LOGIC COLLOQUIUM 78 M. Boffa, D. van Dalen, K. McAloon (eds.) 0 North-Hollmd Publishing Company, 1979
AXTRACT LOGIC AND SET THEORY. I. DEFINABILITY
Jouko Vaananen Department of mathematics University of Helsinki Hallituskatu 15 SF-00100 Helsinki 10 Finland
Definability in an abstract logic is compared with defina- bility in set theory. This leads to set theoretical character- izations of implicit definability, LBwenheim-numbers and Hanf- numbers of various abstract logics. A new logic, sort logic, is introduced as the ultimate limit of abstract logics definable in set theory.
§ 0. Introduction
The aim of this paper is to bring together, in a coherent framework, both old and new results about unbounded abstract logics (a logic is unbounded if it is able to characterize the notion of well-ordering). Typical problems that can be asked about any logic are:
(1) Which model classes are implicitly (with extra predicates and sorts) defina- ble? (2) Which classes of cardinals are spectra? (3) What is the Lawenheim-number? (4) What is the Hanf-number?
In the case of unbounded logics these problems are particularly relevant as such logics fail to be axiomatizable and mostly lack workable model theory. An attempt to shed light on (1)-(4) is the main purpose of this paper. Out method is to build, right from the beginning, a close connection between abstract logic and set theory. The basic notion of the whole paper is that of symbiosis. We say that an abstract logic L* and a predicate P of set theory are symbiotic if, roughly speaking, the family of A(L*)-definable model classes coincides with the family of model classes which are A1 (P). For example, second order logic L1' 1s. symbiotic with the power-set operation, or, what amounts to the same,
I1 A(L ) = {Kithe model class K is A2}.
In Chapter 2 we give a new proof of the following result (essentially due to
39 1 392 J, VGNLNEN
Oikkonen [lo]): If L* and P are symbiotic, then
An(L*) = {Klthe model class K is An(€')}.
As a corollary we get for n > 1:
An(Lww) = {Klthe model class K is .An}.
Consideration of the logics An(Lww) leads very naturally to what we call sort logic. To grasp the idea of sort logic, let us consider a typical many- sorted structure
M =
M consists of three kinds of objects: universes M., relations Ri and indi- viduals ai. TO quantify over the individuals we have first order logic; to quantify over relations we have second order logic; but to quantify over uni- verses (i.e. sorts) we need a new logic. Accordingly, let sort logic Ls be the many-sorted logic which allows quantification over individuals, rela- tions and sorts. It is clearly impossible to define the semantics of sort logic in set theory, but it can be done, for example, in MKM (Morse- Kelley-Mostowski) theory of classes. It follows readily from the above analysis of An(Lww) that
Ls = (Klthe model class K is definable in set theory]
(stated in [81 p. 174). The rest of Chapter 2 is devoted to an analysis of the non-syntactic nature of the A-operation. We show, for example, that the set of LII-sentences which I1 give rise to A(L )-definitions, is Il - but not -definable in set theory. This 3 13 result reflects the difficultness of finding a simple syntax for A(LI1). Chapter 3 is concerned with a restricted A-operation, A:, which does not allow the use of new sorts (or universes). This operation is clearly related to L1' as we may think of LI1 as A'(w)(Low). The key notion of this chapter is that of a flat formula of set theory. We obtain the following characterization of generalized second order logic: If L* and P satisfy a strengthend symbiosis assumption, then
A' (L*) = {;(Ithe model class I( is defined by a flat formula of the (W) language {E,PII.
In particular
L1' = {Kithe model class K is defined by a flat formula of set theory} ABSTRACT LOGIC AND SET THEORY 393
These results are proved in a level-by-level form. In Chapter 4 we extend the analysis of the set theoretic nature of model theoretic definability to spectra and Lowenheim-numbers z(L*). We characterize the spectra of symbiotic logics and prove for L*, symbiotic with P,
I(A (L;I)) = sup {ala is n (P)-definable with parameters in A}
z(An(LA)) = sup {ala is An-definable with parameters in A} (n > 1).
A similar analysis of Hanf-numbers h(L*) is carried out in Chapter 5. The non-preservation of Hanf-numbers under A necessiates the introduction of a bounded A-operation AB, and respective set theoretical notions I:, If and A:. The main result says: If L* and P are symbiotic in a sufficiently bounded way, then
h(L;t) = sup {ala is l:(P)-definable with parameters in A) and for n > 1,
h(An(LA)) = sup {ala is rn(P)-definable with parameters in A).
In the rest of Chapter 5 we consider the numbers
2 = sup {ala is n -definable}
h = sup {a/a is ln-definable].
Note that In = I(An(Lww)) and hn = h(A(Lww)) (for n > 1). It turns out that for n > 1,
I = sup {ala is A -definable} and
In < hn = In+, .
In particular, we get
I(Ls) = h(Ls) = sup {ala is definable in set theory].
This paper is based on Chapter 2 of the author's Ph.D. thesis and the author wishes to use this opportunity to express his gratitude to his supervisor P.H.G. Aczel for the help he provided during the preparation of the thesis. This work was financially supported by Osk. Huttunen Foundation. 394 J. VZNXNEN
§ 1. Preliminaries
We give at first a rough sketch of the preliminaries, which should he enough for a casual reader familiar with [21 and [a]. More detailed preliminaries then follow. Our abstract logics are defined roughly as in [2]. If Q is a generalized quantifier LQ is like Lwo[Q] in [a]. I is the HBrtig-quantifier, W the well-ordering quantifier and % the Henkin-quantifier. L1' is second order logic. All logics are understood to be many-sorted. The logic which is obtained from LI1 by adding quantification over sorts is called sort logic and denoted Ls. If L* is an abstract logic, c(L*) is the family of PC-classes of L* in the sense of 181. Il(L*) consists of the complements of PC-classes of L*. cn(L*) and Iin(L*) are obtained by iterations of the 1- and Ii-operations. A (L*) refers In to the intersection of ln(L*) and Xn(L*). The families c:(L*), nn(L*) and Ai(L*) are defined similarly but the PC-definitions are not allowed to introduce new sorts. This ends the sketch.
1.1. Abstract logics
For many-sorted logic we refer to [5]. Types are sets of sorts, relation- symbols and constant-symbols. If L is a type, the class of all structures of type L is denoted Str(L). If M E Str(L) and K is atype such that KCL, then MIK denotes the reduct of M to K. If x E L, then 2' denotes the interpretation of x in 11. IMi denotes the union of the universes of M. A quasilogic is a pair L* =
(Ll) If cp E L* (that is Stc*(L,cp)), then L is a type and cp is a set called an L*-sentence, (L2) If M k* cp (that is I*(M,cp)), then there is a type L such that M E Str(L) and cp E L*, (L3) If M k* cp and M Z N, then N I* cp.
