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Finitely Generated Magari Algebras and Arithmetic Lex Hendriks Dick de Jongh Department of Mathematics and Computer Science University of Amsterdam August , Some consequences are studied of Shavrukovs theorem regarding the Magari algebras diagonalizable algebras that are embeddable in the Magari algebra of formal arithmetical theories. Semantic characterizations of faithfully interpretable modal propositional theories in a nite number of propositional letters are given, in particular for nitely axiomatizable ones. Supported by this theory computer aided calculations on the theories of lowest complexity in one propositional letter were executed leading to a complete list of formulas that axiomatize such theories under which the maximal ones of particular interest. Abstract Introduction This paper discusses Magari algebras often called diagonalizable algebras over a nite number of generators. Magari algebras are the algebras corresponding to the provability logic L GL in , PRL in . According to Solovays theorem on provability interpretations the theorems of the provability logic L are precisely those modal formulas that are provable in PA under arbitrary arithmetical interpretations interpreting as the formalized provability predicate in PA. Here, we are concerned with nitely generated Magari algebras that are embeddable in the Magari algebra of Peano Arithmetic. Shavrukov characterized these subalgebras as having the socalled strong disjunction property. Research supported by the Netherlands Organization for Scienti c Research NWO In this paper the terminology of propositional theories i.e. sets of propositional modal formulas closed under Modus ponens and Necessitation is more convenient. Rephrased in that terminology, we study those propositional theories T over L in a nite number of propositional variables that are faithfully interpretable in PA. Theories correspond to lters in the free Magari algebras and interpretability to embeddability as a subalgebra. Interpretable theories T in p pn are those propositional theories in p pn for which there is a sequence of arithmetical sentences A An is a theorem of such that an L formula is an L consequence of T i PA in the arithmetical interpretation in which the atomic formula pi is interpreted as Ai see e.g. , or . Written out T axiomatizes an arithmetically interpreted theory An The faithfully interpretable propositional theories T in Ln i.e., L restricted to the language of p pn are according to Shavrukov the conf j T L g f j PA A g sistent recursively enumerable r.e. theories that satisfy the strong disjunction property T L implies T L or T L . Parenthetically interpretable theories in in nitely many propositional variables need not be r.e. The strong disjunction property may be thought of as being composed out of the simple disjunction property T L implies T L or T L , and consistency T L implies T L . An older concept to which this can be related is the concept of exact provability introduced in see also in the terminology used here a formula can be de ned to be exactly provable if it axiomatizes an interpretable theory. That means that an exactly provable or exact formula of L is a formula which axiomatizes an arithmetically interpreted propositional theory fj L g f j PA A An g The object of our research is to to get an overview of exact formulas of low complexity aided by computerized calculations. For that purpose di erent semantic characterizations of interpretable theories and exact formulas in terms of Kripkemodels have been developed which are of interest in their own right. It turns out that an important role is played by maximal exact formulas, i.e. exact formulas that are not implied by any other exact formulas, and, more in general, by maximal theories with the strong disjunction property. The characterizations of these concepts discussed in this paper make heavy use of the relationship between exactly provable formulas in provability logic and sets of nite types of modal formulas see , rst introduced as characters in . The paper is built up as follows. After a preliminary section , characterizations of interpretable theories and exact formulas are given in section . Then maximal theories with the strong disjunction property and their relationship to maximal interpretable theories, in general, are discussed in section , and maximal exact formulas, in particular, in section . In the last section , it is shown how the theory was applied to calculate the exact formulas in one propositional variable of modal complexity , and the maximal ones among them. We owe considerable thanks to V. Shavrukov for advice and for heeding us from some mistakes. Preliminaries The provability logic L is the modal propositional logic with as its axioms the ones of classical propositional logic as well as all formulas of the forms and , and the inference rules Modus ponens and Necessitation. As usual is de ned as , and we will use the abreviation for the formula . Note that L will mean that is derivable from using the axioms and rules of L including Necessitation but not