The Fraenkel-Mostowski method, revisited

64 N otre Dame Journal of Formal Logic Volume 31, Number 1, Winter 1990 The Frαenkel- ΛΛostowski M ethod, Revisited N ORBERT BRUNNER* Permutation models generated by isomorphic topological groups satisfy the same choice principles (Boolean combinations of injectively bounded statements). As an application the group zyxwvutsrqponmlkjihgfedcbaZYXWVUTS Jp of p- adic integers is characterized: A monothetic linear group G generates a model that satisfies the same choice principles that hold in the model corresponding to Jp iff the G- model satisfies the well- orderable selection principle, and ACq holds, q prime, iff q Φ p. The main result is a strengthening of a previous theorem of Pincus: All Fraenkel- Mostowski- Specker independence proofs concerning choice principles can be proved in finite support models. Abstract 1 Introduction In this note we comment on some aspects of the structure theory of permutation models which are related to their historical development fifty years ago. Following some ideas of Fraenkel, permutation models were invented by Lindenbaum and Mostowski [16], [17] as a device for proving independence results on the axiom of choice in ZFA set theory (a weakening of ZF which permits a set A of atoms). In [17] Mostowski used models with only finite supports. Later he extended this method to models with infinite supports [18], and finally Specker [24] presented the most general construction of a permutation model from a group- generated H ausdorff topological group. This line of research seems to have been based on the conviction that more general construc*Preliminary versions of this paper were presented on several occasions; in particular, at a Meeting on Nonwellfounded Sets in 1986 in Mons and at DMV- Tagung 1987 in West Berlin. The author would like to express his gratitude for many suggestions which helped to improve the paper, including comments by M. Boffa, U. Feigner, Y. Rav, W. Ruppert, and K. Svozil. Thanks are also due to the referee. Received September 9, 1987; revised May 2, 1988 FRAENKEL- MOSTOWSKI METH OD 65 tions are needed for more sophisticated independence proofs. We will show that all Fraenkel- Mostowski independence proofs for ZFA which concern the choice principles in Jech [12] can be performed in finite- support models. In fact, one can stipulate the existence of a support function, though not for least supports. Hence, the flexibility gained from Specker's construction does not materialize in provability strength. As was first announced by Pincus ([23], p. 137), for each Π 2 sentence in a Specker model there is an infinite- support Fraenkel- Mostowski model that satisfies it. 1 LI Notation Our surrounding set theory is ZFC, (V,G ,= ) denoting the real world. Following an idea of Truss, we define a ZFA- universe as follows: For XG V we set Y(X) = U jVα :α G On}, where Vo = Xx {0} and Vα = [(A, a): a minimal, such that A Q U{V^: β G a}}. We define x Gx j>, if x G A and y (A,a) for some a > 0 and a > A> and we set x = xy iff x = y in V. (V(X), G *, = x) satisfies ZFA + AC, and 0X = ( 0, 1) , Ax = (V0,l) is the set of atoms. If there is no danger of confusion, we shall omit the subscript. A faithful representation d:G - • S(X) (G a group, S(X) the full symmetry group, and d an injective homomorphism) is extended to all of V(X) as in Specker [24]: (dg)(x,0) = ((dg)x,0) for x G X, and we recursively set (dg)x = {(dg)y: y Gx x] (within V(X)) for x G V(X)\ V0 , defining an G - automorphism. We write sym^jc = sym^x = {g G G : (dg)x = x] for the stabilizer of x G V(X) and iΐ x dx = Π {sym dy:y G x x] for the pointwise stabilizer. A topological group (G , ,G ) (G the topology) generates the permutation model PM = PM (tf,G , , G , *) = [x G Y(X) :vy G x TC({JC}) :sym y G G }, where TC( ) is the transitive closure in V(X); a proper V- class C ^ PM defines a class (C,On) of PM iff sym C G G. Then (P M ,G *,= *) satisfies ZFA. We shall assume that Ax Q PM . If possible, we shall not mention d. This approach to permutation models is more flexible than that in Brunner and Rubin [7], insofar as it allows us to compare models on different sets of atoms. Also, we prefer to work with atoms instead of with irreflexive sets, since the latter can be used as in Feigner [8] to introduce additional information on the G - structure which cannot be reconstructed from the group action. G is linear (group- generated), if 1 (i.e., the identity) has a neighborhood base consisting of open groups. For example, G fin, which is generated by {fix e: e G [ AJ < ω j , and Gnat> which is generated by {sym x: x G PM }, are group- generated Hausdorff topological groups. If G generates PM , so does Gnat ^ G» whence it suffices to restrict one's attention to linear H ausdorff groups. We shall always assume, then, that G = G n a t . 