This definition is somewhat weaker than the definition of a system of logics in [21, and substantially weaker than the definition of a logic in [a]. 1 The quasilogic L is defined as usual. If Q Qn are generalized mu ,..., 1 quantifiers, we let Lmu(Q ,. . . ,Qn) denote the quasilogic which is obtained 1 from Lmu by addition of the new quantifiers Q . . .Qn. Second order infinitary I1 . logic, which is denoted by Lmw, is obtained from Lmw by addition of quanti- fication over (finitary) relations. The following generalized quantifiers play a special role in this paper:
Hartig-qudntifier: IxyA(x)B(y) ++ card(A) = card(Bj, ABSTRACT LOGIC AND SET THEORY 395
Well-ordering-quantifier: WxyA(x,y) t-+ A well-ordnrs its domain,
Regularity-quantifier: RxyA(x,y) t+ A orders its domain in the type of a regular cardinal, Henkin-quantifier: 9 xyuvA(x,y,u,v) VfVg3x3yA(x,y,f(x) ,g(y)). H - Note that our Henkin-quantifier is the dual of the original one. If L* is a quasilogic we let Lz be the quasilogic the sentences of which are those cp E L* for which L E A and cp E A, and the semantics of which follows that of L*. For example, (Lmw)H(K) will be LKW if the syntax of is defined in the usual set theoretical way (see e.g. [3]). We denote Lmw L;F 1 I1 by L:w and in general by LZw. LWw(Q ,..., Qn) and Lww are shortend to L(Q1 ,..., $) and L". As usual, LA denotes (LmW)A. For w < h E A, we use LAh to denote (LmA)A. Similarly LAG denotes (LmG)*. If K < A, then LKK added with the weak LKX does not make much sense, but we redefine it as second order quantifiers 3X( 1x1 5 a A.. . ) for a < A. The obvious set theoret- ic definition gives LKX c H(K + lal) whenever X = K,. A class of structures of the same type is called a model class if it is closed under isomorphisms. If K C Str(L) is a model class, then the model - class Str(L) - K is denoted by K. A model class K is L*-definable if there are L and cp E L* such that K = Mod(cp) = tM E Str(L)IM c* cp}. We say, as usual, that a quasilogic L* is a sublogic of another quasilogic L+, L* 5 L+, if every L*-definable model class is L+-definable. L* and L+ are equivalent, L* - L+, if they are soblogics of each other. An abstract logic is a quasilogic L* such that
(Lb) If L and L' are types such that L C L', then L* 5 L'* and for cp E L*, M E Str(L'),
M k* cp if and only if MI, I* cp.
(L5) For every rudimentary set A, type L E A and cp, $ E Li, there are cp A and cp v $ in Li such that Mod(qx$) = Mod(cp) flMod($) and Mod(cpAjr) = Mod(cp) fl Mod($).
(L6) For every rudimentary set A, types L,L' E A and (0 E L'* there are 3ccp and Vccp in Li such that if L' - L consists of the constant-symbol c, then 396 J. VZN~EN
1 n I1 It is obvious that Lmw, Lmu(Q ,..., Q ), L,,, and their frapents are abstract logics. If the analogue of (L5) and (L6) holds for the negation, we call L* a Boolean logic. When we are only interested in the definable model classes of an abstract logic L*, we sometimes write
L* = {KI the model class K is.. . } meaning that an arbitrary model class K is L*-definable if and only if K is...
1.2. Sort 1oDic
The class of formulae of infinitary sort logic L:u is obtained if the I1 following formation rule is added to the recursive definition of L -formulae: mu
If cp is a formula and s is a sort, then 3scp and Vscp are formulae .
To define the semantics of Ls we have to work in the MKM theory of classes mu or in any other theory in which satisfaction for formulae of set theory is defi- nable. If L is a type, s a sort, s f? L and L' = L U {s], then for any M E Str(L) we define
M k 3sq1 if and only if 3N E Str(L')(N = M & N k cp) IL
M Vscp if and only if VN E Str(L')(NIL = M -+ N cp).
This defines Lf, as an abstract logic. We denote L;~ by L'. It appears that sort logic has not been singled out as a logic before, although it has been studied in a semantical form in [lo]. Let lI1(Lw), n 2 1, be the sublogic of L:, the formulae of which have the form
3SlVS2.. .3(V)Sn'p
where sl, ...,s are sorts and cp E L::. Let IIn(Lmw) be the sublogic Of L:, consisting of formulae of the form
VS13S 2...v(3)sncp
I1 where sl, ..., s are sorts and cp E L,,. For each n < m, the abstract logics ln(Lmw) and nn(L ) are definable in ZF. Note that &(Lmu) and Iln(Lmw) mW are closed undpr second order quantifiers. Let A (L ) be the sublogic of ln(Lmw) the formulae of which are equiva- n mw lent to ll (L )-formula. An(LW) is an abstract logic but it does not seem to n mw ABSTRACT LOGIC AND SET THEORY 39 7
have such a simple syntax as ln(Lmw) and nn(Lmw). In fact the class of A (L )-formulae reflects to a certain extent the properties of the underlying n mu model of set theory and changes when the model is changed. Note that A,(LWw) is just A(Lmw) in the sense of [a]. More generally, A(&(Lmu) U nn(Lmu)) - An+l (Lmw). The fragments &(LA), nn(LA) and An(LA) are defined similarly.
1.3. Extension-ogerations
We review the definition of the A-operation from 181 because the definition naturally leads to both more general and more restricted operations.
If cp E L*, L' C L and M E Str(L'), let
~(M,cp) = IN E Str(L)INlL' = M & N I*cpl.
A model class K of type L' is 1-defined by cp if L - L' is finite and
K is n-defined by tp is L - L' is finite and
K = [M E Str(L')IVN E Str(L)(NIL, = M + N I* @)I.
K is I(L*)-definuble (n(L*)-defiwble) if it is 1-defined (n-defined) by some cp E L*. Finally, K is A(L*)-defhzble if it is both l(L*)- and I[(L*)-definable. A(L*) gives rise to the semantics of an abstract logic, but to find a syntax for that logic seems as difficult as finding a syntax for A,(Lmw). Note however, that in special cases A(L*) has a beautiful syntax (see e.g. [81 $4). To be specific let us agree that A(L*) is the abstract logic the sentences of which are &-tuples
logics are sublogics of A 3 (Lmw ) and therefore the operations An, n > 2, are relatively uninteresting, apart from their relation to sort logic. If the above definition of l(L*) and iT(L*) is modified by requiring that 398 J. V&hN
L has no new sorts over and above those of L', the essentially weaker notions 1 of n;(L*)- and ll(L*)-definability are obtained. Let Ai(L*) be defined as 1 1 A(L*) above. More generally we define f.;+,(L*) = I;(nn(L*)), nn+,(L*) = ni(l:(~*)) and A~+,(L*)1 = A,(z~(L*)11 u nn(~*)).
1.4. Set theory
Our set theoretical notation follows mostly that of [41. However, we write Ra for the a'th level of the ramified hierarchy. Cd(x) is the predicate "x is a cardinal number (initial ordinal)", Rg(x) is the predicate "x is a regular cardinal" and Fw(x,y) is the predicate x = P(y), where P is the power-set operation. Card(x) is the least ordinal which has the same power as X. HC(x) = max(card(TC(x)),so). We sometimes use snas a predicate, meaning the predi- cate "x E fin", of course. PK(y) is the set {x _C yI 1x1 < K} and pW,(x,y) is the predicate "y = PK(x)". The sets of zn(P)- and nn(P)-formulae are de- fined as usual. A predicate is ln(P) (nn(P)) w.p.i. (= with parameters in) A if it is definable with a P) (PI)-formula w.p.i. A. A predicate is In( (nn An(P) w.p.i. A if it is both yn(P) and nn(P) w.p.i. A. An ordinal a is ln(P) (nn(P),An(P))-definablew.p.i. A if the predicate "x E a" is.