substitution. This means that L is equivalent to L . We say is interderivable with quot and write for the conjunction of L and L . Note that this implies that always . We reserve the terminology is equivalent to quot for L . . From this point on we will usually abbreviate L to By its completeness theorem, L is the logic of all nite, transitive and irre exive Kripkemodels as proved in and . In the sequel we will frequently use facts about the semantics of modal logic that can be found, for example, in . The Kripkemodels are supposed to be nite, transitive and irre exive, unless stated otherwise. We consider our Kripkemodels to be triples W lt with W its set of worlds or nodes, lt its accessibility relation and its forcing relation. We will sometimes use the notations k l l gt k and k l l k . h i g quot f j quot f j g De nition The level of box nesting of an L formula is denoted by the inductively de ned function p atom p max if . fg f g The fragment Ln will be the fragment with P n p pn as its set m of atoms and the nesting of the box operator restricted by the condition m. fg Fact The Lindenbaum algebra of Ln is a nite Boolean algebra. m In case of a Lindenbaum algebra we will call the atoms of the Boolean algebra irreducible elements. Note that is irreducible in Ln i is not m a contradiction and, for all Ln , implies or m . Such a Lindenbaum algebra is, of course, always a Magari algebra which is de ned to be a Boolean algebra with an additional operator that , and . satis es the laws As noted in , the irreducible elements in Ln are precisely the satm is able formulas of the form ab V i W i where b is an irreducible element of the classical fragment CpLn and the i range over some set of irreducible elements of Ln . m mulas in Ln we have in every Ln Kripkemodel K for every node k m . k forces exactly one of the elements of An . m . If An is forced by k, m then, for every Ln formula k m , Facts Writing An for the set of irreducible equivalence classes of form . De nition The An such that k will be denoted by n k. m m Fact In any Ln Kripkemodel K , n k depends for any k K only on m the set of atoms forced at k and the n k of the successors m k of k in K . De nition Let k be a node in an Ln Kripkemodel K . Then the n mtype of k, tn k, is de ned by m tnk p P n k p tn k p P n k p tn l l K amp k lt l . m m n The set of all such n mtypes is written Tm . hf j g i hf j g f j gi Although we prefer this semantic de nition of n mtype in the context of this paper, we agree that the distinction between the semantic n mtypes and the n k essentially what is called the n mtype of k in is just a m matter of point of view. If t Q X , we write j t for Q and j t for X . The following facts are easily veri ed and re ect the close relationship between the tn k and m n k. m hi Facts Let K be an Ln Kripkemodel and k l be elements of K . n . k is of exactly one of the types in Tm . n . If t Tm is the type of both k and l, then for every Ln formula k , l . m . The n mtype of k depends only on the set of atoms forced at k and the n mtypes of the successors of k. Facts , and combine into Fact The set of n mtypes corresponds exactly with the set An of irrem ducible elements of Ln in the sense that m lKl nkm , tn l tn k m m Another useful, easily veri able fact is the following Fact The n mtype t of k K uniquely determines the n mtype of k K . We write t m for this n mtype. Lemma immediately leads to Corollary If K is a Kripke model such that each n mtype occurs ex k K k fjg actly once in K , then the subsets of K correspond exactly with the equivalence classes of Ln . That is, the following function is an isomorphism m It can be proved that such an exact model, in which each subset corresponds to a formula and vice versa, exists for each Ln . m for Ln , i.e., for each U K , there is a formula in Ln such that m m fk K j k g U , and K is n mcomplete, in the sense that for all n Lm fk K j k g fk K j k g i . Proof. We apply the socalled Henkin method to the up to equivalence nite set of formulas in Ln , which is closed under taking subformulas. This m gives one a Kripkemodel with the maximal consistent sets as its worlds, with lt de ned by for each , both and are elements of , and pi by pi . The maximal consistent sets can be replaced by their conjunctions which are exactly the irreducible elements of Ln . A subset of m the model corresponds then to a disjunction of irreducibles, i.e. an arbitrary formula of Ln . Obviously, nonequivalent formulas are forced on di erent m subsets of the model. a Theorem For each n and m there exists an exact Kripkemodel K In general, unlike in the case of intuitionistic propositional logic see , , not all the exact models of Ln are isomorphic. For the in nite fragment m Ln there is no such exact model in which all subsets determine a formula, but there is a canonical in nite model which is ncomplete. Here we baptise this canonical ncomplete model ExLn . This model gives considerable insight into the free Magari algebra over n generators. Its construction which was rst given in and is given presently somewhat sloppily but, we hope, clearly after the introduction