1.2 G Φ G n a t is possible: the "second Fraenkel model" [12] is generated by ω G = Z% with the product topology P of 2 . H < G is the direct sum of countably many Z 2 's. We let G be the group topology generated by P U {H}. Then H is dense in G with respect to P , whence H and the product topology P 1 H generates the Fraenkel model (see [7]). Since / / is an open subgroup of (G , ,G) with the relative topology G 1 H = P 1 / / , (G , ,G) generates the Fraenkel model too. But G n a t = P Φ G. Moreover, the assumption Ax <Ξ PM is not automatically true. 66 NORBERT BRUNNER 2 Persistency As was shown by Ahlbrandt and Ziegler [1], under some additional conditions countably categorical structures with isomorphic topological automorphism groups are each interpretable in the other. For permutation models the topology is more closely related to the structure: permutation models with isomorphic topological automorphism groups satisfy the same choice principles. To assign a meaning to this statement, we observe that the group of E- automorphisms of PM <Ξ V( ^ ) naturally corresponds to a subgroup Aut PM of S(X) and that zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDC Autnat generates PM . Moreover, as motivated by Pincus (5.1 of [21]) we specify that a "choice principle" is a negation of a Jech- Sochor bounded statement. In [6] such a result was announced for some examples of choice principles, but the proof could be completed for the axiom of choice, AC, only. 2.1 A sentence φ of the ZF- language is persistent if the following holds: If (G, ,G) G V is a topological group, djiG- * S(Xt), i G 2, are injective representations, and PM/ = PM(flfz,G, ,G,X/ ) are the corresponding permutation models with G- relations G/, such that Ax. <Ξ PM 7 and Gnk = G, then P M 0 satisfies φ iff PMi does (with G replaced by the respective G / 's). If φ is persistent, then its validity depends only on the group, whence we say that G satisfies φ iff some/ all permutation models generated by G satisfy φ . In view of 4.3 this makes sense for all group- generated Hausdorff topological groups. Theorem sistent. Boolean combinations of Jech- Sochor bounded statements are per- As an application, it follows that in the "second Fraenkel model" there is a complex vector space which has Hamel bases of different cardinalities. For as ω was shown by Lauchli [14], the group (Z 2 ® Z 2 ) satisfies this statement. The theorem follows from the following lemma and the proof of the Jech- Sochor Theorem (see [13]). 2.2 Lemma For each σ G On there is an Go —E\ - isomorphism F: PQ(A ) - * Pf ( F M 0 ) in V. Here σ is the -σ th ordinal in V(0) (which is the same as the -σ th ordinal in PM/ ), Ao = AXo and Pf(- ) is the -σ fold iteration of the power set operation in PM/ . 0 Proof: We first defineFon AQ. We let {orb0 aa:a G\ }be the partition of Aol here orb/X = {(dig)x'g G G j, sym/ = sym^., a - * aa in V. As was shown by Truss [25], for each open H <G there is an x G PMi such that sym! x = H. For since G = G &t there is a y G P M ! such that H ^ sym! y, and in P M ! we may se t x= [{d1g)y:gEH}. Since He V, xe V(Xχ) is a set and since x Q PM X and sym! x = H, x G P M i. With H = sym otfα , in V we get a function a - > xa G PMi such that symiXQ, = symotfα. Modulo some obvious manipulations, we may assume that xa is of the form xa = {S a x (K + a)}, where S a Φ φ and K exceeds λ and the V- cardinality of Pσ (X0). We now define F1 Ao as the orbit F = {((dog)aa, (d\ g)xa) :g G G and a G λ) . One easily verifies that F is a oneto- one function F:A0 - > PMi in V such that symi/ fa = sym o# and {dχg)F(a) = F{{dog)a). We recursively extend F to Pd(A0) by Fx = F"x\ F:Pδ (A0) Y(Xχ) . F(x) £ F(a) for a GA0, for if x G Ao this follows from the definition of F(x) FRAENKEL- MOSTOWSKI METHOD 67 as a singleton, and ifzyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA x £ Ao then F(x) G F(a) implies that F"x = Fx = S a X (K + α ) has more elements (with respect to cardinality in V) than in Y(X\ ) Pδ {F"A0) 2 F"x > K. From this one proves by induction on max{rkx, rky} that x = yiϊF ϊ x = Fy;xey