5 2. The basic representations
In this chapter we define the symbiosis of a logic and a predicate of set theory, and prove the main result about symbiosis (Theorem 2.4). The chapter ends with some remarks on the non-absolute nature of the A-operation. By its very definition an abstract logic determines two predicates of set theory: Stc and b. It is convenient for our purposes to establish a converse relation, that is, associate every predicate with a generalized quantifier. Suppose P = P(xl,....x ) is a predicate of set theory. Let
K[P] = {MIM E
Let % be the generalized quantifier associated with K[P] and ABSTRACT LOGIC AND SET THEORY 399
Proof. An elaboration of the proof of the well-known fact that kL is mu A, (see e.g. [31 p. 83) gives (1). (2) is trivial as KEPI is even L[PIWW- definable. 0
The above lemma shows that L[P] and P are in a sense definable from each other. We take a slight weakening of this property as the definition of symbiosis.
Definition 2.2. Suppose L* is an abstract logic, P a predicate of set theory and A a transitive class. We say that L* and P are symbiotic -_on A if the following two conditions hold:
Q L*, then Mod(cj?) is A (Sl) If E 1 (P) w.p.i. {Q,LI (5’2) K[P] is A(Li)-definabZe.
The logic L* is symbiotic on if there is a predicate P # 0 such that L* and P are symbiotic on A. If A = HF, we omit the clause “on A”.
Note that symbiosis on A implies symbiosis on any transitive A’ ?A.
Examples 2.3. The following pairs are symbiotic on HF for any rudimentary A:
(1) LDIA and P, 1.2) LQA and Q, if Q is a generalized quantifier and LQ is unbounded, 131 LWA and On, (4) LIA and Cd, (5) LRA and Rg,
16) Lil and Fw, 171 L,, and pww . n n The folZowing pairs are symbiotic on H(K) for any rundsmentary A 2 H(K)
and K = h’, t w:
(8) LAK and pwKJ 19) LAG and On.
Proof. The proof of (Sl) is similar to the proof of 2.1(1) in any of (1)-(9). As a typical example of the proof of (52), let us consider (4). Recall that W is ~(LI). NOW
A Vxyz( zEy A yEx A xEa + zEx) A A Vz(zEa + - Ixy(xEz)(yEa))A A Vxy(Vz(zEx - ZEy) -+ x=Y) .o 400 J. VL~NLNEN
Note that, if L* is s$mbiotic with P (# 0) on A then Lz is unbounded, in fact, as K[P] is A(L*)-definable, it suffices to observe that A
Wxy(xAy) c+ there are M, E and a,, ...,a such that
Conversely, it is by no means the case that every unbounded logic is symbiotic on some A. We shall indicate later why the logic L(W,Q l,...,Qn,... )n -Theorem 2.4. Suppose A and A t A are transitive classes, P a predicate, and L* an abstract logic extending LA and symbiotic with P on Ao. Then the jollodng are equivaZent for any model class K of type L E A: (1) K is l(Li)-definabZe, 12) K is Z,(P) w.p.i. A. Proof. (1) + (2): Suppose cp E Li 1-defines K. Let Lo E A such that cp E LE. Now K = {M E Str(L)13W E Str(Lo)(NIL= M & N I*cp)}. Hence K is I,( P) w.p.i. A. (2) + (1): Suppose Vx(x E K f-f cp(x,a)) where cp(x,y) is ll(P) and a E A. Let a' = TC(Ia>) (E A). Let E be a binary predicate symbol not in L and i a sort not in L. For any formula $(xl ,..., x ) of set theory let [$(x,,... ,xn)IE be obtained from $(xl ,...,xn) by replacing atomic formulae tcu by tEu (i.e. E(t,u)) and changing all bound variables to variables of sort i. Let Lo be L extended with E and i. As L* is symbiotic, there is an L1 3 Lo and '8 E LtA such that '8 1-defines the class of well-founded extensional structures Let $(x) be the LE-sentence which says, using E instead of E, that x is a structure of type L in which any atomic R(x l,...,x ) is satisfied by ele- ments a1 a if and only if R(al a is true. Finally, let be ,..., m ,..., m ) 5 the conjunction of Now E, E YA and we prove that it 1-defines K. Suppose at first that M E K. Hence cp(M,a) is true. Let N be a transitive set which reflects cp(M,a) and P. M can be expanded to a model of 5 by letting N serve as the universe of sort i elements. For the converse, suppose N k 5. Let N Y M = <... ,M. ,€,. ..> such that Mi is a transitive set. As M k 5, Mi reflects P. Clearly c, is interpreted as a in M. Let A E Mi such that Then A E Str(L) and V(A,a), whence A E K. As K is closed under isomorphism, MLEK.I Corollarv 2.5, For any rudimentary class A: (1) A(LWA) = {Klthe model class K is A1 w.p.i. A}, = A(L~~) (2) A(LIA) = Nthe mode2 c2ass K is Al(Cd) w.p.i. A), I1 (3) A(LA ) = {Klthe mode2 class K is A2 w.p.i. A}. (4) A(Lw ) = {Klthe mode2 class K is Al(Pww )>, n If A 2 H(:), K = A+, X 2 w, then: (51 A(LAK) = {Klthe mode2 class K is Al(Pw ) n.p.i. A}. In [ll] a predicate P(x) is called local if it is of the form 3a(Ra k cp(x)) for some formula ~(x)of set theory, and a proof is sketched to the effect that a predicate is local if and only if it is equivalent to a -predicate. Combining 12 this with Theorem 2.4 yields: Corollarv 2.6. A model class is I(L")-definabZeA if and only if it is de- fined by a local property w.p.i. A. Theorem 2.4 can be immediately iterated to yield a result about ln-defina- 402 J. VGNLNEN bility. A form of the following corollary was first proved by J. Oikkonen in [lo] with a different proof. Corollarv 2.7. Suppose A and A. 5 A are transitive classes, P a predi- cate, and Li a Boolean logic extending LA and symbiotic with P on Ao. Then the following are equivalent for any model class K of type L E A and for any n < w: (I) K is ~,+~(~i)-definabLe, (2) K is Zn+l(P) W.p.i. A. Proof. We use induction on n. If n = 0, the claim follows from 2.4. Suppose then K is ln+l(Li)-definable and n > 0. Let cp E nn(Lz) 1-define K. By induction hypothesis Mod(cp) is nn(P) w.p.i. A. Now K = {M E Str(L)13N E Mod(cp)(NIL = M A N \ cp)}, and therefore K is ln+l(P) w.p.i. A. For the converse, suppose (2) holds. Let S be a Il (P)-predicate such that K is zl(S) w.p.i. A. By 2.4 K is 1-defined by some cp E L[SIA. By 2.1 Mod(cp) is A1(S) w.p.i. A; and therefore An+l(Li)-definable. Hence K is In+,(Li)-definable. Corollarv 2.8. For n > 0 and fGP any rudimentary class A: (1) An(LWA) = {Klthe model class K is An w.p.i. A}, (2) An(Lfil) = {Kithe model class K is An+, W.p.i. A}. Note that K[Pw] is II(L )-definable, whence L[Pw] 5 A (L ) and there- ww 2 wo I1 fore A(LA ) 5 A2(LA). On the other hand ) 5 A2(LWA) 5 A(Lil). Hence in A 2A(L I1 I1 fact A(LA ) (L ), and