Closed Fragments of Provability Logics of Constructive Theories

Closed Fragments of Provability Logics of Constructive TheoriesAuthor(s): Albert VisserSource: The Journal of Symbolic Logic, Vol. 73, No. 3 (Sep., 2008), pp. 1081-1096Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/27590316 .

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The Journal of Symbolic Logic Volume 73. Number 3. Sept. 2008

CLOSED FRAGMENTS OF PROVABILITY LOGICS OF CONSTRUCTIVE THEORIES

ALBERT VISSER

Dedicated to Craig Smorynski on the occasion of his 60th birthday

Abstract. In this paper we give a new proof of the characterization of the closed fragment of the

provability logic of Heyting's Arithmetic. We also provide a characterization of the closed fragment of the

provability logic of Heyting's Arithmetic plus Markov's Principle and Heyting's Arithmetic plus Primitive

Recursive Markov's Principle.

?1. Introduction. Friedman's 35th problem was to give a decision procedure for the closed fragment of the provability logic of Peano Arithmetic, PA. See [5]. It was indepently solved by van Benthem (see: [12]), Boolos (see: [1]) and Bernardi & Montagna (unpublished). The story of this result is told in [3]. The solution of Friedman's question would seem to be the end of the story of closed fragments: the characterization given is simple and definitive. Moreover, as we now know, it is amazingly stable. The same characterization works for all 2^-sound recursively enumerable extensions of Buss' theory S2.1 Very roughly speaking, one could say that it works in all theories in which one can arithmetize employing the usual recursive presentations of syntax. Thus, it almost works for all theories for which the question makes sense at all.

However, the situation changes dramatically, when we go on to consider the prov

ability logics of constructive theories. New obstacles arise to the proof. We proceed to the next level in difficulty. Different theories will have markedly different closed

fragments. In my preprint [14], I gave characterizations and corresponding decision

procedures for the closed fragments of H A and HA*.2 The characterizations are rea

sonably simple. However, in the case of HA, the verification of the characterization involves a complicated algorithm. I was not happy with its presentation and tried to improve it in [15]. Finally the result was published in [16]. Unfortunately, this final presentation still employs the original unperspicuous algorithm.

Received October 15, 2007.

1991 Mathematics Subject Classification. 03F30, 03F45, 03F50.

Key words and phrases, provability logic, constructive arithmetic.

I thank Rosalie Iemhoff for her insights and comments. I am grateful to Nick Vaporis for his very useful systematic list of closed formulas. I thank the referee for his comments and questions.

lrrhe case where we do not have ̂ -soundness just gives rise to a trivial variation.

2For information about HA* see [13], [4], [16] and [17].

? 2008. Association for Symbolic Logic 0022-4812/08/7303-0023/S2.60

1081



1082 ALBERT VISSER

In the present paper. I will give a new proof of my old result by formalizing a Smory?ski style Kripke model argument. It is somewhat surprising that the

possibility of such a formalization was so long overlooked. One reason that it was so easy to overlook, is that one embeds classical methods inside a purely constructive

argument. The new proof, with minor adaptations, also yields a characterization

of the closed fragments of the provability logics of HA + MP and HA -f- MPpr.

Why are these results interesting? We can view the question from two perspec tives. The first perspective starts with the arithmetical theories. These theories are natural theories. This study throws light on a salient property of these theo ries. The second perspective is the perspective of the closed fragments themselves. The description of the closed fragment is a direct generalization of the second in

completeness theorem. In the context of the results for constructive theories, the classical case appears in a new light. We understand it as a special case of a more

general pattern. Thus, enlarging the scope of our enquiry to the constructive case

tells us something essential about closed fragments. Finally, apart from any spe cific perspective, one can say that a beautiful underlying mathematical structure is revealed.

1.1. Prerequisites. We presuppose a background in provability logic. See e.g., [2] or [9]. We also presuppose basic knowledge of constructive arithmetic. See e.g.,

[10] or [11]. The chapter [8] of [10] will be especially relevant.

