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, ,. .. ,Mn;R,,. ,Rm;a,,. ,al;>. 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* = <Stc*,k*> such that (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.