU(x) themselves are closed with respect to passage to substructures we have:

If I can be embedded as a chain of indiscernibles into a member of ~{'0(t~) (or NP(~)) then either I itself or I* must be in ~ro(~c ) (or ~X#(~c) resp.). Therefore the domains guaranteed for the EM functors in this generality in the above theorems are maximal. Note the interesting interdependence between algebraic properties of the quantifier and the maximal domains for EM functors.

We now turn to some observations concerning the r61e of co-robustness. Recall that a logic L is said to be [col-compact iffit offers no characterization of (co, < ) as a relativized projective class.

Lemma 6. Let K be local and L= L~o,(Qr). For L to be [col-compact, QK must be o-robust.

(That the reverse implication is false is exemplified by Example 4 above, which for c fx > co yields a logic that is not even countably compact, though ~o-robust.)

Sketch of proof Otherwise let (Ai) i < co be a chain in the complement of K with union A in K. We assume A to be disjoint from co and introduce sets of constants for the elements of A and co, _A={_ala~A} and ~={_nln~co}, resp. For a new unary predicate P and binary predicates W and < set

S:=zw{P, <, W}wcowA. B is taken to be the following S-structure on the set A wco :

(P, <, (_n).~)s := (co, < , (n).~co)

(B[zu_A)[A:=(A,(a)a~a) and WB:={(n,a)]n~co, a z A . } .

It follows that the first-order theory of B together with the L-sentence

)~:=VxeP-nQK x ; R ~ A /~ WXXiR~ i<nR

forces the (P, <)-part to be isomorphic to (co, <). []

Let us say of a local quantifier QK of type z that it is connected iffthe subclass K o of K witnessing its locality can be chosen in such a way that for any A e Ko and a + a' in A there is an R ~ z and an d e R A that contains both a and a'. (This is the connectedness of the Gaifman graph of A.)

Of our examples all but the cardinality quantifiers satisfy this condition. While co-robustness is not strictly necessary in some instances of Theorem 3 (cf.

the cardinality quantifiers) we nevertheless have the following:

Lemma 7. Let Qx be local and connected, of type z, z of bounded arity. Then L = Low(QK) is bounded with respect to pinning down ordinals iff K is co-robust.

Sketch of proof We show that a counterexample (A~)~<~, to co-robustness can be used to turn the class of well-orderings into a projective L-class.

364 M. Otto

Let z* be a disjoint copy of'c; P, W, < , _A as in the proof of the last lemma and set S:='cwz*w{P, <, W}w_Aw{f~lneco} for function symbols f~, (n+ 1)-ary, for

n e ( _ o .

For infinite ordinal t/we construct an S-structure B [q] with (P, <)B" = (t/, <) as follows:

Fix q; let (C, C) be the tree of decreasing sequences in q; for s ~ C let Isl denote the length of s. For s ~ C turn

{(a, s') ~ a N x cls, cs, M =min {nr A,}}

into a z*-structure Ds by demanding the canonical projection onto Ais I to be a (z, z*)-isomorphism.

Set D: = [_) Ds whereby all the D~ become substructures of D. s ~ C

Finally B[t/] I(zuAuz*u2{P, < }) is taken to be the disjoint union of(A, (a),~A), D and (11, <). W is to be interpreted as the relation

{((a,s)(a',s'))~D2lscs ' or s'Cs}

of"belonging to the same branch"; the fn are chosen such that for s = {(i, ~i) 1 i < n} �9 n B in C, [sl = n, the truncation ( f ~ _) ]A~ is just the canonical (z, z*)-embedding of A~

onto D~. The common L-theory of all the B [t/] contains the following sentences, which in

their turn imply that the (P, <)-part is well-ordered: �9 "(P, <) a linear ordering". �9 The elementary diagram of (A, (a),~a). �9 Forn, m~co, a~A~: V"~"~ i n ( P , < ) J,+m ~ ~+'~x _a=j~xa."~ * �9 F o r n ~ o , a , a ' ~ A j g~'-~ i n ( P , < ) Wfnxa_f~xa_." " '

n i t , n rill 111t �9 For n e e ) , R ~ z , a ~Ari: V~'~ in (P, < ) ( R f~x a~-~Ra). �9 And the crucial sentence, relying on the connectedness of QK:

( n~ R*~c~A )Re," --nQK x; A Wxjxj, [] j , j ' < n R

Magidor-Malitz quantifiers

Magidor-Malitz logic L(Q <'~ is obtained from Lo~o~ by adjoining a countable family of strictly local quantifiers Qn = Q r ~ (n E co), which in the x-interpretation are characterized by

K~n): = {(a, R~)[3B C A, IBI = x, RAn 3 ~B~n}.