?2. The framework. The results of the paper fit a reasonably simple framework. In this section we describe this framework.

The question we are studying is to characterize the closed fragments of the

provability logics of certain arithmetical theories. This introduces two kinds of theories and their connection. We have closed constructive provability logics and arithmetical theories. The connection between the kinds is formed by translations of the language of modal logic without propositional variables into the language of arithmetic. In our framework we will employ a third kind of theories, to wit theories of degrees of falsity. The theories of degrees of falsity are connected to closed provability logics via a translation of their language into the language of modal logic without propositional variables. We proceed to introduce these three kinds of theories and the relevant translations.

2.1. Degrees of falsity. The intended arithmetical meaning of the degrees of

falsity is: iterated inconsistency statements for the given arithmetical theory T. Let co+ := co U {oo}. We let a, ?, ... , range over the degrees of falsity. We use

m, n, ... , for the finite degrees of falsity, i.e., the natural numbers. We equip oo+

with the usual ordering and define oo + 1 := oo. Note that the successor function remains injective under this extension.

The language QJ of theories of degrees of falsity is the language given by:

</>::= a | (0 A c?) | (0 V <j>) \ {</> ^ </>).

The theory Basic is given by intuitionistic propositional logic with 0 in the role of _L and co in the role of T, plus the principles h a ?> /?, for a < ?. A theory of

degrees of falsity is any extension of Basic in 9J. We will use A, T, ... , to range over arbitrary theories of degrees of falsity. We identify -i 0 with (0 ?> 0).

CLOSED FRAGMENTS OF PROVABILITY LOGICS OF CONSTRUCTIVE THEORIES 1083

An extension of Y of Basic is called p-sound iff it does not imply any sentence of the form a ?? ?, for ? < a. We will be mainly interested in p-sound theories. Here are some further important notions.

A theory Y is decent if, for every c? and for every n larger than all finite m

occurring in 0, we have Y \- n -^ c? implies Y \- c?. ar(0) := max{a | Y h a ?>

c?}. If there is no such maximum, c?r(4>) is undefined.

The decency of Y is a rather natural sufficient condition for the totality of ap.

Theorem 2.1. If Y is decent, then, for any c?, we have that ar(</0 is defined. Moreover, ay((?) is either oo or a finite k occurring in <j>.

Proof. Suppose Y is decent. Let n be the maximal finite number occurring in c?. In case Y h (n + 1) ?> c?, we have, by decency, that cxr((?)

= oo.

Otherwise, there is a maximal k < n such that T h k ? > c?. So ay((?) ? k.

We show that k occurs in (?. Let c? be the result of replacing every occurrence of

any m > k in </> by oo. We have Y \~ k ?> (c? <-? 0). It follows that T h fc ?> 0, and, hence, that Y h c?. Let ? be the smallest s > k occurring in c?. Clearly, r h k ?> (0 <-? 0), and hence F h ? ?? 0. So, by the maximality of k, we have that k = k, and, hence, that /c occurs in c?. 3

Here are some salient extensions of Basic.

Strongl?b := Basic + {((a -> ?) ^ ?) \ ? < a}, Stable := Basic + {^^a ?+ a \ a e +}, Classical := Basic + {a V -i a | a G co+}.

Each of these theories is p~sound and decent. (In this paper we will prove the

p-soundness and decency of Basic and Stable.) 2.2. Closed constructive provability logics. We turn to closed constructive prov

ability logics. The modal language Jzf^ of closed logics is given as follows.

0::=?|T|(0A0)|(0V0)|(0^)|D0. The theory /GL?, intuitionistic L?b's Logic on zero variables, is the 2? -theory axiomatized by intuitionistic propositional logic plus the following axioms and rules.

LI. h (Uc? A H((? -

if/)) -

Dy/, L2. h U(? -> DD0, L3. h D(D0 -></>)

- D0,

L4. h 0 => h Bc?.

A closed provability logic is any extension of /GL? in 2? that is closed under L4. We let A, 0, ... , range over closed provability logics. We define the modal degrees of falsity as follows.

