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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
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