therefore An(LA ) An+l(LA) for all n > 0. If - A 2A - this is combined with 2.8, the following obtains: Corollarv 2.9. For n > 1, An(LA) = {Kithe model class K is An w.p.i. A}. Therefore in MKM: Li = {Kithe model class K zk definable in set theory w.p.i. A}. The second part of the above corollary was stated on page 174 of [El. A As 1 (Lww ) is just the usual first order logic Lww, and A2(Lu,) is A(L1'), that is, essentially second order logic, it would be tempting to conjec- ture that A (L ) is essentially third order logic. This is not the case, 3 ww however. By familiar methods (see e.g. [91) one can prove that for any analytical ABSTRACT LOGIC AND SET THEORY 403 (this can be improved, see [91) ordinal a the a'th order logic in A-equivalent to second order logic. It seems plausible to put As(LA) above the whole notion of higher order logic and consider it rather as a fragnent of a quite new power- ful logic, sort logic. Similarly it seems implausible to call 1 a second order quantifier, or even a generalized second order quantifier, as 1 leads to some- thing far beyond second and higher order logic, viz. set theory. We return to the problematics of second order logic in the next chapter. The A-operation can be used to give a very near characterization of symbiosis: Proaosition 2.10. Suppose Li is a Boolean logic extending LA and P a predicate. Then AfLi) - A(LIPIA) if and onZg if ISZ) and (~1)~:If $ E L;, then Mod($) is A1(P) w.p.i. A. Proof. Suppose at first that (S1)A and (52) hold. By Theorem 2.1 every L[P] -definable model class is Al(P) w.p.i. whence by 2.4, A A, L[PIA 5 A(Li). Therefore A(L[PIA) 5 A(Li). On the other hand, if K is Li- definable, then by (Sl) K is AlfP) w.p.i. A, whence by 2.4 K is A(LIPIA)- A definable. Hence A(L2) - A(L[PIA). The converse is immediate in view of 2.4. We can use 2.10 to show that the logic L* = L(W,Q1, ...,Qn, ...)n A(L2) = {Klthe model class K is A1(Pn) w.p.i. An for some n < u}. Thus the range of Theorem 2.4 extends to many non-symbiotic logics. For example: d(L(W,Q1,..., Qn ,... = {Klthe model class K is Al(H ,,..., 8,) for some n < w} = {Klthe model class K is A1 w.p.i. {&l,..., yn,-..u. Note however, that Lw w(W,Ql ,.. .,Qn ,... )n (Al) Stc* is I,(P) and cp v ji, cp A Ji, 3ccp and Vev in (L5) and (L6) can be found with l1(P)-functions. (A2) There is a A,(P)-predicate S such that if cp E L*, then The conditions (Al) and (A2) together form a natural notion of P-absoluteness of L* generalizing the notion of an absolute logic in [el. Note that (A2) -+ (Sl). The following lemma is obvious: Lemma 2.11. (11 If L* and P are symbiotic on A and L+ 5 A(L*), then L+ and P A satisfy (SIJA. (21 If L; and P satisfy (A2) and L+ 5 A(L;), then L+ and P satisfy (sl)A. The next result generalizes a theorem by Burgess (Theorem 2.2 in [81) which says that no unbounded absolute logic is A-closed. The proof remains almost the same. Theorem 2.12. Suppose L* is a Boolean logic symbiotic with P on A and L+ N A(L;). Then L+ and P satisfy (SIIA but not (A21. Proof. Suppose S is a A (P)-predicate such that if cp E L+, then 1 Wl E Str(L)(M 1' cp ++ S(M,cp)). Let K = IMlM P K is clearly A,(P). By 2.4 there is a cp E L+ such that K = Mod(cp). Let M = Corollarv 2.13. Suppose L* is a Boolean logic, L* and P are symbiotic on A and they satisfy fA2). Then L1; is not A-closed. Corollarv 2.14. The following logics are not A-closed: (1) LIPIA where P is a predicate of set theory, (2) LQA where Q is a generalized quantifier such that LQ is unbounded. (3) LWA3 LIA. LRA. LY, LWIWl, LWIG. 1 Hence, if Lt\ is a symbiotic logic extending LA, there are no Q ,. . . ,Qn such that A(L~;)- LQ1 . . .Q:. ABSTRACT LOGIC AND SET THEORY 405 The following theorem gives another aspect of the failure of syntactical methods in constructing A-extensions. Recall the definition of Stc in $1. A(L*) Theorem 2.15. Suppose A and A 5 A are transitive classes, P a predi- an abstract Zogic such that there a embedding cate and L; is l,(P)-function LA into I,; and L; is symbiotic with P on Ao. Thex the predicate Stc is n2(P) but not 1,(P). Therefore A(L~)and P do not satisfy A(L;) (A1I. Proof. By definition StCA(L;)(L,X) ++ 39$L0L1 E TC(X) (stc*(Lo,cp) & stc*(L1,q) & = VM E Str(L)(3N E Str(Lo)(NIL = M & N I*cp)* (VN E Str(Ll)(NIL = M + N k* $I)))) This proves that Stc is n2(P). To prove that Stc is not A(L;) A(L;) I,(P), let R(x,y) be a II (P)-predicate which is not 12(P). We construct ll(P)-func- 2 tions f and g such that K[Tl is ll(P) whence by 2.4 there is an L*-sentence (P(C~,C~,C~)which 1-de- A fines K[Tl. For any x let qx(y%E) be the L -formula (see the proof of 2.4) -W Let 9(E) be an L*-sentence which 1-defines the class of models (9(E) & qX(c,3) & qy(c2,E)) -+ dc,,c2,c3). By (Al) we may assume there is a type L' such that I- E L'* and the predi- XY XY xyA cates z = I- and z = L' are [,(P). Let L be the subtype of L' Xy XY XY xy associated with c3,c2,c, and E. Let 6 be an arbitrary valid L*-sentence. We define 406 J. V%&NEN f(X,Y dX,Y Now (*) holds as is not too difficult to see. It follows, for example, that there is no -formula which decides whether 13 a given L’I-formula 1-defines the same model class as another given L’I-sen- mw WW tence n-defines. So, although LEw has a primitive recursive syntax, it seems I1 unlikely that any similar syntax can be found for its fragment A(L ) or for MW A~(L*). We end this chapter with some remarks on decision problems of symbiotic logics. For simplicity we only consider logics of the form L:u. The decision problem of L* is the set Val(L* ) = {cp E HFlcp E L:w and cp is valid}. It WW ww I1 is known (see [13] and [12]) that Val(LWw) is the complete Ti -subset of HF. 2 More generally, if L* is symbiotic with F and L* is sufficiently syntactic (e.g. L* = LQ for some Q), then Val(Liu) is the complete Il (P)-subset of HF. 1 A proof of this can be found in [l3]. For results about Val(LIuw) and Val(LRww) see [141. § 3. Flat definabilitv and second order logic In this chapter we construct the part of set theory which coincides with second order logic in the same way as the whole set theory coincides with sort logic. Definition 1.1. Quantifiers of the form (1) 3x(HC(x) 5 HC(yl U...U