D?J_ := _!_,

nn+l?_ := DD"1,

d??_L := T.

We translate QJ into Jz?S via the translation em bo given by the following clauses.

embo(a) := Da_L,

em bo commutes with the propositional connectives.

1084 ALBERT VISSER

Consider any closed logic A. We define:

TDFA := {0 G QJ | A h embo(0)}. The theory TDFa is the theory of degrees of falsity of A.

For 0, y/ in QJ: 0 KA ty :<^> A h embo(0) ?> embo(^). The relation h a is the relation of provably deductive consequence w.r.t. A.

We write ?A for the induced equivalence relation.

Note that TDF is monotonie w.r.t. theory extension.

Since em bo is a fixed embedding we will, when no confusion is possible, omit it.

Thus, e.g., we will treat TDFa as if it is a fragment of A. We will write aA for ajDFA We will say that A is p-sound if TDFa is p-sound and that A is decent if TDFa is decent.

We prove a number of basic theorems. Consider any closed provability logic A.

Clearly, ^tdfa and h a are preorders on the language of QJ of theories of degrees of falsity. Moreover, Ka extends ^tdfa- I.o.w., the identity mapping supports a

functor n a from the preorder category I~tdfa to the preorder category KA.

Theorem 2.2. Consider any p-sound closed provability logic A. Suppose nA has a

left adjoint <Da whose range consists of degrees of falsity. Then we have:

1. <Da is a functor, i.e., a is monotonie.

2. <DA(0) hTDFA 0 3. 0 KA<M0). 4. OA(0) ~A 0 5. Oa(/?)

= ?.

6. The function aA is total and aA(4>) ?

OA(0).

Proof. Items (1), (2) and (3) are well known facts about adjunctions, noting that we have suppressed nA, since it is supported by the idenity mapping. See [7]. Item (4) follows from (2) and (3). We prove (5), By (2), we have Oa(/?) ^tdfa ?> So, by p-soundness, we have:

?Da QS) < ?- Moreover, we have: ? KA ?\(?). SoAhD^ DOA(]J), I.o.w., A h (? + 1)

- (Oa(/?) + !) By p-soundness, we have ? + 1 < Oa(/?) + 1, so

? < *a(?S). We prove (6). It is sufficient to show that Oa satisfies the conditions for aA. First

note that OA(0) I~tdfa 0. Suppose ? ^tdfa 0. It follows that ? KA 0, and, hence, Oa(/?) ^tdfa ^a(0)- Since ? =

0A(/?) and A is p-sound, it follows that

? < 4>a(0). H

We state a sufficient condition for the existence of a left adjoint of 7rA.

Theorem 2.3. Suppose A is a p-sound closed provability logic. Suppose further that aA is total and, for all 0 in Q, we have 0 Kac?a^). Then, aA is the left adjoint ofnA.

Proof. Let A be a closed provability logic. Suppose that aA is total and 0 Ka

aA(0). Suppose aA(<p) ^tdfa ty- Then, 0 KA aA(0) and aA(0) KA ty. We may

conclude that 0 KA *//.

Conversely, suppose 0 Ka ^- Since, oA(0) ^tdfa 0, we have aA(0) KA 0. So, we find: aA(0) KA 0 and 0 Ka ty and j// Ka cxA(ty). We may conclude:

<^a(0) I~a ol\(w)- By thep-soundness, wefindaA(0) < a\(y/). We may conclude

ol\() hTDFA OLK(x/j) and aA(y/) hTDFA V- Hence, aA((?) hTDFA W- 3

Remark 2.1. Note that the proof did not use all the data of the theorem. We did not use the maximality of qa but only the fact that a\((?) hTDFA (?.

Let r be a theory of degree of falsity for which oly is defined. We define the function

nfr: ̂S ~^3f.

nfr(-L) := 0, nfp commutes with the propositional connectives,

nfr(n</>) := ar(nfr(0)) + 1.

We write nfA for nfTDFA- We have the following theorem.