(Cf. our Example 2 above.) For an overview of the model theory of L(Q <'~ we refer to [Ka 85] and the

original source [Mag-Ma177]. As an immediate application of the results in the last section we have the following:

Proposition 8. Let tr be infinite, L= L(Q 2) o r L(Q < ~') in the x-interpretation. Then 1. L is bounded with respect to pinning down ordinals iff cf ~ > co. 2. precisely for cf x > co it is true that any L-theory with arbitrarily large models

admits an EM functor on o~r(x).

* Here we use the abbreviation :} ~ to mean that :~ is strictly decreasing


Notice the contrast between L(Q) and L(Q 2) displayed in the first part: According to [Lop 66] L(Q) is bounded in any interpretation.

We shall in this section further show that (a) for countably compact L(Q<~ in an interpretation ~ of uncountable

cofinality, the Hanf number is exactly "~o(~); and that (b) there is no EM theorem analogous to Ebbinghaus' result for L(Q) to be

expected for L(Q <~~ in the N~-interpretation.

(a) The countably compact case

For this paragraph we fix x with cf x > co and assume that L = L(Q <'~ is countably compact in the x-interpretation. Let further an L[S]-theory T in a countable language S be given that affords sufficiently large models. W.l.o.g. Skolemization in the sense of Lemma 2 has already been achieved: for the unary predicate D in S we know that T has sufficiently large models with cardinality of the D-part at most x; and from A ~ T, BcA, BDD a, it follows that B~LA.

An analysis of the proof of Theorem 4 shows that the following situation provides the data that are sufficient for the construction of an EM functor for L and T:

For some as yet unspecified set C of new constant symbols suppose there is a coherent sequence of atomic S u C-types (r,0})), ~ and an accompanying sequence of SuC-structures (B,),~o~ such that for n ~ co:

�9 B.~T. �9 {cn"lc~C}3Dn"; �9 1r coCB,; (Note that ~. ~ replaces x+.) �9 For increasing ~ in ~c-co and any atomic SwC-formula q~(~) we have

B. # ~b [~3 iff ~b(#) z r.. Recall how compactness is used in the classic theorem of Ehrenfeucht and

Mostowski to produce a chain ofindiscernibles that simultaneously represents the whole of a coherent sequence of types. The idea here is to imitate that procedure using countable compactness and the special nature of our Skolemization to produce a sequence of indiscernibles where indiscernibility is understood with respect to parameters from the D-part. It is essential for this step that we are dealing with a Skolemization technique that does not rely on individual constants and can thus keep the language countable!

Lemma 9. Let T as above admit models of cardinalities above any ~ < :l,~(~c). Then there exists a set of constants C and an SuC-structure B such that the above four conditions are met for B, : = B, n ~ co.

Proof. W.l.o.g. we assume that the given language S contains a binary predicate < and unary predicate P, for which T already expresses that (P, <) is a linear ordering into which (tc. co, <) can be embedded. This can indeed be done by an L(Q)-sentence: using an extra predicate, it is formalized that there is an infinite chain of intervals of ~c-like order type. We now extend T to T' by the following sentences, for q~(~, X) e L[S]:

V ~ D V~, ~ in (P, <)Vff'/" in (P, <) (6(~,~)~b(~,2')).

Since T was required to admit sufficiently large models with D-part of cardinality x, the theorem of Erd6s and Rado can be applied to show that any finite number of

366 M. Otto

these sentences can be satisfied in models of T. By compactness we have a model B' of T' whose (P, <)-part contains (x. ~o, < ). With C: = {did e D B'} a set of constant names for the elements of the D-part of B', we can set B : = (B'IS, (d)d~DB,). []

As an immediate consequence we have:

Proposition 10. Let L= L(Q <~ be countably compact in the x-interpretation for some ~ of uncountable cofinality.

Then any countable L-theory T whose spectrum is not bounded below ~1o~(~ ) admits an E M functor for L and T on o,~r(x). In particular "~o,(~c) is the Hanf number for countable L-theories.