iff Fx G Fy; and the identity (dλ g)F(x) = F((dog)x). This implies that the image of F is contained in P M ^ for if g G sym o x and x G Ao, then g G syn^F x from the definition of F 1 ^40> and if x ^ ^40 then (vy G x)(3z G x) : (<iog).y = z, whence (ά xg)Fy = Fz£Fx and (ά\ g)Fx c Fx, proving that sym o x c syn^F v. So we have that F is an G- isomorphism from Pg(A0) into P f( F M 0 ) . That it is onto follows by induction from the following claim: lίX= {Fy.ye Y], Y£ P M 0 in V(X0 ), then sym 0 Γ 2 s y m ^ b y a previous remark this gives sym o x = syn^F*:), whence preimages of P M r subsets of im F are in P M 0 . For if g G s y m ^ , then (Vj G F )(3z G F ) : (^i<?)F( j ) = F(z), whence F ((rf0 ^)^) = F ( z) , and by injectivity (^o^)^ = z, proving that (dog)Y c y. 2.5 2.1 can be improved to show that injectively bounded statements 3xφ ( x) are persistent, where φ ( x) s V^((.y* < &(x) ΛT C ( X) Π y = 0)- +c(x,y)) 9 b9c pOί( <x) {x) and c(x,y) ++ are bounded, i.e., for some absolute term a, b(x) <=^ b pa{xUy) c (x9y) 9 and y* is the H artogs number. We sketch a proof. We assume that in P M 0 Φ ( x) holds. Then with σ = (σ x) and F:Pfi(A0) - > PMi of 2.2 we have φ ( Fx) in P M ^ σ(x) is an ordinal such that for all u and α a y with y* < 6(x), P ( «) G- isomorphic to P (x), and TC(w) 0 ^ = 0 , there is a t; such that v* < 6(x), TC(y) Π x = 0 , and P α (jc U ϋ) is G- isomorphic to Pa(uU y) with the isomorphism being an extension of the previous one. The proof resembles that of C15 in [21]. It follows that b(x) = b(Fx) = β. We let = σ (Fx) and G:P{(AX)_- + P M 0 be the mapping of 2.2. Then the following r sets are G - isomorphic: Pf (Fx U y), where y is any set in P M ! such that y* < jS and y Π TC(JC) = 0 ; P<?(GFxU Qy), where (Gy)* < jS; P^(G F x U υ ) , where ϋ is defined from C14 in [21] as at p. 730 in [21] to satisfy TC(ΛΓ) Π V = 0 in addition to y* < jS; Pg(λ: U w), some IVsuch that w Π TC(x) = 0 and w* < ]8. Since c(x,w) holds by φ (x), c(Fxr,.y) follows, proving φ ( Fx). Examples for statements covered by 2.3 but not by 2.1 are "every field has an algebraic closure" and "there is a set of 2*° representatives for the Dedekind- finite cardinals". 2.4 Choice principles seem to behave poorly under products. If G = Π <G ;: / G /> is an infinite product (| G, | > 2 and / infinite), then G does not ω satisfy AC (i.e., PM does not satisfy AC "). For if G, generates PM, in V(Xi) and di: G, - • S(Xi), then we let rf: G - > ^ = U {A^ x {/}: / G /} be the sum representation and PM the corresponding model which meets our assumptions Ax c PM and G n a t = product topology. If we set Pt = ( ( ^ x {/}) X (0},l) (in V(X) the set of atoms that comes from A7), then in PM < P, : / G /> is a wellordered family of nonempty sets which has no infinite subfamily with a choice function. Also, in V there is an G- isomorphism F :P M / - + {x E PM : Ax Π TC(x) <Ξ pi]9 whence G satisfies all Jech- Sochor bounded statements holding in Gh 68 NORBERT BRUNNER IfzyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA G is a topological ultraproduct, modulo a countably incomplete ultrafilter, then G is a P- space, i.e., G is closed under countable intersections, whence the principle of dependent choices DC holds. Hence the bounded statement AC ω is not inherited by continuous open homomorphisms. 2 However, the axiom of choice for families of finite sets, AC fi n , fails. Theorem IfGis abelian or a P - space, then G does not satisfy AC fin , unless G is discrete. Proof: The common property of P- groups and abelian groups which we use is: For x G PM and g G G, iϊ x{(dg k )x: k G Z) G G . We show by an application of Howard's argument [11], that if G satisfies AC fin , it satisfies AC, whence G VfO is discrete. If X G PM , we le t / be a choice function on [X] \ [0] (which exists by AC WO ). Then fix X> sym (X,f) and X is well- orderable. We assume, on the contrary, that x G X and g G sym (X9f)\ sym x. Then Y = {(dgk )x: k G Z) G P M . Since F is wo (well- orderable), we can form y =f(Y) G Y. Since contradicting (dg)y = ((dg)f)((dg)Y) = y. So in parg <£ sym x, (dg)yφ y, ticular the set A of atoms is well- orderable, whence {id} = fix A G G. 3 The translation problem The problem of finding explicit topological characterizations is open for most persistent statements. In 1967 Mathias observed that G satisfies AC iff G is discrete, and recently Blass has given a combinatorial (in terms of Ramsey groups) characterization of the Boolean prime