yn) & ‘.o(x,yl ,...,y,)) (2) Vx(HC(x) 5 HC(yl U. ..U yn) + cp(x,yl ,. .. ,yn)) are called fZat quantifiers. The set of fZat fomlae of set theory is the smallest set containing ~o-formulae and closed under &,v,- and flat quantifiication. b The lE(P)- and nn(P)-formulae are defined by induction on n as follows: b b Io(F)- and Ro(F)-formulae are just the lo(P)-formulae. (;)-formulae are b formulae of the form (1) where cp(x,y l,...,y ) is nn(P). IIn+l(P)-formulae are b b ZFC- formulae of the form (2) where cp(x,yl ,yn) is ln(P). P)ZFC- and nn(P) ,. . . In( formulae are defined as usual. b It is easy to see that the set of 1 (P)ZFC-formulae is closed under &,v,~,3xEy,VxEy and (1) above. Note that by Levy‘s theorem “41 p. 104) every ll-formula is ABSTRACT LOGIC AND SET THEORY 407 The whole point of flat formulae is the following reflection principle: Lemma 3.2. Suppose cp(xl, ...,x ) is a flat fomla and a an arbitrary set. Then there is a tpansitive set M such that a E M, HC(M) = HC(a) and M reflects cp(x, ,. . . ,x 1. Proof. By the usual reflection principle ([41 p. 99) there is a transitive set N containing a such that N reflects cp(x l,...,x ). For any subfornula $(yl,...,ym) of cp(xl.....x ) and for any b2, b E N let ..., m f=f (b ..,bm) E N such that HC(f) 5 HC(b2 U...U bm) and *(Y,, . . . ,Ym) 2” N 3y1(HC(y1) 5 HC(b2,...,bm) & IL(yl,b2 ,...,b,)) -+ $(f,b2,..., b m ). Choose M to be the smallest transitive set containing a and closed under the functions f where $(yl,...,ym) runs through the subformulae of $(Y1,.-.,Ym)’ dX13.. . ,xn). 0 The above lemma shows, among other things, that every flat formula is AEFc (using Theorem 3.7.2 of [41). Definition 3.3. Suppose L* is an abstract logic, A a transitiue class and P a predicate of set theory. L* and P are strongly symbiotic -_on A if the following two conditions are satisfied b (SSl) If cp E L*, then Mod(cp) is Al(P) w.p.i. {Q,L}. (SS2) K[P] is Ai(Lfi). Strong symbiosis is harder to come by than symbiosis. For exmple,W is 1 not A,(LI)-definable (essentially because in countable domains I is redudent and Theorem 7.3 of 131 can be used), whence LI is not strongly symbiotic. This failure can be regarded as an indication of the incompleteness of the definition of LI, rather than as a characteristic property of LI. The situation is different with second order logic which seems to resist strong symbiosis in an essential way, as we shall prove in a moment. Examules 3.4. The following pairs are strongly symbiotic on HF for my rudimentary A: (1) LIPIA and P, (2) LA(W,Q) and Q, if Q is any generalized quantifier, 13) L~(w)and On, (4) LA(W,I) and Cd, (5) L~(w,R) and Rg. Theorem 7.5. Suppose A 5 HC and A E A are transitive sets, P a predi- cate, and Li an abstract logic extending LA and strongly symbiotic with 408 J. V&~NEN P on Ao. Then the following are equivalent for any model class K of type L E A: (1) K is li(Li)-definable, b (2) K is Z,(P) w.p.i. A. Proof. We follow the proof of 2.4. The implication (1) -+ (2) is obvious. For (2) + (l), suppose cp(x,y) is l:(P), a E A and Vx(x E K t+cp(x,a)). Let a' = TC((a)). Let v be the conjunction of 5 (as it is defined in the proof of 2.4) and the first order sentence which says that there is a bijection which maps all elements of the sorts in one-one to elements of the sorts in L3 L. Using Lemma 3.2 one can prove that v still 1-defines K. But every model of p has the same power as its L-reduct. Hence the new universes introduced by L3 can be dispensed with in favour of new predicates, and therefore v can 1 be converted into a A (L*)-definition of K. 0 1A The proof of Corollary 2.7 carries over immediately and we have: -Corollarv 3.6. Suppose ASHC and A S A are transitive sets, P a pred- icate, and Lz a Boolean logic extending LA and strongly symbiotic with P on Ao. Then the following are equivalent for any model cZass K of type L E A and for any n < w: (1) K is li+,(LI)-definable, b (2) K is ln+l(P) w.p.i. A. Corollary 1.7. For A 5 H(ol ): 1 b (1) An(LA(W)) = iK[tke model class K is An w.p.i. A}, 1 (2) An(LA(W,I)) = {Kithe mode2 class K is Ab(Cd) w.p.i. A}. The following corollary is proved mutatis mutandis as Proposition 2.10: Corollarv 3.8. Suppose Li is a Boolean logic extending LA and P a predicate. Then A:(Li) - A;(L[PIA) if and only if (SSZl and \SSZiA: If $ E Lz, then Mod($) is A1(P)b w.p.i. A. It follows that second order logic is not strongly symbiotic, because there 1 I1 1 is no Q such that L1' (- Al(L )) - A1(LQ). Second order logic is rather 1 the closure of first order logic under the -operation. More exactly, let us 11 define for any abstract logic L*: (L*) = {KI the model class K is AA(L*)-definable for some n < to}. ABSTRACT LOGIC AND SET THEORY 409 1 1 1 Clearly, L1' = A:u)(Lwu) = A(w)(LW) = A(u)(LI). Note that A(u)(L*) satis- fies the single-sorted interpolation theorem. The following characterization of A;u)(L*) and second order logic follows from 3.6: Corollarv 1.9. Suppose Lz is a Boolean logic extending LA and P a predicate such that Li and P are strongly symbiotic on A 5 H(ul). Then the following hold: 1 111 A(w)(LI) = {Klthe model class K is definable with a flat fomla of the language {E,P} w.p.i. A]. (2) Lil = {Kithe model class K is definable with a flat fomla w.p.i. A}. To sum up, second order definability corresponds to flat deflnability in set I1 theory, implicit second order (that is A(L )-) definability corresponds to A - 2 definability in set theory, and finally, definability in sort logic corresponds to definability in set theory. Recall that by Theorem 3.7 of [2], first order definability corresponds to Ay-definability. 