of the notion of depth of a node. De nition Let K be a Kripkemodel and k K . The depth of k, k, is de ned by k if k is a terminal node k max l k lt l otherwise. fjg tively de ned ExLn for m . m fg De nition ExLn with its lt and , is de ned as the union of induc ExLn p pn , the elements of ExLn are all ltincomparable, and Q p p Q h ExLnS Q X Q p pn m X i m ExLn X ExLn X closed upwards m i QX ltY Y X , and Q X p p Q ExLn Si ExLn . i fh i j f g i , h i , g The above de nition is such that ExLn will contain the elements of ExLn i of depth exactly i. This immediately brings along with it that ExLn is conversely wellfounded. It is obvious from the construction that each n mtype will be realized by some k ExLn . The latter ensures the ncompleteness of ExLn . It is convenient to us to execute most of our constructions inside this model. For many of these constructions this would not have been necessary. Example The construction of ExL and ExL yields pv S C S C CS CS CS C S C S C S C S C S S C C S p Cv Sv p Cv S v v pv v Let us write and n n . The following facts about the nodes of ExLn will be useful in the sequel. Facts k m m . If k l m, m gt and tn k tn l, then k l m m . If k m and l n k m m , then k l. m .km , These facts suggest a kind of normal form for the irreducible formulas corresponding to the elements of ExLn . Then n k n k m m . m De nition Let k ExLn and assume k m. From the ncompleteness of ExLn we conclude that for k ExLn the are the irreducible elements in Ln . However, from the counterexamples in we know there is no nice bound for the m one has to use when one wants to write a formula as a disjunction of irreducible formulas of Ln . nk Interpretable theories and exact formulas in Ln As stated in the introduction, Shavrukovs theorem in gives a characterization of the exactly provable formulas in L or exact formulas for short. Fact A formula L is exact i strong disjunction property is s.d. is not a contradiction and has the L or The property in this fact is called steady by Shavrukov . Whether a formula Ln is steady or not, Shavrukov also proved, depends only m on its behavior with regard to other formulas in Ln m Fact A formula for formulas in Ln m Ln is exact i m is not a contradiction and is s.d. Ln m or For a simple proof of this last fact see . We will transform this characterization of exactness into a semantic one. That will turn out not to work for interpretable theories in general. The characterization will apply however to the maximal interpretable theories of the next section. If T De nition n k ExLn k . fjg is a propositional theory in Ln , then nT fk ExLn j k T g Obviously n and n T will always be closed upwards in the sense that, if e.g. k n and k lt l, then l n . Theorem A formula Ln is exact i n is nonempty and extendible downwards, i.e. k l n h n h lt k amp h lt l . Proof. Let be an exact formula in Ln . As unequals the contradiction by de nition, we have n by the completeness of ExLn . To prove the second condition, let k l n , and n k n l be representatives of the irreducible classes in Ln corresponding to k and l. Assume that, if h n and h lt k, then h lt l. Then, again by the completeness of ExLn , we would have nk n l, or equivalently n k nl. As is supposed to be exact, would either prove nk or n l, in contradiction with the assumption that k l n . Hence, there should be an h n such that h lt k and h lt l. n and , and assume there are k l n such Let L that k and l . By the last condition of the theorem, there is an h n such that h lt k and h lt l. As we would then have h , we obtain a contradiction. Hence, we proved that or . a By the completeness of L noninterderivable and give rise to distinct and n . This is in general not so for theories. An example is the theory axiomatized by p on the one hand, and the theory T axiomatized by m p for each m, on the other. The sets p and T are the same, consisting of all nodes that together with all their successors force p, but clearly the theories are not p is not a consequence of T . Similarly, the theory T m p p for each m can be shown to have the strong disjunction property. But not all pairs of nodes in T have a common predecessor in ExL , because T consists of those nodes that together with all their successors force p and those nodes that together with all their successors dont force p. This shows that the semantic characterization of exactness does not generalize to interpretability of non nitely axiomatizable theories, at least if one doesnt freely use in nite models. There is a restricted class of theories that does respect the characterization. De nition T has the nite model property f.m.p. i T implies that there is a nite Kripkemodel for T on which is falsi ed. Fact T has the nite model property f.m.p. i for each k nT k T If T has the f.m.p., then T is determined uniquely by n T . Also the semantic characterization of exactness immediately generalizes by the same proof. Theorem A theory T in Ln with the f.m.p. is interpretable i nT is nonempty and extendible downwards. n f j g Later we will encounter the nite model property in a syntactic form. Lemma An L theory T has the