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Under set-theoretic hypotheses, it is proved by Magidor and Malitz that logic with the Magidor-Malitz quantifier in the Ki -interpretation is recursively axiomatizable. It is shown here, under no additional settheoretic hypotheses, that this logic cannot be axiomatized by finitely many schemata. Magidor and Malitz [2] introduced the ^-variable-binding quantifiers Q. The language L(Q") is formed by adding Q to first-order predicate logic. For an infinite cardinal K, QnX\X2.. .xnφ may be assigned the so-called /c-interpretation in a structure ΐPίί, wherein QX\.. .xnφ is satisfied if there exists a n A c ΐPίί of power K that is homogeneous for φ, i.e., for any au... ,an E A, φ(ctι,... 9an) holds in 9K. Among many other results Magidor and Malitz establish, under the set-theoretic axiom 0K l, a completeness theorem for L(Q") in the Kt-interpretation (hereafter LίQ^)). Unfortunately, the complete axiom system for L(Qκj) exhibited in [2] lacks the simplicity of, e.g., Keisler's set of axioms for L(Q^) (cf. [1]). This paper, a sequel to [3], demonstrates that this failure of simplicity is not without reason. It will be shown here, without additional set-theoretic hypotheses, that LίQ^) cannot be axiomatized by finitely many schemata. Even more strongly, we prove: Theorem 1 No collection of axiom schemata of bounded quantifier depth suffices to axiomatize L(Q^). This result is due to the first author (Shelah). He communicated it via notes to the second author (Steinhorn), who then prepared this paper. The proof of •Partially supported by the United States-Israel Binational Science Foundation. **Partially supported by a grant from NSERC of Canada and the N.S.F. Received February 6, 1984; revised October 29, 1986 and June 16, 1989 2 SAHARON SHELAH and CHARLES STEINHORN Theorem 1 follows roughly the same plan as that of Theorem 3.1 in [3]. For any possible bound k on quantifier depth of axiom schemas for L(Q^), a model Oil will be constructed, whose L(Q,Q)-theory under a nonstandard interpretation of Q and Q does not have a standard model (one of power Xλ with the Kj-interpretation for Q and Q), but in which every valid LίQ^Q^-schema of quantifier depth less than k holds. For the convenience of the reader, we include below the definition of an axiom schema from [3]. It will not be used until the very end of the paper. For examples, consult [3]. Definition 1 Let <£ be a logic and let Rx(υ\9..., v^),... 9Rn(v"9..., υ^n) be relation variables. A schema is an £(Ri9...,/^-formula Φ(/?i,... 9Rn) in which each of the variables v) for / < n and j < m, is bound to a quantifier of <£. The quantifier depth of the schema is |{ vj: / < n ΛJ < W;}|. Now we embark upon the sequence of definitions and lemmas leading to the proof of Theorem 1. Definition 2 Let k > 3 and n > k (for definiteness, let n = k). A kdegenerate structure ΐίϊί = (MU ω, P9<9R9F9G) is an L-structure, where the nonlogical symbols of L are a unary predicate symbol P, a binary relation symbol <, a binary relation symbol R, an /z-place function symbol F, and a £-place function symbol G, satisfying: ( a ) M Π ω = 0 , P(9K) = ω (b) < linearly orders M (c) R is a relation o n M x M (d) F\M-+ P ( 3 t t ) , G : M k ^ P(ΐίϊί) (e) there is no sequence <#/: 1 < / < 2n) from M such that ax < a2 < . . . < α2Λ, R(ai9aj) holds for all / *./, F ^ , . . . ,an) = F(an+U... ,a2n), and <^: 1 < / < n) = (aj,: n + 1 < j < In) (mod G) (i.e., for any 1 < ix < i2 < . . . < ik < /i, GCα/j,... ,aik) = G(an+iι,... 