(b) A negative result

In this paragraph we restrict attention to the Nl-interpretation of the Magidor Malitz quantifiers. We start from an observation of Ebbinghaus that shows that his EM theorem for L(Q) [Eb71] cannot be extended to L(Q z) in any straightforward way: that construction automatically leads to automorphisms with an uncountable set of fixed points. The negative square bracket relation N17~ [N1]~l (cf. [Todor 87] for a proof not using CH) can be formalized in L(Q z) so as to give a sentence which does not admit such automorphisms:

Let f be a binary function symbol and set

4o : = Qxx = x ^ Vz ~ QE(xy; f xy :~ z).

Visualizing f as a colouring function it is obvious that ~b o allows no nontrivial automorphisms that fix uncountably many points. Satisfiability of q~0 is a consequence of the above partition result.

We present two strengthenings which yield sentences enforcing rigidity. One is for L(Q z) and uses CH, the other needs no extra assumptions but is only for L(Q4).

In the case of CH, one can use a sentence of L(Q 2) that characterizes certain rigid Boolean algebras; namely those which have no uncountable chains or antichains while any non-zero element has uncountably many predecessors. The existence of such Boolean algebras, as well their rigidity, is established in [Bau-Kom 81] under the assumption of ~ w ; this assumption can be weakened to CH on the basis of [Sh 81], as is pointed out in [Mild 90]. We nevertheless present a different argument, simply because it offers a common starting point for both the following claims.

Lemma 11. For the N~-interpretation: 1. (CH) Assuming CH there is a satisfiable sentence of L(Q 2) that enforces

rigidity. 2. There is a satisfiable sentence of L(Q 4) with just rigid models.

Proof ad 1: Assuming CH, Bonnet [Bon 80] constructs a subtype of the order type of the reals admitting no uncountable partial ordermorphisms without fixed points. Towards a formalization in L(Q 2) we choose symbols <, p for a binary relation and function, resp., and ZOo, rq for unary functions.

Let ~b~ express the facts that < linearly orders the uncountable universe and that p is a bijective pairing function with inverse maps ~z o and rq.

And choose q~2 : = "-I Q2(XX'; lp(X, X')) where

~p(x, x'):="rq(x), rc2(x), rq(x'), rc2(x' ) are pairwise distinct"

^ ~(x) < ~ ( x ' ) ~ ~2(x) < ~2(x').


The sentences q~ and (D2 together express the fact that < admits no uncountable partial ordermorphisms whose domain and range are disjoint. So q~:=~b0/xq~l^q~ 2 is satisfiable under CH and excludes all non-trivial automorphisms.

ad 2: Using the same language as above, we try to give a formalization of the non- existence of ordermorphisms with few fixed points. This can be done because under ~bo any {f}-automorphism is completely determined by its restriction to any uncountable subset. Whether the formalization using Q4 is optimal, we do not know.

Let ~b 0 and ~b 1 be as above; let q~a say that < is Nl-like and put

~4 := Vz-1Q4(xox~x'ox'~; z(z, Xo, xl, x;, xl)) ! ! .

for Z(z, Xo, xl, Xo, x0 . = (f(niXo, ZhXx)= f(zC2Xo, nzx i )~f ( lhXo, nixl) < z) ! t ^ tO(Xo, xl, Xo, x0 ,

where ' ' �9 l])(X0, X1, X0, Xl) . = 7~1X 0 :# 7~lX 1 A ~2X0 =~ 7~2X 1

A (fOhXo, rqXx)<f(zqx'o, zhx'l)~-~f(rCzXo, rczxl)< f(rc2x'o, 7r2x'l)).

To understand the content of q54, observe that under q~o, q~l, and q~a there is a close correspondence between homogeneous sets for ~p and { <,f}-automorphisms:

If B is homogeneous for h0 then the map

~ = {(f(nlbo, lrlbO, f (r%bo, rczb0) l bo 4= b~ e B}

is a partial ordermorphism with domain f(E~Zl(B)~2). If, in addition, B is uncountable, then so is its image under ~l. Thus, by q~o, f(~gi(B)~ 2) is equal to the whole universe, and Qn is an automorphism.

On the other hand, from any { <, f}-automorphism a homogeneous set for ~v is easily obtained.