f.m.p. i , for each , T , for each n T n . Proof. Assume T has the f.m.p. First let T . By its f.m.p. T has a nite model with a node k falsifying . Supposing k has depth m ensures that T m . The other direction is trivial. If T , then the right hand side implies that T n for some n, i.e., there is a model for T of depth n falsifying , a nite model. a One sees from the proof that actually the negative formulation of the right hand side of the above lemma is the natural one. Note that the n of an exact Ln is in nite by the conditions of the characterization. On the other hand there is a simple correspondence between such an in nite set and a nite set of n mtypes in ExLn n De nition Let be an Ln formula. Then Tm tn k k . m fjg By the results in the previous section this is a sound de nition and, in consequence, in particular every exact Ln formula corresponds uniquely to m a nite set of n mtypes. Grantedly, this de nition is somewhat unnatural by the insertion of the . The reason for this is, of course, that we are interested in the types of the nodes in n k ExLn k . fjg Lemma Let and be Ln formulas. m n n Then Tm Tm implies . n n Proof. Let Tm Tm , and assume k . Then n k T n T n . Hence tn k tn k for some k that forces . tm m m m m So, k and, since Ln , k . The other direction is the same. a m Lemma Let be an Ln formula. m n n Then Tm ft m j t Tm g. Proof. Obvious, considering fact . a n closed subset K of n such that the elements of K exactly realize Tm . Lemma For each Ln formula and each m there is a nite upwardly Proof. Just take any nite subset of n such that its elements exactly n realize Tm . The upward closure of this set will do, because its elements also force . a To nd the sets of n mtypes suitable for exact formulas , we have to translate the conditions on the n of exact into conditions on the n underlying set of n mtypes. For example, for a nite Tm to correspond n , it is necessary that some type in T n can have itself to an in nite m as a successor. To describe this kind of re exivity we introduce the notion of a re exive type. n De nition A term t Tm is called re exive if t m j t. The following theorem is related to lemma . of . n t Tm Theorem A formula type. Ln with m gt is exact i there is a type m n , which, of course, makes t a re exive such that j t Tm n Proof. Let Ln be an exact formula. Note that Tm is a m n be nite and closed upwards such that nite set of types. Let K n n ftm k j k K g Tm , as guaranteed to exist by lemma . According to theorem we can nd an h n below all of the elements of K . By lemma this h must have a type as required. n Assume and t to ful ll the conditions given. As Tm , also n . Suppose k l n . Let K be a nite upwardly closed n subset of n such that k l K and ftn k j k K g Tm m compare lemma . Consider a world h just below this K such that n j h pg j t. It will be clear that tn h t and since is asfp P m sumed to be an Ln formula this proves h n . Of course, h lt k and m h lt l, so the conditions of theorem apply to n . a The theory developed in this article has enabled us to calculate the exact formulas in L . This will be explained in more detail in the last section. It will be shown that already in this very rst small fragment there are noninterderivable members. It turned out that it was worthwile to single out the maximal elements of these . More general than maximal exactness the concept of maximal s.d. theory turns out to be in itself an interesting one. The next section will be devoted to this last concept and the one after it to maximal exact formulas. Maximal theories with the strong disjunction property and maximal interpretable theories In this section it turns out to be fruitful to not restrict ones attention to r.e. s.d. theories, but also include nonr.e. ones, i.e. noninterpretable theories. A theory T is maximal s.d. if T is consistent and s.d. and, for no consistent U with T U , U is s.d. An Ln formula is maximal exact if is exact and, for each exact Ln formula such that , also A theory T is consistent if T implies T A theory T is maximal consistent if T is consistent and, for no U with T U , U is consistent A theory T is of in nite height i.h. if, for no n , T n A theory T is maximal i.h. if T is i.h. and, for no U with T U , U is i.h. A theory T is maximal interpretable if T is interpretable and, for no U with T U , U is interpretable. Note that the concept of a maximal interpretable theory is a very natural one. Such a theory, and in particular also a maximal exact formula, describes a maximal condition that arithmetical sentences can be required to satisfy. Lemma Each i.h. theory is contained in a maximal i.h. theory. Proof. By Zorns lemma. a De nition nT Lemma T is maximal i.h. i , for each , either T fg Proof. Assume is such that not T , and, for no n T f g n. Then T can be properly extended to an i.h. theory containing which means that T is not maximal i.h. Assume T is not maximal i.h., i.e., for some i.h. U , T U . Let be an element of U , but not of T . Then not T , but not T f g n for any n either, because for no n, U n . a n , or, for some Lemma If T is s.d., then T is consistent. If T