ideal theorem. In [3] he discovered an axiom SVC (small violations of choice) which holds in all permutation models, and also gave translations of choice principles in terms of SVC- witnesses. We show that such translations automatically establish the corresponding global (class- ) forms of these choice principles in the model, and also give equivalent properties of the group, though they are clumsy to state. A χ- set S is an SVC- witness, i.e., for each set x there are a G On and a surjective f:axS- >x. Dually, S is a γ- set, if for each x there exist a G On and an injective/ :x- > a X S. Each γ- set is χ, and if S is χ then P(S) is 7. SVC states that there is a χ- set. For example, in ZFA, AC is equivalent to the existence of a χ- set S with a Dedekind- finite P(S). It is an open problem to determine the status of the assertion that every χ- set is 7 in the hierarchy of choice principles in ZFA + SVC. The partition principle (if P is a family of disjoint nonempty sets, then P < U P) implies this assertion. 3.1 Theorem G = G ^t, and Let PM be a permutation model generated in V(X) by G, SePM. (1) The following statements are equivalent: (i) Sisx (ii) There is a surjective function F:S X On - > PM in PM (iii) There is an open H < G, such that {sym//X : x E S | is a neighborhood base of id in H (this is the reason for coining the term χ- set), where sym//jc = H Π sym ^ x. FRAENKEL- MOSTOWSKI METH OD 69 (2) zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA The following statements are also equivalent. (i) 5 is a y- set (ii) There is an injectiυ e function F: PM - *S x On in PM (iii) There is an open H < G such that {sym/ / x:x E S] is the set of all open subgroups of H. Proof: We avoid mentioning d. The first observation is abstracted from [3]: There is a set B E P M , such that {sym x:x G B] = {H< G:Hopen}. As was observed in the proof of 2.2, for each H <G there is an xH E PM, such that sym xH = / / , H- + xH in V. We set B = U{orb xH:H < G], which is a set in V{X) since G E K As B c PM and sym B = G, B E P M . We now prove (1). (ii) => (i) follows from replacement. For (i) => (iii) we set H = symGf wh ere/ :a X S- + B is onto. For (iii) => (ii), we first observe that oτ b H(s,x) is a surjective mapping from oτ b Hs onto orb//jc, if symHs < sym//X. Hence if o r b ^ x^ α E On, is an enumeration of the / / - orbits of PM and symHsa < sym/jXα, where a - > (sa9xa) is in Y(X), then F = U f o r b ^ s ^ a ) , 4 > : α G θ n ) i s a mapping from dom Fc S X On onto PM and sym F = H. The proof of (2) proceeds similarly. For (i) => (iii) one sets H = sym G / for some one- to- one/ : B - > α x >1 sym/ / / (x) 3 sym//jc is trivial and "<Ξ" follows from / (x) =f(dhx) for Λ E sym/ / / (x) and injectivity. For (iii) => (ii) we take symHsa — sym/z^o; and observe that oτ b H(xa,sa) is a bijective mapping between the orbits, whence F = U {orb/ / <xQ;, ( 5 α , α ) ) : α G θ n ) : PM - • S X On is a symmetric one- to- one mapping. 3.2 It follows from (2) that PM is covered by a well- orderable class of finite sets (the class form CMC of Levy's axiom of multiple choice MC), iff PM satisfies MC. Consequently, as a corollary to [7], we obtain: G satisfies MC, iff G is locally bounded. In [7] it was shown that PM satisfies CMC iff the automorphism group Aut is locally compact. If Aut is locally compact, it is locally bounded and hence satisfies CMC. And if Aut satisfies MC, it is locally bounded and therefore its Weil- completion G is a locally compact subgroup of S(AX) (working with the natural representation of Aut). Since each g E G corresponds to a Cauchy net in Aut n a t , for x E PM a value g(x) E PM can be defined, thus identifying g with an automorphism, i.e., G = Aut. 5.3 The first independence theorem which used an uncountable set of atoms (V- cardinality) appeared in Mostowski [20], whose proof showed that AC* (AC for families of infinite cardinality at most K E On) does not imply well- orderable choice AC WO . Moreover, as follows from the following cardinal inequality, countably many atoms do not suffice. Concerning a possible converse of this inequality, it was observed by Levy [15] that the character χ ( G n a t ) can be arbitrarily high, and G would still satisfy the same persistent statements as S(ω) with the product topology. Theorem / / G satisfies AC" and χ (G n a t ) < K, then G satisfies AC WO . 