Theorem 2.4. Let A be a p-sound closed provability logic for which a a is a left adjoint ofn\. We have Ah</>^ nf a (</>).

Proof. The proof is a simple induction on c?, using Theorem 2.2. 3

Theorem 2.4 yields a characterization of closed provability logics A satisfying the conditions of the theorem in terms of nfA, since we have A h 0 iff nf a( </>)

= oo. In its turn nfA is completely determined by a a, which is fixed by TDFa- This result will be central in our characterization of the closed fragments of HA, HA + MP and HA + MPpR.

Remark 2.2. Note that we can view the result as follows. Suppose A satisfies the conditions of Theorem 2.4. We can now enrich the language of A with new constants a?now not considered as defined?and we can add axioms ha <-> ?a _L

to A. We extend the necessitation rule to the system with the new axioms. Let's call the resulting system A+. Now Theorem 2.4 tells us that we have box-elimination inA+.

We introduce a way to construct closed provability logics from p-sound theories of degrees of falsity. Consider any theory of degrees of falsity Y for which ap is defined. We define:

ALr:={0e^oD |rhnfr(0)}. Theorem 2.5. Let Y be a p-sound theory of degrees of falsity for which ap is

defined. Then, ALp is a closed provability logic. Proof. Since nf r commutes with the propositional connectives, we find that ALr

is closed under the propositional tautologies and modus ponens. Consider any c? and y/ in 2q . Let 9 := nfr(0) and v := nfr(^). We show that ALr satisfies LI. We want to show that:

T h nfr((?0 A U((? -> y/)) -> Di//).

By p-soundness, this translates to:

min(ar(0) + l,ar(0 -> v) + 1) < ar(v) + 1.

This follows from:

min(ar(0),Qr(0 ~^ v)) ^ or(v).

This last desideratum is immediate from the fact that, for any ? e +,

Y ^ ? -+ 6 an?Y Y- ? ^ (6 -* v) ̂ Y ^ ? -> v.

1086 ALBERT visser

We verify L2. We have, using p-soundness:

nfr(nn0) = ar(nfr(n0)) + 1 = ar(ar(nfr(0)) + 1) + 1 = ar(nfr(0)) + 2 =

nfr(?0) + l.

The desired result is an immediate consequence. We turn to the verification of L3. Note that:

nfr(?0^0) = (K(?) + l)^?). Clearly, F h ar(6)

- ((ar(6) + 1)

- 9). On the other hand, if ar(6) / oo:

TF (aT(6) + 1) -> ((ar(0) + 1) -> 0). Hence, whether q?y(9) = oo or not, we have c*r(nfr(n0 ?? 0)) =

or(nfr(0))- The desired result follows immediately.

Finally we verify L4. Suppose Y h nfp(0). It follows that e*r(nfr(0)) = oo. So

nfr(D0) =

ar(nfr(0)) + 1 = co. Hence T h nfr(D0). H

Example 2.3. The operation AL is not monotonie w.r.t. theory extension. E.g., we do have (a) ALBasic H (-^ _L ?> ?_L) ?> !_, but, on the other hand, we have (b) ALStabie V- (->-> !.

- D_L)

-? DDL

Here are the expected consequences of our definition.

Theorem 2.6. Suppose that Y is p-sound and that cxy is total. We have:

1. TDFALr=r, 2. ay is left adjoint to nAi_Y, 3. A = Al_r iff OLY Is left adjoint to nA, Proof. We prove (1). Suppose 9 is in QJ and suppose ALr I- embo(#). Then,

T h nfr(embo(?)). It is easily seen that nfr(embo(#)) = 9.

We prove (2). Let 9 be in QJ. Note that ALr h??^ rJar(0), since:

nfr(ae) = ay(0) + 1 = nfr(nar(0)). We may now apply Theorem 2.3.