is consistent, then T is i.h. Proof. Trivial. consistent. a Lemma If T is a maximal i.h. theory, then T is s.d. and hence Proof. Let T be maximal i.h. To show that T is s.d., assume T . Now apply lemma to both and to obtain that, either T or, for n , and the same for . It is su cient to exclude the some n , T possibility that T n for some n as well as T m for some m. In that case it would follow that T n and T m which implies T maxn m , contradicting the in nite height of T . fgfgfg a mal s.d. Lemma T is maximal i.h. i T is maximal consistent i T is maxia Proof. From the previous lemmas. Corollary . Each s.d. theory in Ln is contained in a maximal s.d. theory. . If T is r.e. and maximal s.d., then T is recursive. . Not each interpretable theory in Ln is contained in a maximal interpretable theory. Proof. Part is trivial. Part follows from lemmas and . Part follows from part by considering two recursively inseperable r.e. sets of natural numbers A and B and then constructing the theory Un f n pnA j g f n n pnB jg The theory U is s.d., since any two models can be joined by a new root taking care only that the new root may need a speci c valuation for p, and r.e., and hence interpretable. A maximal recursive extension of U would seperate A and B and, therefore, cannot exist. a One would expect it to be easy to prove that each maximal interpretable theory has to be maximal s.d., but we have not been able to show this, so we leave it as an open question. Open Question Is each maximal interpretable theory maximal s.d. Lemma If T is maximal s.d., then T has the f.m.p. Proof. Assume T for a maximal s.d. theory T . By lemma , for some n T n. Lobs theorem implies then that T n . Now apply lemma . a The next theorem uses the above to provide a semantic characterization of maximal s.d. theories. k n T , there is an m such that for all l n T of depth m, l lt k. Theorem T in Ln is maximal s.d. i T has the f.m.p. and, for each Proof. Let T in Ln be maximal s.d. That T has the f.m.p. is lemma , so it su ces to check the second point. For that purpose, consider k nT . Obviously n k is not a theorem of T . So, for some m , T nk m . Equivalently, T m nk nk. This implies that, for all l n T of depth m, l lt k, since n k uniquely describes k. It is su cient to check the condition of lemma under the assumption that the right hand side of the present theorem is satis ed by T . So, assume T . As T has the f.m.p., there is a nite model for T that falsi es , given as quotk for some k ExLn . Since, for some m , all nodes with depth m in n T are below k, T m . By contraposition, T m which was to be proved. a A natural question concerning L is whether there are any maximal interpretable theories that are symmetric with respect to p and p. Rossersentences are interpretations of maximal exact formulas see the last section, but they are not examples of such symmetry, since p p is in the theory of the form of the Rosser sentence, and p p in the dual theory. It is easy to see that no s.d. theory can contain both p p and p p without containing p or p itself use the excluded middle. In the next section it will become clear that an exact formula in p cannot be symmetric in p and p, and maximally exact. So, an example will necessarily have an in nite axiomatization. Consider the subset K of ExL de ned as follows S K KS K K p Km p Si m Ki i m i m m The theory T k for all k K is maximal s.d. it is easily checked that it satis es the conditions of theorem and symmetric. An axiomatization of T is given by ff g g fhf g i h ig f j g Example A symmetric maximal interpretable theory in p. ppp ppp pp pp p p This axiomatization is somewhat more perspicuous than the equivalent one formulated with the p . A sentence interpreting p such a theory in PA is a kind of symmetric Rossersentence which can, of course, neither be , nor . Note, in fact, that p p as well as p p are strongly nonprovable for such a sentence, in the sense that e.g., p p is provable for it. Note also that p p couldnt be provable for an arithmetic sentence. Proof that the axiomatization is complete for T . Assume T . There exists a possibly in nite Kripkemodel K satisfying T on which is falsied. It su ces to show that each mtype realized in K can be realized in Km Si m Ki . For types this is obvious. Assume some mtype t is realized in K . All types in j t are, by induction hypothesis, realized in Km . Assume that of the elements in Km realizing types in j t, k is in the Ki with highest index i. It will be obvious that the axioms of T imply that there has to be a type present in j t that is realized by the only other element of Ki . This implies that t is realized by an element of Ki , which will do. a Maximal exact formulas This section will be devoted to maximal exact formulas. First we will have to sharpen our semantic characterization of exact formulas. Let us exploit the relationship between irreducible classes and semantic types to write n t m for the n k with tn k t. m m De nition W C be a set of n mtypes. Let Then n C m f n t j t C g. m fj n Recall that Tm tn k k m . g Lemma If Ln , then m nTnmm . a Proof. Immediate from lemma . n