70 NORBERT BRUNNER Proof: Since there will be no confusion, we write a instead of a. S = (S a: a E λ> is a well- orderable sequence of nonempty sets, and H = sym S E G n a t . We construct a choice function σ E P M . There is a neighborhood base of id of the form (sym^XQ,: a E κ>, a - + xa E PM in V(X). From AC* we obtain a choice function/ E PM for the transfinite sequence (oτ b Hxa :a E K) E PM, such that / ( α ) E orbf/ jt^. In V ^ ) , where AC holds, we let s be a choice function for S, s (α ) E S α , and we define φ E κ λ c PM such that sym x φ ( α ) < sym s( α ) , α E λ. T h e n F = U {orb/ / «*0 ( α ) ,α > , s(a)):a E λ) is a function, F E PM , and we define (σ a) = F(f(φ ( a)), a) E orb/ jS^α) <Ξ S a9 a choice function of S in PM . We nowinvestigate structural properties of permuta4 Support functions tion models that are not persistent. PM c V(X) is a finite support model, if there exist a group G and a representation d:G- >S(X), such that Gfin gener<ω is a support of xG PM , ates PM (i.e., PM = PM (tf,G , ,Ggn,ΛT)); e E [ AJ <ω such that if sym x 2 fix e. In V(X) there is a function 5 :P M - > [Ax] sym x Ώ fix S(x). If, in addition, S is a class of PM , it is called a support function for G. PM has a support function iff there is a group G with a support function, such that G fin generates PM . PM is an M- model, if every x has a least support supp(x). This notion was introduced in Mostowski [18]. We start with a first- order characterization of the finite- support- model property. 4.1 Theorem (1) PM is a finite- support model iffA<ω = U{An:nGω} <ω <ω is a y- set. (2) PM has a support function iff [A ] is a χ- set Moreover, if PM is generated by G, then it is a finite- support model iff for some open H< G H f i n = H n a t , and it has a support function iff there is a support function for some open H < G. From this one easily deduces that the usual permutation models constructed from normal ideals of infinite sets cannot be obtained from a different group and the ideal of finite sets of atoms. Proof: (1): If G fin generates PM for some group G, then {sym G / :/ E A<ω ] is a local base of id for G f i n , whence A<ω is χ. If conversely {symHf:fE A<ω ] is a local base of id for some H < G in G n a t , where G n a t generates PM , then {fixH (im / ) : / E A<ω } <Ξ H f i n is an open base of id for H n a t . Hence H n a t = H f i n , and since His open, it generates PM too. <ω <ω is a 7- set, and F(x) = <T(x), ax}9 x E We now prove (2): If [A ] PM, is the mapping from 3.1.(2)(ii), then symHT(x) = sym^x for H = symGT and S(x) = U {i m / : / E T(x)\ is a support function for H n a t : sym^x 2 fixHS(x). Hence H n a t = H f i n , and since H is open H n a t generates PM , which <ω therefore has a support function. If conversely S:P M - » [^4] is a support function, then we set H = sym G S. By the proof of 2.2, every open subgroup K of His of the form sym^*, for some x E PM . We let / : n - • S(x) be any bijecn tion and set F = oτ b κ f K = sym//X. Then K = sym/ / F, and since F Q S(x) is <ω <ω finite, Fe [A ] . 4.2 If G is totally bounded, then every permutation model PM generated by G has a support function S for G . F or by 3.4 of [7] G n a t = G f i n . We set O = FRAENKEL- MOSTOWSKI METH OD 71 {orbzyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA a: a G Ax] c [v4 x ] < ω , and observe that in PM [ O ] < ω can be well- ordered by some relation < , sym < = G. Hence S(x) = Ue, e G [O]<ω the <- least element such that sym x Ξ> fix(U e), defines a support function for G, sym S = G. But in general a group has a representation without a support function. Example There exists a finite support model without a support function. Proof: We work in V(Z X ω), Z the integers, and let G = Z ω operate on the set ^4 of atoms through dg(z,n) — (g(n) + Z,JZ). PM is the model generated from G fi n . If PM has a support function, then there is an open H <G such that there is a support function S for H f i n = H n a t . Since S is a support function for each open subgroup of H, we mayassume that H = fixG ^ = sym^S for some <ω e G [v4] . We define the following sets of atoms of PM : Pn = {(z,n,o):zG 2], E n= {(z,n,O):z even}, On = Pn\ E n, and we observe that sym G P rt = symG{E n, On}=G9 whence f(n) = S(E n)U S(On) G [A]<ω defines a function in PM , sym/ / / = H. If e Π P Λ = 0 , then/ (/ i) Π P Λ Φ 0 ; for otherwise g G fix// S(E n), where #(AZ) = 1 and g(i) = 0 for / Φ n, despite the fact that dgE n = On. We nowobtain a contradiction: A standard permutation argument shows that there does not exist a function f:ω - > [;4] < ω such that f(n) Π Pn Φ 0 for infinitely many n. 4.3 To complete the picture, we show that every linear group generates some finite- support model. This together with 2.1 proves that, for the purpose of independence proofs, the class of finite- support models with arbitrarily large sets of atoms (in view of 3.3) is sufficient. With a proper class of atoms the situation might be different. For every group- generated Hausdorff topological group (G , ,G ) there exists a permutation model PM in some V(X) and a representation d of G as a subgroup of S(X), such that Gfin = G generates PM , Λ x c PM , and PM has a support function Sfor Gfin such that S(x) is a minimal support ofx. Theorem Proof: We begin with an adaptation of a standard result from permutation group theory (see, e.g., [10]). If (G , ,G) is a group- generated Hausdorff topological group, then there is a permutation model QM in some Y(X) and a representation t of G as a subgroup of S(X), such that Gfin = G^at = G generates PM . We set X= {h H:h G G and H < G open} and let G operate on X by (tg)(hH) = ghH. In Y(X) we form the permutation model QM generated by Gfin. Then Λ x g QM, Gfin = G ^ t , and since symtG(hH,0) = hHh~\ G fin = G . Since G is Hausdorff,- 1 is an injective representation ofG. We let B be the set found in the proof of 3.1 for QM. In V we form V(B) and define the operation dof G on B by (dg)(b) = (tg)b, tg the G- automorphism of QM. PM is generated from G ?in in V(B). Since [symdG(b90) :b G B] = {symtGb:bGB] = [H<G:Hopen}, Gfin = G and AB is a γ- set in PM; hence 1 there exists a support function S: PM - • f ^ ] " in PM (as in the proof of 4.1). Obviously, S(x) is a minimal support ofx. M- models cannot be obtained in this way. As follows from [4], if PM is an M- model then AC^S holds, the axiom of choice for well- ordered families of 72 NORBERT BRUNNER well- order able sets, as does P AC WO , which says that every family of wellorderable sets has an infinite subfamily with a choice function. Therefore, a compact group generates an M- model iff it is discrete.3 4.4 The groups that appear in the literature are usually the automorphism groups of some first- order structure on the set of atoms which contains all the information about the permutation model, as in Pincus [22]. To give another example of the above construction, we consider the group Jp of p- adic integers, p a prime number. Each g G Jp is thought of as a formal series oo Σ 8n'Pn> Sn G p, and addition is defined accordingly (cf. 10.2 of H ewitt and Ross [9]). The topology is defined by a nonarchimedean metric, the groups L n = [g G Jp: gk = 0, for k < n] forming an open neighborhood base of the identity. In fact, each open subgroup of Jp is an L n. Jp is a compact linear monothetic Hausdorff group. For a G Jp, n > 0, we set x(a,n) = [x G Jp:xk = ak for kGn] a n d X= {x(a,n):a G / p , n > 0}. An operation dof Jp is defined as (dg)x{a,n) = x(a + g,«), giving sym(x(α ,«),0) = L n9 whence PM(d9Jp9 +9Jp,X) is the finite- support model that results from 4.3 when applied to Jp. As an application, we add some weak forms of AC to an independence result of Zuckerman [26]. Lemma IfE is a finite set of primes, then G = Π ( J p:pGE) satisfies MC, WO KW (the well- orderable selection principle, that for each well- ordered family F of sets F with at least two elements there is a selection S: 0 Φ S(F) c F and F\ S(F) Φ 0 ) , and ACn (choice for families of n- element sets) iffn is not an integer combination n = Σ n(p) • /?, n(p) G ω, of E. PGE Proof: PM is a model generated by G. If H < G is open, because of the special structure of G it is a product of the L n(p)'s, whence the index [H:K] is a product of primes in E, K < H < G open. In particular, for K = H Π sym x, the cardinality |orb/ / x| = [H:K] has all its prime factors in E. If \ S\ = n and n is not an integer combination of E, there is som exGS such that | orb/ / x| = 1, H = sym S, because S = U{orb/ / X:x G S], i.e., sym x ^ sym S. It follows from [11] that every family of ^- element sets has a choice function. If, on the other hand, n = Σ n(p) • /?, 0 Φ F c E, then F = U{orb Fn:nGω}, Fn = pGF U{n(p) x oτ b HiP9n) L H(p,n) = L n x Π *q> n < JP> and s y m ^ = xg:p<ΞF], qφ p H(p, n + 1), is a counterexample to PAC Λ (every family of 7?