1 It is well-known (see e.g. 191) that the II -part of second order logic has 1 already the whole implicit strength of second order logic. Another way of saying 1 the same is Lrr 5 A(LQ,), because % is II,-definable (see [TI). This fact has the following more general analogue: Prooosition 3.1C. Suppose A 5 HC and A. 5 A are transitivz classes and L* is a Boolean logic symbiotic on Ao. If IIi(Luw) I A(Li), then A;~,cL;) c A(L;). Proof. Suppose L* is symbiotic with P on A . In view of 3.9 it suffices to prove that if K is definable by a flat formula of set theory in the language {E,PI w.p.i. A, then K is A,(P) w.p.i. A. Suppose cp(x,y) is a flat formula in the language {E,P} and a E A. Then cp(a,b) holds if and only if there is a strong limit LY such that R(a) reflects P, a and b are in R(a), 1 and R(a) cP(a,b). As the assumption Il (L ) 5 A(LZ) implies that Pw is 1 1 ww A (P) w.p.i. A, the above equivalence shows that cp(a,y) is P) w.p.i. A. 1 I,( Similarly icp(a.y) is Z,(P) w.p.i. A. Proposition 3.10 can be improved by considering suitably defined Aa ( w)-Opera- tions, where a is an ordinal definable in finite order logic or LY E A (see t91). The results about absoluteness of symbiotic logics in the previous chapter carry over to strongly symbiotic logics as follows: I,et us consider the following properties: (SAl) Stc* is $bp) and Qv$, W$, 3ccp and Vccp in (L5) and (L6) can be found with (P)-functions. 11 410 J. VGhN b (SA2) There is a A (P)-predicate S such that if cp E L*, then 1 t/M E Str(L)(M k* cp ++ S(M,cp)). The condition (SA1) and (SA2) together from a notion of strong P-absoluteness of L*. Note that if P is omitted, strong absoluteness coincides with the notion of absoluteness, because every I1-predicate is 1: . So the difference comes only when some non-trivial predicates P are considered. For example, second order logic is &-absolute but not strongly Pw-absolute (see the remarks after 3.8). The following theorem is proved as 2.12: Theorem 3.11. Suppose Boolean L* and P are strongly symbiotic on A and L+ 5 Ai(L2). Then L+ and P satisfy (SSIIA but not (SA2l. Corollary 1.12. Suppose Boolean L* and P are strongly symbiotic on A 1 and satisfy (SA21. Then L2; is not Al-closed. 1 Corollarv 1.13. The following logics are not A -closed: 1 (1) LIPIA, where P is a predicate of set theory, 12) LA(W,Q), where Q is a generalized quantifier. Hence, if Li is a strongly symbiotic logic on A extending LA, there are no generalized quantifiers Q' . . .Qn such that 1 A~(L);) - L~(Q,..., Q~). Also the proof of Theorem 2.15 carries over: Theorem 1.14. Suppose L* and P are strongly symbiotic on V, satisfy b (SAll, and there is a I1(P)-functionwhich embedds Lmw into L*. Tken b b the predicate Stc is I$,(€') but not 12(P). (L*) This theorem shows how difficult it is to find a syntax for A 1-extensions, 1 1 whereas the full A(w)-extension has a simple primitive recursive syntax. The situation is hence -irnilar as in the case of A-extension. 5 4. LGwenheim numbers The purpose of this chapter is to transfer the definability results of 5 2 from the level of model classes to the level of spectra and in particular minima of spectra, that is Lowenheim numbers. Definition 4.1. Suppose L* is an abstract logic and u) E L*. The spectm ABSTRACT LOGIC AND SET THEORY 411 of cp, Sp(q), is the class {card()!) (M k* cpl. The indexed family is called the family of L*-spectra. Examples 4.2. (1) The class of successor cardinals and the class of limit cardinals are LI-spec tra. (2) {x~~IK(K+ x 5 pK)> is an LILspectm. (3) The cZass of regubr cardinals and the class of weakly inaccessible cardinals are LR-spectra. (4) {2KI K a cardinal1 is an LI'-spectm. (5) {zK(K is measurable} is an LII-spectrum. For other examples of spectra see [131 and [141. The following problem is called the spectrum problem for E: Is the comple- ment of an arhitrary L*-spectrum again an L*-spectrum? The spectrum problem for LW, for example, has a negative solution because {K~K5 fi 1 is an LW-spectrum but {K/K > No} is not. The spectrum problem for LI can have a negative answer - this will be discussed later. The spectrum problem for L1' has a positive solution for a rather trivial reason: if C is an L'I-spectrum, then C = Sp(cp) for some identity-sentence Q and the complement of C is just Sp(7 cp). This fact has a more general analogue. At first we note the following trivial lemma: Lemma 4.3. Suppose C is a class of cardinals and C' is the class of structures , where card(A) E C. Then C is an L*-spectrum Cf and only if C' is I,(L*)-definable.1 If this is combined with Theorem 3.5 and Corollary 3.9, the following charac- terization of spectra yields: Theorem 4.4. Suppose A 5 HC and A. 5 A are transitive classes, P a pred- icate, and Lz a Boolean logic extending LA and strongly symbiotic with P on Ao. Then b Sp(Lt\) = {C 5 CdlC is &(P) w.p.i. A}, 1 SP(A(~)(L~))= IC S CdlC is definable by a flat formula in the language {E,P] w.p.i. A}, 412 J. V&hII?,N SP(L;') = {c E CdlC is definable with a fkt formula of set theory w.p.i. A). Definition 4.5. Suppose L* is an abstract logic. The L8wenkeiwnumber Z(L*) of L* is the least cardinal K such that min(c) 5 K for every C E sp(L*), if any such K erist. Equivalently, l(L*) is the least cardinal K such that if cp E I,* has a model, then cp has a model power 5 K. It is well-known that l(L*)A exists if A is a set. Theorem 4.6. Suppose A and A 0-cA are transitive classes, P a pred- icate, and i$ a Boolean logic extending LA and symbiotic with P on Ao. Then for any n < w: 2(An(Li)) = sup {K~Kis IIn(P)-definable w.p.i. A}. If Z(AJL$)) is a limit cardinal (e.9. n > 1 or LI 5 A(Li)j, then moreover l(An(LA+)) = sup {ala is Kn(P)-definable w.p.i. A 1. m. Suppose at first that a is II (P)-definable w.p.i. A. Suppose cp(x,y) is a 1 (P)-formula and a E A such that VB(B 2 a ++ cp(5,a)). Let K be the class of linearly ordered structures the ordertype of which is an ordinal 2 a. K is clearly ln(P) w.p.i. A, whence K is ln(Li)-definable. But every model 'of K has power 2 card(a). Hence a 5 min {card(M)lM E K} < Z(An(Li)). For the converse, suppose K < l(An(Li)). Let cp E Lz such that K X = min(Sp(cp)) < ~(A~CL;~)). Now VB(B > X - 3y 2 83M( IM] = y & M 'F* cp)) whence X is II (P)-definable w.p.i. A. 0 Corollam4.7, For my rudimentary set A and n > 1: (I) l(LIA) = sup {ala is n,(Cd)-definable w.p.i. A), II (2) l(LA = sup {ala is n -definable.w.p.i. A}, 2 (3) l(An(LA)) = sup {ala is II -definable w.p.i. A), (4) l(Li) = sup {a(a is definabk in set theory w.p.i. A) (in MKM). ABSTRACT LOGIC AND SET THEORY 413 Part (2) of the above corollary was proved earlier but independently in 161. LGwenheim-numbers can also be characterized in terms of a notion of de- scribability. This notion is related to the notion of