We turn to (3). From left to right is immediate from (2). We treat the right to left direction. Suppose ar is left adjoint to nA. By Theorem 2.2(6), it follows that or = cxA, and, hence, that T = TDFa and nfA = nfr. By theorem 2.4, we have:

Ah0^>AhnfA(0) ^TDFAhnfA(0) ^>Fhnfr(0) & ALr r- 0. H

Remark 2.4. The box of ALr is in a sense the minimal box of a closed provability logic which has T as its theory of degrees of falsity. Suppose ap is total. Let A be any closed provability logic such that TDFA = r. We have, for 0 in Q:

Ah (or(0) + l) -> nor(0) ->

D0.



Question 2.5. Let Y be p-sound and suppose that ap is total. Consider all logics A with TDFa = r ordered by theory extension. It is easy to see that z'GL + {embo(</>) | Y h <?} is the minimal element of this structure. Moreover, since every formula of

ALp is equivalent to embo(0) for some c? in QJ, we have, for every strict extension 0 of ALp, that TDFe is a strict extension of Y. Ergo, ALp is a maximum of our structure. It would be interesting to have more information about this structure.

E.g., are there other maxima?

Question 2.6. Can we axiomatize AL?asic and ALstabie? I conjecture that such an axiomatization will demand extended schemes involving finite disjunctions and

conjunctions of variable length.

2.3. Arithmetical theories. Let T be any constructive ce theory with a designated interpretation K of iS\, the intuitionistic version of Buss' S3>. We write UTA for the formalization of: A is provable in T. This formalization is supposed to be executed 'inside' the interpretation K. We interpret the formulas of 2? in T, via an

interpretation embf that commutes with the propositional connectives and sends

0 to BT embf ((?). We define the closed fragment of T as follows.

CFr :={</> | T\- embf(0)}. Note that CF need not be monotonie w.r.t. extension of theories. We define an

interpretation of embf o? 2$ into the language of T by:

embf := embf o embo.

The degrees of falsity ofT are the formulas ?^_L := embf (a). We write TDFj for:

TDFcFr,and h 7-for: hcF7 We will study the closed fragments of three salient theories: HA, HA + MPpr,

and HA + M P. To prepare our results we will consider a specific class of theories

containing the target theories. Let's say that T is a ha-theory if T is a theory in the language of arithmetic that is HA-verifiably a n!>-conservative extension of HA.

Note that this notion is intensional. the arithmetical formula defining the axiom set of a given theory should be part of the data specifying the theory. This formula is needed to make sense of the question of a theory being a ha-theory. We can easily provide examples of pairs of theories that are extensionally the same of which one is a ha-theory and of which the other is not. For the theories we are considering like HA + MP, we assume that they are defined by a natural formula.

The ha-theories include HA, HA*, HA + MP. HA + MPPR, HA + ECT0, MA := HA + ECTo + MP and PA. Note that ha-theories are Efi-sound. Note that it follows that the closed fragment of any ha-theory is p-sound.

For all ha-theories U. V, and W, we have U h ?^_L <-> E^-L, since any

ha-theory is a HA-verifiably a II^-conservative extension of HA. So. modulo U

provable equivalence, the degrees of falsity of any of our theories are the same as the corresponding degrees of falsity of any other of our theories. This justifies us in suppressing the embedding embf in the context of any ha-theory. We will treat

TDFr simply as a fragment of T. We use aT for aTDfT, etc.

For the theories HA, HA*. HA + MP. HA + MPPR and PA, all the information about the closed fragment is contained in the theory of the degrees of falsity. In

fact, for each target theory T of this list, we have CFj = ALjDFr. Here are the

theories of the degrees of falsity of our target theories.

1088 ALBERT VISSER

1. TDFha =

Basic,

2. TDFha* = Basic + {((a -> ?) -> ?) \ ? < a} =: Strongl?b,

3. TDFha+mp = TDFha+mPpR

= Basic + {^a ?> a \ a e co+} =: Stable, 4. TDFpa := Basic + {a V ^a \ a G co+) =: Classical.