- element sets WO has an infinite subfamily with a choice function). We now prove KW . In view of MC it suffices to prove K W^ , and we may assume that the family has the form F = (Fa: a G K), Fa = orb/ / xa, H = sym F , where H = Π ^(P )ί(Π ^ pGE L\ p (/ 7) / ) ^ l andsymxα = J[L miP9a).U J m(p,a)7>n(p) peE for all p G E, Fa is a singleton, a contradiction. Hence K = U{Kp:p £ E], Kp= {a G κ :m(p,a) < n(p)}, and we define the selection function on Kp as where H(p) = L n{p)_ x X J[ L n{q). Since G is follows: S p(a) = orbHip)xa, qφ p FRAENKEL- MOSTOWSKI METHOD zyxwvutsrqponmlkjihgfedcbaZ 73 abelian, sym (orb/ / zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCB xa) = <//, sym xα >, the group generated by H and sym xa, and as (H, sym xα > =£ </ / (/ ?), sym xα > in the/ rth coordinate, S p(a) Φ Fa. Despite their importance in topological algebra, the p- adic groups seem to have escaped attention by set theorists. They prove the following new independence results: (i) Mostowski's independence results [19] can be augmented WO by KW + AC< Λ φ AC m , if m,n do not satisfy the condition ( 5) ; (ii) G = Π (Jp:p prime) is a monothetic group, whence it satisfies AC^° for each n > 1 (cf. [7], p. 158), but by 2.4 and the lemma no instance of PAC,,, n > 2, holds. With respect to persistency, finite products of Jp are the only linear monothetic groups which satisfy KWWO. For if G is linear and monothetic, it is discrete or totally bounded, since all nontrivial subgroups of Z have a finite index. Hence by 25.16 of [9] G satisfies the same persistent statements as Π<G P :p prime), where Gp is Jp or Z (pnip)). If there are infinitely many factors GpΦ ly PKWω fails (counterexample: oτ b Gxh xt E P, from 2.4). Otherwise, if G is not discrete, it contains finitely many factors Jp and their product is an open subgroup of G, generating the same model. If G is discrete, it satisfies the same persistent statements as 1 = the empty product. We may conclude: A monothetic linear group G satisfies the same persistent statements as some finite product P = U(Jp:p E E) iff G satisfies KWW0. Corollary Moreover, p E E iff AC^ fails and, as follows from the proof of the lemma and 2.4, P KWF is false for Jp and hence for P, if p E E. Another application of the /7- adic groups is motivated by quantum mechanics. Benioff [2] has considered several extension mappings T - • T of quantum mechanical operators T on a Hilbert- space H of some model PM c Y(X) to T on a corresponding Hilbert- space Hin the "real world" V(X); e.g., if .fiΓis the completion of H in Y(X) then T is the unique extension of T to H. Benioff has 4 applied his results to solve ontological questions. Here we investigate the spectral behavior under this extension for the following spaces. If G is a nondiscrete linear monothetic group, it satisfies the following statement: "There is a H ubert space / / which is not finite dimensional while each orthonormal system is finite". Example Proof: As follows from the previous remarks, PKWfin is false; let (Fn: n E M ) be a counterexample. We set H = Π {Ker φ n:nEM}9 where φ n: 12(U{F Λ: n E M}) - > C is the continuous linear functional φ n(x) = Σ x(a). If (xn:nE: ω) is a sequence in H, a<ΞF n then s = U[s(xn) :n E ω) is finite, where s(xn) = {a:x(a) Φ 0}, whence / / is locally sequentially compact. F or φ n(x) = 0 implies Fn\ s+(x) Φ 0 , s+(x) = {a: Re x(a) > 0 or (Re x(a) = 0 and Im x(a) > 0)}, and s(x) Π F nΦ 0 implies s+ (x) Π FnΦ 0 , whence an infinite s defines a selection function S(n) = FnΠ s+(xm), m minimal with s(xm) Π Fn Φ 0 , with an infinite domain dom S = <ω {n eM:s Π FnΦ 0 }. If D is an orthonormal subset of H, then [D] is Dedekind- finite (cf. [5]), and the axiom of multiple choice implies that D is finite. Since G satisfies MC, D = U{Fn:n G M] can be represented as a well- 74 N ORBERT BRUNNER orderable union of finite sets, whence there is a diagonal operatorzyxwvutsrqponmlkjihgfedcba S on h(D) for which the topological approximate point spectrum U(S) Φ σ p(S), the point spectrum (in the model PM generated by G ). If Γis the restriction of S to H, then the sequential approximate point spectrum σ a(T) = σ p(T) Φ Π(T) (see [5] for the background). For H we have σ a(f) = Π ( f) = Π (Γ);_hence σ a(T) Φ σ a(f) changes. If His the nonstandard hull of Hin V(X) and H = H, the nonstandard hull in PM (with respect to a countable ultrapower), then for the exten- sion ftoHwQ have σ p(T) Φ σ p(f) 2 Π (Γ). We finally note that the results of this paper do not apply if the permutation model is formed in a subclass of V(X), the most interesting case being V! (X) = {x<Ξ V(X): TC(x) Π Ax is finite), which is due to Levy. N OTES 1. In [23] the result is stated for Σ 2 . In a letter sent on May 5, 1988, Pincus communicated to me a proof that works for sentences Π 2 in the power set operation. 