indiscribability (see [4] p. 268) but differs mainly in that less parameters are allowed. Definition 4.8. Let D be a set of fomlae of set theory. An ordinal a is D-describable w.p.i. A if there are a ~(x)E D and an a E Ra rl A such that and R6 q(a) for rk(a) B < a. The predicate P is R-absolute if euery Ra reflects P. Lemma 4.L Suppose P is R-absolute and a is ll,(P)-defimbZe w.p.i. A. Then there is a 6 > a such that 6 is ll(P)-describubZe w.p.i. A. Proof. Suppose cp(x,y) is (P)-formula and a E A such that 11 V6(B 1 a * d6,a)). Let $(y) be the ll(P)-formula 3xcp(x,y) and B the least 6 such that R6 k $(a). Then y -> f3 + R k $(a). Hence $(a) describes B. RE 1 $(a) Y clearly implies 6 > a. Lemma 4.10. Suppose P is R-absolute and a is ll(P)-describabZe w.p.i. A. Then a + 1 is n,(P)-definable w.p.i. A. Proof. Suppose dx) is a ll(P)-formula and a E A such that RB k @(a) if and only if 6 ? a. Let $(y,x) be the ll(P)-fonuula which says that cp(x) is true in a transitive set which reflects P and the ordinal of which is < y. If $(@,a), then (because P reflectslfor some y < 6 Ry 1 cp(a), whence 8 > a. On the other hand, if 6 > a, then Q(6,a) as one can choose Ra as the required transitive set. Corollarv 4.11. Suppose A and A E A are transitive sets, P an R-abso- lute predicate, and Li an abstract logic extending LA and symbiotic with P on Ao, and l(Lz) is a limit cardinal. Then l(L*) = sup Iala is ll-describable w.p.i. A}. :A Proof. The claim follows immediately from 4. 9, 4.10 and 4.6. 0 Lemma 4.12. Suppose a is first order describable (that is described by 414 J. VEGNEN a A. some formula of set theory) w.p.i. A. Then is II 2 -definable w.p.i. Proof. Suppose cp(x) is a formula and a € A such that for B 1 rk(a) Rg 1 cp(a) if and only if B 1 a. Let $(y,x) be the 12-formula "Ry 1 cp(a)". Then JI(B,a) if and only if B 2 a. Corollarv 4.11. For any set A: I1 l(LA ) = sup {ala is first order describable w.p.i. A}. Another way of formulating Corollary 4.11 is the following: Prooosition 4.14. Suppose A and A, C A are rudimentary sets, P an R- absolute predicate, and Li an abstract logic extending LA and symbiotic with P on A. and l(Li) is a limit cardinal. Then l(Lz) = the least a such that In particular, -_ l(Lkl) = the least a such that The above result suggest the study of ordinals a such that Let us denote the predicate (*) of a by Dn(a). The following lemma will be most useful: Lemma 4.15. The predicate Dn(a) is IIn, for n > 1. Proof. Let S(x,y) be the 1 -predicate which is universal for 1 -formulae with one free variable y (see e.g. [41 p. 272). Let F(z) by the A,-predicate "z is a 1 -formula with one free variable y". If F(z), let f(z,a) be the relativization of z to Ra. f is clearly A2. Let So(x,y) be the A1-predi- cate which is universal for 1 -formulae with one free variable y. Now we have: Dn(a) ++ VY E RaVz E w(F(z) + (So(f(z,a),y) v 1 S(z,y))), and therefore Dn(a) is TIn. 0 ProrJosition 4.16. If o is nn-definable w.p.i. A, then there is a 8 2 o such that 6 is An-definable w.p.i. A (n > 1). Proof. Let cp(x,y) be a n,-formula and a E A such that ABSTRACT LOGIC AND SET THEORY 415 Let $(u,v) be the A -predicate "Dn-l(~) & Ru 1 3x - cp(x,v)". Note that D,(a) may not be 111 but by the proof of 4.15 it is A2. Let e(w,v) be the An-predicate 3u 5 w$(u,v). We claim that 1 e(w,a) defines an ordinal 1 a. By reflection there is an ordinal 6 such that $(6,a). Let 6 be the least such 6. If y 1 5 then B(y,a). On the other hand, if e(y,a), then $(&,a) for some 6 5 y, whence 6 5 6 5 y. 0 Corollarv 4.17. For any rudimentary set A and n > 1 : I1 l(LA ) = sup Ia/a is A2-definabZe w.p.i. A). Z(A~(L~))= sup {ala is A -definabZe w.p.i. A}. The predicate Dn( a) is actually equivalent to a LBwenheim-Skolem-theorem, as the following theorem shows: Theorem 4.18. The following are equivalent for any n > 1: (11 Z(A~(L~~))= K, (21 Corollary 4.19. If K is supercompact, then l(A (L )) = K. If K is 2 KW extendible, then Z(A~(L~~))= K. Proof. If K is supercompact, then D (K); if K is extendible, then 2 D7(K). These facts are proved in [ll]. 0 § 5. Hanf-numbers Hanf-numbers can be characterized in the same way as LGwenheb-numbers. One has to bear in mind, however, that A does not preserve(Hanf-numbers (see 1151). Therefore we introduce a new notion of definability, bounded definability, which is neat enough to preserve Hanf-numbers but still almost as powerful as A- or A,- 416 J. V*hN definability. This notion was first studied in [151. The main result of this chapter is Theorem 5.6. The chapter ends with a discussion on definable ordinals and sort logic. Definition 5.1. Let P be a predicate of set theory. A predicate B S(xl ,.. .,x ) of set theory is 1,(P) w.p.i. A if there are a Io(P)- fomk (P(X l,...,~n,~,~)and a E A such that vxl.. .vxn(s(x,,. . . ,xn) f-t 3xcp(xl,. . , ,xn.x.a)) and vx l...~x (IX~(P(X, ,...,x ,x,a)I is a set). s(xl,..., x ) is II~(P)w.p.i. A if - s(xl ,...,xn) is Z:(P) w.p.i. B B A. S is nAl w.p.i. A if S is both ll(P) and II:(P) w.p.i. A. B An example of a A (Cd)-predicate which is not (provably) A,(Cd) is given 1 in [151. Note that every I1-predicate is 1; by Levy's theorem. From the fact ~X(P(X) ++ 3x(dx) & W&rk(y) c rk(x) -t - cp(y)) B it follows that every I,(P)-predicate is I,(P,Pw). Therefore there is no need B to define 1 (P)-predicates for n > 1 - they would coincide with the ln(P)- predicates. Now we define the model theoretic analogues of the above notions. wition 5. 