n Lemma If C Tm m gt , then C Tm for an exact formula . there is a nite upwardly closed K ExLn such that C ftn k j k K g we will call C upwardly closed realizable m . there is t C such that t C t m j t. Moreover, in that case type for C . n C . Such a type t will be called a enveloping m Ln i m n Proof. If Ln is exact, then Tm will have the required property m n by the de nition of Tm , property by theorem , and satis es the nal requirement by lemma . n C is an exact formula. To apply theorem to We prove that m n C , we have to nd an appropriate re exive n mtype. By the assumpm tion on C , there is an n mtype t C such that t C t m j t. Let K be the upwardly closed realization of the types in C as assumed in the rst conditon of this lemma. Note that K realizes precisely the n mtypes in ft m j t C g compare lemma . Let k be a new root immediately below K such that k forces exactly the elements of j t. Then n tn k t. So, k n C and, hence, t is a member of Tm n C C m m m and a type appropriate for the application of theorem . a We will prove that the maximal exact classes in Ln correspond to what we will call tail models in ExLn . Clearly this is a result that to a large extent is bound to the particular model ExLn . De nition K ExLn is called a tail model i . K is closed upwards . there is an m such that fk K j k mg is linearly ordered by lt and all nodes of this set force the same atoms. Our de nition of tail model slightly di ers from the one in , in that Vissers tail models are equipped with a minimal in nitedepth element. equipped with a tail descending from k with forcing of the atoms de ned as on k. De nition If k ExLn, then k is the tail model consisting of k quot quot Lemma If quot k n. Ln , k n and k has a re exive n mtype, then m Proof. First note that all elements of the tail have the same n mtype as k. Hence, all these nodes force , and consequently . a Lemma If K ExLn is a tail model, then K n for some in Ln. Proof. Let K be k , k having depth m. Let be tion of the atoms and negated atoms as they are forced W K n for de ned as the conjunction of m n k. and m quot f the conjuncon k. Then n k k k m jga Theorem If model in ExLn . Ln , then is maximal exact in Ln i n is a tail proof. Assume n is a tail model and Ln . Since n is in m n nite and Tm nite it is obvious that the tail has to contain elements appropriate for an application of theorem . This shows that has to be exact. Assume to be an exact formula such that , i.e., such that n n . Then, because n is nonempty and extendible downwards it has to contain the tail elements from a certain node onwards, and, because it it is closed upwards it has to contain all other elements of n , which means that and are interderivable. Hence, is maximal exact. n Let Ln be maximal exact. Assume t Tm is a re exive type m with the properties guaranteed to exist by theorem . Now, take as in the proof of lemma k n with n mtype t such that n tn k Tm . By lemma , k n . By lemma , there exists m n k n . Since was asssumed to be maximal exact, a with this means and the tail model constructed is n . quot quot quot a From this theorem the remark we made that maximal exact formulas in p cannot be symmetric with regard to p and p becomes immediately obvious. The tail is always asymmetric. We follow with some more properties and problems concerning maximal exact formulas. n Lm is maximal exact, then there is precisely n . Moreover, T n j t. one re exive type t in Tm m Proof. The last part follows immediately from theorem . Assume Ln m quot quot Theorem If a formula with m gt is maximal exact. Assume s and s to be two distinct n mn types in Tm . If k and k in n realize s and s respectively, then, by lemma , k and k are two distinct tail models within n . This contradicts the fact that n is a tail model. a Examples of nonmaximal exact L formulas with exactly one re exive type will be given in the table in the last section. It is unclear to us whether exact Ln formulas that are maximal with respect to Ln formulas m m are maximal overall. De nition An exact Ln formula is called mmaximal exact i , for m all exact Ln such that m , . It will turn out in the last section that the maximal exact formulas in L are maximal exact. The fact that, by lemma , Ln formulas are m determined by their n mtypes leads to the following insight. C that contains exactly one re exive n mtype t and for which C is minimal upwardly closed realizable, in the sense that, C is upwardly closed realizable, but this is not the case for any proper subset of C containing t. Fact The mmaximal exact Ln formulas are the ones with a set of types m A conjecture is that the set of n mtypes of an arbitrary exact Ln form mula is the union of the sets of types of the mmaximal exact Ln formulas m from which is derivable. That such a union always is the set of types of an exact formula if at least an enveloping type is present follows immediately from the next lemma. Ck of n m types corresponding to exact Ln formulas with an enveloping type t common to all k m n C Ck , then there exists a Ln such that Tm C . m Proof. It su ces to note that, if K Kk are upwardly closed