2. This is to be expected in view of the following fact: If x e PM and P M (x) is the V(ΛΓ) of Section 1.1 relativized to PMi then P M (x) is a permutation model, generated by the quotient group sym (x)/ fix (ΛΓ). Conversely, each G/ H, H a closed normal subgroup of G, generates some P M (x). Hence quotients preserve countable choice. 3. The Pincus construction may be strengthened as follows. There is some ordinal β such that each open subgroup H of G is H = sym (Λ:) , for some x E PM of rank rk (x < β) . The Pincus model QM is constructed in V({ΛΓG P M :rk (Λ:) < β}) with the group action induced by the action of G in PM . 4. For instance, it was observed by W. Boos (The Journal of Symbolic Logic, vol. 53 (1988), p. 1289) that in appropriate generic extensions the extended model contains hidden parameters for the ground model. REFEREN CES [1] Ahlbrandt, G . and M. Ziegler, "Quasi finitely axiomatizable totally categorical theories," Annals of Pure and Applied Logic, vol. 30 (1986), pp. 63- 82. [2] Benioff, P. A., "Models of ZF as carriers for the mathematics of physics," Journal of Mathematical Physics, vol. 17 (1976), pp. 618- 640. [3] Blass, A., "Injectivity, projectivity and AC," Transactions of the American Mathematical Society, vol. 255 (1979), pp. 31- 59. [4] Brunner, B., "Dedekind- Endlichkeit und Wohlordenbarkeit," Monatshefte fur Mathematik, vol. 94 (1982), pp. 9- 31. [5] Brunner, N ., "Linear operators and Dedekind- sets," Mathematica Japonica, vol. 31 (1986), pp. 1- 16. [6] Brunner, N ., "Automatic transfer between permutation models," pp. 61- 68 in Contributions to General Algebra 5, edited by E. Eigenthaler, 1987. [7] Brunner, N . and J. E. Rubin, "Permutation models and topological groups," Rendiconti del Seminario Matematico della Universitά di Padova, vol. 76 (1986), pp. 149- 161. FRAENKEL- MOSTOWSKI METHOD 75 [8] Feigner, U ., "Choice functions on sets and classes," pp. 217- 255 in zyxwvutsrqponmlkjihgfed Sets and Classes, edited by G . Mϋller, N orth H olland, Amsterdam, 1976. [9] Hewitt, E. and K. A. Ross, Abstract Harmonic Analysis, Springer, Berlin, 1963. [10] H odges, W., "Categoricity and permutation groups," pp. 117- 151 in Logic Colloquium '85, edited by the Paris Logic G roup, 1987. [11] Howard, P. E., "Limitations on the Fraenkel- Mostowski method," The Journal of Symbolic Logic, vol. 38 (1973), pp. 416- 422. [12] Jech, T., The Axiom of Choice, N orth H olland, Amsterdam, 1973. [13] Jech, T. and A. Sochor, "On 0- models of set theory," Bulletin Academie Polonaise Mathematique, vol. 14 (1966), pp. 297- 303 and 351- 355. [14] Lauchli, H ., "Auswahlaxiom in der Algebra," Commentarii Mathematici Helvetici, vol. 37 (1963), pp. 1- 18. [15] Levy, A., "On models of set theory with urelements," Bulletin Academie Polonaise Mathematique, vol. 8 (1960), pp. 463- 465. [16] Lindenbaum, A. and A. Mostowski, "Unabhangigkeit des AC," Comptes Rendus Sciences Varsovie, vol. 31 (1938), pp. 27- 32. [17] Mostowski, A., "U ber den Begriff der endlichen Menge," Comptes Rendus Sciences Varsovie, vol. 31 (1938), pp. 13- 20. [18] Mostowski, A., "Unabhangigkeit des Wohlordnungssatz vom Ordnungssatz," Fundamenta Mathematicae, vol. 32 (1939), pp. 201- 252. [19] Mostowski, A., "AC for finite sets," Fundamenta Mathematicae, vol. 33 (1945), pp. 137- 168. [20] Mostowski, A., "The principle of dependent choices," Fundamenta Mathematicae, vol. 35 (1948), pp. 127- 130. [21] Pincus, D ., "Z F consistency results by Fraenkel- Mostowski methods," The Journal of Symbolic Logic, vol. 37 (1972), pp. 721- 743. [22] Pincus, D ., "Two ideas in independence proofs," Fundamenta Mathematicae, vol. 92(1976), pp. 113- 130. [23] Pincus, D ., "Adding D C , " Annals of Mathematical Logic, vol. 11 (1977), pp. 105- 145. [24] Specker, E., "Zur Axiomatik der Mengenlehre," Zeitschrift fur Mathematische Logik und Grundlagen der Mathematik, vol. 3 (1957), pp. 173- 210. [25] Truss, J., "On successors in cardinal arithmetic," Fundamenta Mathematicae, vol. 78(1973), pp. 7- 21. [26] Zuckerman, M. M., "Choices from finite sets and choices of finite subsets," Proceedings of the American Mathematical Society, vol. 27 (1971), pp. 133- 138. Institut fur Mathematik und Angewandte Statistik Universitάt fur Bodenkultur Gregor Mendel- Strasse 33 A- 1180 Wien Osterreich