2. Let L* be an abstract logic. A model class K is lB(L*)- definable if it is 1-defined by an L*-sentence cp such that VA ~KVBE E(A,cp)(card(B) 2 K). K is IIB(L*)-definable if i? is lB(L*)-definabZe. K is AB(L*)-definable if it is both lB(L*)- and IIB(L*)-definabZe. A' A' is a natural operation on logics and resembles A-operation so much that it is in fact not at all obvious that there is any difference between them. For a treatment of AB see 1151. We pick up some of the results of [151 to the following lemma (note that (1) below fails for A): Lemma 5.1. (I) A' przserves Lthdenheim- and Hanfnumbers. (2) lB(~*)- Z(L*) if L" 5 A~(L*) or L* is one of the following logics (or a fragment of onel L,~, Lmw(w), L,w(~a), L,~- Hence AB(L*) - A(L*) for such L*. (3) v = L (4) If Con(ZF), then Con(2FC + AB(LI) + A(L1)). Related to the bounded notions of definability is a new notion of symbiosis as well: Definition 5.4. Suppose L* is an abstract logic, P a predicate of set theory and A a transitive class. L* and P are bowdedly symbiotic a A if the following two conditions are satisfied: B (BSZ1 If cp E L*, then Mod(cp) is A,(P) w.p.i. fcp,L} (BS2) K[P] is AB(LI)-definable. The pairs of example 2.3 are all boundedly symbiotic. We omit the proof of the following theorem because the proof would be mutatis mutandis as that of 2.4. Theorem 5.5. Suppose A and A. E A are transitive classes, P a predi- cate, and “2 an abstract Zogic extending LA and bowdedly symbiotic with P on Ao. Then the following are equivalent: (2) K is lB(Li)-defiwlbZe, B (2) K is I,(P) w.p.i. A. Theorem 5.6, Suppose A and A _C A are transitive classes, P a predi- cate, and Li a Boolean logic extending LA and boundedly symbiotic with P on Ao. Then h(Li) = sup Iala is l:(P)-definable w.p.i. A} and for n > 1: h(An(Li)) = sup Iala is ln(P)- definable w.p.i. A}. Proof. In order to prove the two claims simultaneously, let us agree that B l:(P) for n > 1 means ln(P). Now, let n > 0. Suppose that a is ln(P)- definable w.p.i. A. Let K be the class of linearly ordered structures the order type of which is < a. K is li(P) w.p.i. A and therefore l:(Li)- B definable (using 2.4 and 5.4, ln(Li) for n > 1 means ln(Li)). Hence K is B B l(An(LA))-definable. If n > 1, then L1’ 5 An(Lf;) whence by 5.3 (2) K is BB 1 (An(Li) )-definable. If n = 1, the same conclusion follows trivially. Hence B there is a J, E An(Li) which lB-defines K. As K has models of power 5 card(a) only, J, does not have arbitrary large models. But 11, has a model of power Z K B for every K < a. Hence card(a) < h(An(Li)). It follows easily that a < h(A:(Li)). For the converse, suppose K < h(Ai(L2)). Let cp be in A:(L,) such that K 5 X = sup Sp(cp). Now 418 J. VGNXNEN B Hence X is In(P)-definable w.p.i. A. n Corollarv 5& For any rudimentary set A: I1 (1) h(LA ) = sup {ala is 12-definabZe w.p.i. A}. (21 h(An(LA))= sup Iala is Z,-definable w.p.i. A} (n > 1). (3) In MKM: h(Li) = sup Iala is definable in set theory w.p.i. A} = Z(Li). Part (1) of the above corollary was proved earlier, but independently, in [61 (see also 111). Let us write In for sup {ala is Rn-definable), hn for sup {ala is In-definable}. By what we have already proved: (for n > 1) In the next few lemmas we shall establish the mutual relations of the ordinals ,h .n < w. It turns out that the following notation is helpful: Inn t = the least a such that Dn(a) = the least a such that Trivially In 5 t 5 hn for n > 1. Lemma 5.8. FOP n > 0, In < tn. Proof. Let S(x,y) be the In-formula which is universal for 1 -formulae with the free variable y. Let a = {cp(y)Icp(y) is a In-formula such that -I dy) defines an ordinal}. a E Rw+, and therefore a E Rt . Let $(x,y) be a In-formula equivalent to n Vu E xS(u,y). Now $(a,y) is true for some y, whence Jl(a,y) is true for some y E Rt . This y is an ordinal which is greater than any TI -definable ordinal. n Therefore ln _< y < tn. Lemma 5.9. If n > 1, then tn is &-definabZe, and hence tn < hn. ABSTRACT LOGIC AND SET THEORY 419 Proof. Recall from 4.15 that Dn is nn. Hence the claim follows from Va(a < tntf-D (a) &V6 < aiDn(6)). 0 Lemma 5.10. If n > 1, then h = ln+l. Proof. Suppose a is nncl-definable and cp(x,y) is a 1 -formula such that Let $(x) be a ln-formula saying that x is an ordinal and cp(y,6) holds for all y E Rx and 6 < x. If Vx$(x), then VxQ(x,a), a contradiction. Therefore there are 6 such that - $(6). Let 6 be the least of them. Hence if 6 < 6 then (~(6). On the other hand, if *(B) and y 5 6 then $(y), whence y # 6. Therefore $(x) ln-defines 6. Hence it suffices to prove that a 5 6. Suppose the contrary, that is 6 < a. If y E R6 and 6 < 6, then by (*) ~p(y,B). Hence $(6) holds, a contradiction. Therefore a 5 6. 0 Corollarv 5.1 1. 2, < h2 = l3 < h3 = l,, < h,, = l5 <... If the proofs of 5.8-5.10 are carried out with parameters, the following theorem yields: Theorem 5.12. Suppose A is a rudimentary set and n > 1. Then z(An(LA)) < h(An(LA)) = Z(An+,(LA)). Corollarv 5.13. (MKMI 2(L;F) = h(LiF) = the least a such that Ra( V. Proof. Suppose dxl,..., x ) is a formula of set theory and al,..., a sets in Ra, a = ), such that Q(al a ). Let m < w such that l(LsHF ,..., is equivalent to a -formula dxl x NOW R dxl,..., x 1m ,..., 1. +(al,..., a ) for a sufficiently large k < W. We may assume Dm-l(a). Therefore Ra $(al ,..., a 1. Hence Ra @(al,..., a ). For the converse, suppose Ra V. Then every definable ordinal must be < a. Therefore a 2 Z(LS ) HF * Corollary 5.14. If the required cardinals exist, then 1st measurable < 2, < 1st supercompact < h2 = Z3 < 1st extendible < hg. Proof. The predicate "a is measurable" is 1,. H&ce the 1st measurable is Il -definable and therefore < 1,. If K is supercompact, then D,(K) (see 2 [ll] p. 86), and hence < t2 5 K. The predicate "a is supercompact" is Z2 n2' Hence the 1st supercompact is -definable and therefore < h,. If K is ex- 12 tendible, then D (K) (see [ll] p. 103), and hence l3 < t3 5 K. The predicate 3 420 3. V&bEN "a is extendible" is 113 and therefore the 1st extendible is -definable. 13 Hence the 1st extendible < hj. 0 So we see that the Lowenheim- and Hanf-numbers of even the lowest levels of sort logic exhaust a wide range of large cardinals. This would seem to suggest that the logics A ) are rather strong indeed. In connection with note nA(L 5.14, that the ordinals ln,hn,n < w exist even if there are no large cardinals; they exist in L, for example. It seems to be a rather common phenomenon that the Lawenheim-number of R logic is smaller (often substantially) than the Hanf-number (see e.g. 5.12). However, in the second part of this paper we shall construct a model of set theory where the Hanf-number of LI is smaller than the Lowenheim-number of LI. In that model the spectrum problem for LI has a negative solution, because there is a cardinal K between l(L1) and h(L1) such that {h(h K} is a spectrum, but {hlh < K} is (obviously) not. 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