One checks that the sentence qY:= ~b 0 ̂ ~Pl/x q53/~ ~b 4 satisfies claim 2. []

Remark. In the above context, consider the sentence

r : = ~1 A ---I Q2((xxt ; 7~l(X ) =~ 7~l(X' ) A (~I(X) < ~I(X')~'+~2(X) > ~2(X'))) �9

It rules out (for the x-interpretation) the existence of partial anti-ordermorphic maps ofcardinality ~. Hence it demonstrates the necessity to restrict the domain of EM functors for L(Q2)-theories with arbitrarily large models to some subclass of (9. We do, however, not know whether the class o~(~), as guaranteed by Theorem 4, is maximal for Magidor Malitz logic.

Many models from functors on Jr(s)

In this section we apply investigations of Baumgartner [Bau 763 to the construction of maximal families of mutually non-embeddable models containing definable linear orderings. As mentioned above this is one of the interesting model-theoretic applications of EM functors. The essential and rather restrictive assumption made, is that the given theory provides a definable substructure which is linearly ordered

368 M. Otto

and into which the EM functor is taken to embed its arguments. (Definability here can be understood to allow for a "small" set of parameters and sets of formulae to define both the underlying subset and its order relation; but we shall in the following assume that there is a symbol < of which the given theory says that it linearly orders the universe, largely for reasons of notational convenience.) This assumption is of course much stronger than e.g. the order property of stability theory. On the other hand, our construction works for functors with domain s/g*:= (~ ~(x), which emerges as a natural domain for EM functors in the

c f K > r

context of strictly local and co-robust quantifiers. The power of having a definable linear ordering lies in the close structural

relationship between the order type of the argument of the functor and the linear ordering of the image model. This is described in the following theorem due to Hodges, [Hodg 69], which can be found in [Char-Pou 83].

Theorem 12 (Hodges). Let F be an S-functor on some infinite ~ff C (_9 which is also an E M functor for the theory " < linearly orders the universe". Let further t be a f ixed n-ary term of the language S, n ~ co.

Then for I t ~ the restriction of <m) to the tm)-image of [I]" is almost a lexicographical product over I itself:

There exist a permutation Q ~ S n and a map sgn : n ~ { + 1, - 1 } such that for any I ~ ~/f there is an ordermorphic embedding f of the above linear ordering into the lexicographic product ~ I sgn~~ where t m) o ~ o f is the identity on the given ordering.

i < n

(Here I § 1= i and I - 1 = I* its converse; products are to be understood with dominant right-hand factor; and ff is the obvious operation of ~ on I".)

We now turn to Baumgartner's investigation of a subclass of ~ff(N 1) [Bau 76]. It serves our purposes because it offers a method of construction which for simple instances gives rise to linear orderings in JY'*; and the orderings thus constructed are associated with characteristic stationary sets. The following are severe specializations of concepts and methods in [Bau 76].

For an infinite cardinal 2 we introduce the following: �9 N~:= {f~(o~.n+ 1)2if(0)=0, f monotone and continuous}. �9 For IcN"~ set C(I) :={ f (co .m) l fEI , m ~ n } . �9 For fEN"z let f[ f l] be the maximal initial segment of f whose range is

contained in fl + 1, and set I [fl] : -- {f [fl] [ fc I} for I C N]. �9 On N~ let ~( be the canonical lexicographic ordering: f - < f ' iff for

a = min {fl =< co. n lf(fl) ~: f'(fl)} we have f(~) < f'(~). The linear orderings considered will be subsets I CN~ for some n, linearly

ordered by ~( : we take I to denote this ordering of I. The essential result from [Bau 76] is the following:

Theorem 13 (Baumgartner). Let 2 be uncountable and regular. For i = O, 1 let ni ~ co and I, CN"a ~ be given with II,I =2 and II,[fl]l <2 for all fi<2.

I f there exists an embedding a:Io--+I 1 then C(Io)\C(I1)=O, i.e. this set is not stationary in 2.

For the proof see [Bau76] where, however, higher generality requires additional prerequisites.

Let us now fix 2, uncountable and regular. Specializing further, we shall call I C N"~ simple iff the following conditions are

met:


1. I is countable or 111=2 and lI[f l ] l<2 for f l<2. 2. For any tc of uncountable cofinality, I is in OF(x). For the following we fix, for any ~ < 2 of cofinality co, an element f , ~ N~ whose

limit is ~, i.e. with f~(o))= ~. It is easily checked that for any CC{c~<2[cfc~=og} of cardinality 2 the set

Ic:= {f, l a~ C} is simple and has C(Ic)= C. Our use of simple ! rests on the following

Lemma 14. For i=0 , 1 let IiC N]' be simple, ni e co. Then there is a simple ICN~ ~ for which

I"~I1 . I o and C(1)\C(IouII)=_C(IowI1)\C(I )=_0.