realizations of C Ck , then K Kk is an upwardly closed realization of C , and then to apply lemma . a Lemma If C is the union of sets C It is certainly not true that any union of types of mmaximal exact formulas is the set of n mtypes of some exact Ln formula. A counterm example is provided by the sets of types belonging to p and to p, both maximal exact formulas, which cannot be combined to an exact formula, even for m . A common enveloping type is needed, and is obviously not available for p and p see section . Another, related fact worth mentioning is that any exact formula in Ln is determined by the maximal exact formulas from which it is derivable. This can be derived from the observation that each downward chain in n will ultimately only contain nodes k with re exive types, and that k n for some maximal exact formula from which is derivable, combined with the observation that, if for Ln formulas and , n and n are distinct, then there is a node m with a re exive n mtype in the one, but not in the other. Certainly this does not generalize to interpretable theories. The s.d. theory T axiomatized by n p for each n that was introduced after theorem provides a counterexample. Its only maximal s.d. extension is the one axiomatized by p. quot Calculating exact formulas In this section the calculation of the exact formulas in L will be discussed. It will be shown that already in this very rst small fragment there are noninterderivable members with maximal elements. Of the next fragment L even the cardinality of the set of maximal exact elements has eluded us so far. The fragment L is already de nitely too large to attack in this manner. To calculate the exact formulas in L we worked with the set of types. To start with, there are eight types enumerated in T that may be ordered in such a fashion that the result is an exact model this notion was introduced after corollary . There are more possibilities, but here are two exact models of L , the left one coinciding with the rst two levels of ExL . v vp v p v Q S C S C CS CS CS C S S C C S C S C S S C C S p Cv Sv p Cv S v pv v p vH vH H H HH HH H HH H p v HHv A Q A Q A pA v QQv In the chosen exact model in our case the left one the logical connectives that are used to form formulas correspond with settheoretic operations which are given as input to the algorithm. Systematically then all L formu las are generated with their truth sets, throwing out any formulas the truth set of which has occurred before. Also the nonre exive types and are distinguished from the re exive and . This enables the algorithm to check whether the upward closure of the truth set of the formula has an enveloping type see lemma . If such is the case, then the formula and the upward closure of its truth set is noted down in the list of exact formulas, if at least the upward closure of the truth set has not occurred before. The list of exact L formulas that came out will be given at the end of this section. We rst give the maximal exact formulas among them the ones with minimal sets of types as they were generated by the program, preceded by their sets of types ffgg ff g g ff gg ff gg pppppppppppp p ppppppppppppp For the bracketing has a higher priority than . Of course, which formula from an interderivability class is given by the program depends on the settings of the program like the priority rule for the connectives to be used and a selection on the shortest formula made. We will give the formulas a more informative form. pppppppppppppppppp Formulas and correspond to provable and refutable sentences in PA. Formulas and can be faithfully interpreted by Godelsentences and their duals in PA. Similarly, formulas and correspond to Rossersentences and their duals in PA. The only small surprise is formed by formula and its dual . It is easy to see that is interderivable with p and then, of course, with p . These two formulas are not L , but can apparently interderivably be given as such. Note also that, by the xed point theorem of L see e.g., , , there is no surprise in the fact that in the equivalences to the p in the right hand side can be eliminated in favor of the , but only in the fact that using p one can push down the complexity. The reader will enjoy providing the tail models that show that these maximal exact formulas are actually the maximal exact L formulas. As mentioned above we do not know whether such a state of a airs holds for m gt . The exact L formulas that the program churned out will be given below without comment and with only very slight editing. They will be given with their sets of types that show that each such set is a union of sets of types of maximal formulas. Again, as mentioned above, we do not know whether this is generally the state of a airs for m gt . pp p p p p ffff gggg ppppppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppp fffffffffffffffffffffffffffffffffffffffffffffffffffffff gg ggggggggggggggggggg ggggggggggggg gg ggggggggggggg gggggg p p p p p p fg p p p p p p fg p p p p p f g References J.F.A.K. van Benthem, Modal Logic and Classical Logic, Napoli, . 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