We will provide a new proof for (1) (see also [16]), and we will prove (3) for the first time in this paper. For (2), see [16]. Finally, (4) is an immediate consequence of the classical answer to Friedman's problem. We will show that, for the theories HA, HA + MP, HA + MPPR, the mapping

aT is the left adjoint of nT. Moreover, aT is computable. This implies that nfT is

computable. Since we have, for 0 in J?f?, that p h 0 iff nf j(u0) = oo, we obtain a decision procedure for CF^.

For the proofs of the analogous facts for HA*, see [16]. For PA, see the classical literature.

?3. Small beer. In this section we collect some small facts in the environment of our main result. This section can very well be skipped.

3.1. Structural facts about theories of degrees of falsity. A first observation is that

ap, if defined, is monotonie in T. One can show by a Kripke model argument that the theories Strongl?b and Classical are maximal among p-sound theories.

Theorem 3.1. Classical is the unique maximal p-sound extension of Stable.

Proof. Let T be a p-sound extension of Stable. Suppose Y h 0. We have:

Classical h 0 <-? l\ia ?> ?),

for some selection of pairs a, ?. Since we have the principles of Classical inside double negation, we find: Y \->-> f\(a ?? /?). Ergo, Y h /\(a ?> ?

?>jff), and, hence, Y h /\(a

?> /?), since Y extends Stable. So, we find, by p-soundness, a < ?, for each pair a, ? in our conjunction. It follows that Classical h 0. H

We may conclude that TDFMa ? Classical, thus obtaining an upper bound. One can show that TDFMa is strictly between Stable and Classical. Further study of

TDFma?evidently the most exciting theory of degrees of falsity?will have to wait for another paper.

Question 3.1. 1. Are there more examples of maximal p-sound theories? 2. Is Stable the mimimal theory that has Classical as unique maximal p-sound

extension? 3. Are there interesting subtheories of Strongl?b that have Strongl?b as unique

maximal p-sound extension?

3.2. n^-sentences of theories of degrees of falsity. Suppose HA h 0 ^ P, where P is n!| and where 0 is (a standard interpretation of a sentence) in QJ. By reasoning in Basic, we have HA I?i-h0

<-> f\(a

?+ ->-"/?)> where a,? ranges over a finite set

of such pairs with ? < a. We find that: HAhP^(a-^-n-i^). Hence by the Friedman translation w.r.t. ?:

HAY-PP^ ((a \/?)^ (((/? V ?) -+ ?) -* /?)). Since P is n^ we have HAhP^ P?. Ergo, HAhi>^(?^ ?).

Conversely, we have HA h /\(a ?

?) ?> -i->P. Say P is Vx Sx, where Sx is Zj. By Friedman translating w.r.t. Sa, we get:

HA h f\((a V Sa) -> (/? V Sa)) -? ((Vx (Sx V Sa)

- Sa) -> Sa).

Hence HA h /\(a ^ ?) -^ Sa, and, so, HA h /\(a -> ?) ?> P. Combining we get:

HA h /\(a^ ?) <r+P. Note that our argument works for every ha-theory for which Friedman's transla

tion works with S^-formulas as 'superscript formulas'. For ha-theories T that are extensions of HA + MPpr we can reason more directly.

Let T be a ha-theory that extends HA + MPpR. Clearly for any n^-formula Q, we have rhnngng, Suppose T h P <-> c?, for P in nf? and </> in .2b. We have T \-i->P <-> ->-'/\(a ?> /?), by elementary reasoning. Since both P and

A (a -> jS) are n?, we find: rhP^/\(a^ ?). 3.3. TDF as a mapping of theories. We note that TDF^ is monotonie in T. Here

is another simple insight. Let T be a ha-theory. Then, HA + TDF^ is a ha-theory and TDFpiA+TDFr

= TDF^. It is important to note that to understand the insight properly we have to think intensionally. E.g., the formula defining the theory HA + TDFj goes into the verification of the fact that it is a sub theory of T. We assume that this formula is an obvious formula involving the formula defining the axiom set of P.