Sketch of proof. Let @:N~~215 N~I~N"~ ~ be defined according to fo@f~ := You{@. n0 + c~,fo(cO, no)+A(~))I~__< ~o. n,} and set I : = {fo@fl[f~ ~ Ii for i=0 , 1}.

The projections which invert | on I are readily used to verify the claim of isomorphy. The first condition for the simplicity of I is an immediate consequence of the simplicity of the factors. Membership in the relevant OF(~c) follows from isomorphy with the product ordering and the fact that the classes oF(x) are closed under products.

As the other congruence is obvious, it remains to show that C(I)\C(IowI 0 is not stationary. But C(I) is contained in {?o+7112h~C(Ii) for i=0 , 1} and any stationary subset could therefore be listed without repetitions in the form (7o(C0+?~(e))~<~, where for e < 2 we have yi(a)e C(Ii)\{0}. An application of Fodor 's lemma to the regressive projection onto C(Io) , which would be induced by this representation, yields a contradiction. []

We further observe that the order type of the rationals, t/, can be represented by a simple set in No,+~+a~ "

I~: = {~(co, co + 1 . . . . , o) + ~o) t 1 < n < c~, ~ increasing in co, So = 0}

offers a canonical representation with C(I,) = {0, ~o + e~}. Forming products with t/ therefore leaves the characteristic set C invariant with respect to =. We are now in a position to prove the following:

Theorem 15. Let S contain < ; let 2 > IS] be regular and uncountable. I f F is an Ehrenfeucht Mostowski functor on ~f* : = (~ OF(~c) for "< linearly orders the universe", then the family ~f~>o,

(FO)),+~.. I,1 =~

contains a subfamily of 2 ~ many mutually non-embeddable S-structures of cardinality 2.

Proof. Let {a < 21cfa = co} be split into 2 many disjoint stationary subsets (Dp)a < of 2 according to S olovay's theorem. Let further (E,)~ < 2, be a family of subsets of 2 that form an antichain with respect to C.

Put C~ := U Da and I~ := Ic. for a < 2 )'. Then (F(I~))~ < 2* witnesses our claim.

Indeed, it suffices to show that the existence of an embedding Q of Io into F(I~)I { < } for simple sets I~ C N 1 implies that C(Io)\C(I~) =_ O.

The latter is seen as follows: Since fragments of lesser cardinality are irrelevant; since there are less than 2

many terms in our language; and since equinumerous subsets of simple sets are

370 M. Otto

simple themselves, we may w.l.o.g, assume that for one part icular n-ary term t of our language we have Q(Io)Ctrul~([I1]n). So by Hodges ' theorem, Theorem 12 above, there is an embedding of I o into some 17I I~ gn(~ for suitable

i<n

s g n ' n ~ { + l , - 1 } . Using the fact that both Io and 11 are elements of of(N) converse factors in this product can be eliminated:

Restricting attention to the pre-images of any fixed point in 1-I I~ gnu~ it is io<i<n

seen that the ito h projection of the image of this set under 0 must be countable if sgn(io) = ~ 1; whence in this case it is possible to replace the factor I] g"(i~ by I,.

We conclude that Io can be embedded into I , . I ] (11" I,). Since the latter i<n

ordering can be obtained from a simple 1ON 2n§ whose characteristic set is congruent with C(I1) by Lemma 14, we conclude with Baumgartner 's theorem, Theorem 13 here, that indeed C(Io)\C(It) is not stationary. []

Corollary 16. There are no theories of linear orderings in a logic built upon strictly local ~o-robust quantifiers, which put a non-trivial bound to the number of isomorphism types of models of sufficiently large regular cardinalities.

Acknowledgements. The work presented here is part of my Ph.D. thesis at the University of Freiburg, which has been supervised by Professor H.-D. Ebbinghaus. I am indepted to Professor Ebbinghaus for his advice and numerous suggestions which have been fundamental for this work.

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em constructions for a class of generalized quantifiers

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