9an+ik)). Lemma 1 (i) Passage to substructures preserves k-degeneracy (ii) If (ΐftlβiβ < a) is an increasing chain of k-degenerate structures, then (J 9110 is k-degenerate (iii) Ifΐttli, i = 0,1,2, are k-degenerate structures such that Oίlj Π 9H2 = 3ΐlOί //ιe« OH! α«ί/ 9ϊl2 can be amalgamated over 3H0 /Λ/O a k-degenerate structure 3H w/z'Λ universe 311 j U 3K2 without introducing any new equalities. Proof: The first two assertions are evident, so it remains to verify (iii). For this statement, simply shuffle together the linear orders on Mγ and M2 and stipulate that -ιR(a,b) holds whenever a E M{\MQ and b E M2\M0, or vice-versa. The reader should observe that more subtle ways of amalgamation in (iii) may be possible if ϋϊli and 0H2 satisfy additional conditions (e.g., if ω\(F"(Mf) U F"(Mξ)) is infinite and ^ and 3tt2 are countable). We will have to avail ourselves of this additional freedom in what follows. Let 3) be the set of all quantifier-free L-types with parameters from ω that are realized in ^-degenerate models. We then have: NONAXIOMATIZABILITY OF L ( Q ^ ) 3 Lemma 2 Let / ? G S . Then there is a finite p0 <Ξ p so that, if 911 is kdegenerate, 3Π7S t=A)~*P Moreover, |2) | = Ko. Proof: Given /7(Xi,... ,x m ), let /?0 be the set of those formulas in 77 specifying whether or not P(X;) holds (and if it does, the element in ω equal to x,), the complete < and R diagram for pairs (Xi,Xj) for which -ιP(x/) Λ -IP(XJ) holds, and the value in ω of Fand G for ̂ -tuples and Ar-tuples from [xγ,... ,xm], respectively. Obviously, /?0 is as desired. To see that 2) is countable, notice that there are but countably many possibilities for p0 as above. Lemma 3 There exists a countable existentially closed k-degenerate structure 911*, which is unique up to isomorphism over P(9ϊl*) = ω. Proof: Using amalgamation and preservation under unions of increasing chains, the proof of the existence of 9TC* is routine. Uniqueness follows via a back-andforth argument, using the first statement of Lemma 2 above to ensure that the construction can be continued. Lemma 4 The structure 911* is ̂ -homogeneous and ThL(9TC*) admits elimination of quantifiers. Proof: The proof of the uniqueness of ΐίϊί* yields, mutatis mutandis, ©-homogeneity. Let us now prove the quantifier elimination. Let φ(yϊ9... 9ym) = 3x ψ(x9yu... ,ym) be given, where φ(x9yΪ9... 9ym) is quantifier-free. We show that φ(y\,... ,ym) is equivalent, relative to ThL(3H*), to a quantifier-free formula in at most the variables ^ 1 , . . . 9ym. If j^m) is n o t satisfiable in any λ>degenerate structure, then clearly T h L ( 3 1 Γ ) ¥φ(yu...,ym) <+ y x Φ y γ . T h u s w e a s s u m e t h a t φ(yi,...,ym) is satisfiable in λ:-degenerate structures. L e t ( P = & p ( y 1 , . . . , y m ) : p ( y 1 , . . . , y m ) E S ) Λ ( 3 9 Π ) ( 3 α 1 , . . . , α w G 9 1 l ) [M is ^-degenerate Λ ϋH ^p(a\,... ,am) t\ Ix ψ(x,aι,... ,am)]}. Next, let (Po be the collection of all finite p0 <Ξ p E (P as guaranteed by Lemma 2. For /?oE(Po, let /?! be obtained by "deleting" the parameters from ω in p0, e.g., replace F(yu... ,ym) = r, G(y2,... ,Λ+i) = ̂ where r,s G ω by F ( ^ , . . . , ^ ) { ^ J G ( j 2 > JJΛΓ+I) according to r ( φ\s Each such /?! is finite, and it also is clear that (Pi, the set of all such/?!, is finite. We now claim that ΎhWL*)£lxψ(x9yl9...,ym)++ V ( A) The implication from left to right needs no argument, so we turn our attention to the reverse implication. Suppose that # l 5 . . . ,am E 3H* and for some pλ E (Pi, 911* t= Λ/?I (# 1 , . . . , am). Moreover, let 911 be ̂ -degenerate, c9bι,...9bme 911, and 911 t= ψ(c,bι,... ,bm) Λ Λ Pi(bl9.. .,bm). The definition of (Pi assures us of the existence of 3TC. Since both (a\,... ,am) and ( 6 1 , . . . ,bm) satisfy px, we can use 9H \ ω U {c, b\,..., bm} (by permuting ω) to define a /:degenerate structure TV with universe ω U {au... ,am,d\ —where d is a new element not in 911* —so that