Question 3.2. If we run through ha-theories which theories of degrees of falsity are assumed. I.o.w, what is the range of the mapping TDF? E.g., is every ce r-sound

theory of degrees of falsity extensionally equal to the TDFr of a ha-theory 7?

We provide a modest partial result.

Theorem 3.2. Let Y be an intensionally given ce extension of Stable and suppose that:

HA h Va, ? ( ?HA+MPpR+r?u: ->?)- aHA+MPPR(a -

?) )

Then, HA + MPpr -i-Y is a ha-theory and TDFha+mppr+f = T.

Proof. We assume the hypothesis of the theorem. Reason in HA. Suppose P is n^ and that Dha+mPpr+f^- Then, for some finite conjunction y of formulas from T, DHA+MPpR(j

?> P). It follows that ha+mppr(~,-i};

?? P). Moreover, for

some finite set of pairs a, ?, we have DHA+MPpR(^-ny <-> /\(a ?> /?)). So we have

(i) nHA+MPpR+r A(a -> jff)and(ii) ?HA+MPpR(A(a -* ?) ~> p)- By our hypothesis,

we have from (i): ?ha+mppr A(a -? ?)- So, by (ii), we find Dha+mppr^- Hence, ha^. We may conclude that HA + MPpR + Y is a ha-theory. Suppose HA + MPpr + Y h c?. Then, since Y extends Stable, by the results of the

next section: Y h c?. So, Y = TDFHA+MPpR+r- H

?4. The main result. We first provide Smory?ski style Kripke arguments to char acterize TDFha, TDFha+mppr and TDFHa+mp- Then, by formalizing these argu ments, we prove the existence of a left adjoint of the embedding of h^ into hr, considered as preorderings on 2, for 7 one of these three theories.

1090 ALBERT VISSER

4.1. Kripke model arguments. We show that Basic provides a complete axioma tization of TDFHa- A Kripke model for Basic is a Kripke model in the usual sense

with a G + as atoms for the atomic forcing. Here we use 0 in the role of _L and oo

in the role of T. We demand: k\V n and m > n implies k I h m. A Kripke model for Basic is finite iff it has a finite number of nodes and, for some number N, we have that N is forced in all nodes.

Consider any Kripke model 3? for Basic such that 5? ? 0. Let TV be larger than all finite atoms occurring in 0. We transform our model to a model Jf ' by resetting the atomic forcing to: k Ih' n iff k Ih n or n > N. It is easy to see that 5ff is

again a model of Basic and 5Z1 ? 0. We collapse J?f' by dividing out the maximal

bisimulation, thus obtaining a finite model f of Basic with f ? 0. We have shown:

Theorem 4.1. The class of all finite Kripke models for Basic is complete for Basic.

It follows immediately that Basic is decent. We have:

Theorem 4.2. For all 0 in QJ, ?Basic(0) is defined. Moreover aBasic(0) is com

putable by a multi-exponential algorithm.

Proof. The fact that ?Basic is total, follows from decency. Let N be the smallest finite degree of falsity not occurring in 0. We construct a

finite Henkin model for ordinary propositional logic for the formula

ty := (N A /\{(k ->?)\k<?< N}),

where we take as the nodes of the model saturated subsets of the subformulas of

0 and ty plus oo. One may show that, for k < N, we have Basic h k ?> 0 iff <%? lh k ?> 0. It follows that we can determine a?Basic(0) by running through X. H

We define, for finite Kripke models of Basic:

Vk is the largest n such that k ? n.

Note that in a finite Kripke model the atomic forcing is completely determined by the Vk- Consider any ̂ -formula 0. Suppose Basic Y- 0. We show that HA Y- 0. Consider a finite Kripke model for Basic that does not force 0 at the root. We unravel this model to a finite Kripke tree, say ̂'. We build a model J/'* of HA on

the frame of OF. We associate to every node k a model Jtk of PA + -> vk + (vk + 1) as follows.

To the root b we assign an arbitrarily chosen model of PA + -

v^ + (vj, + 1 ). (Such a model exists by L?b's Theorem and the Completeness Theorem.)

Suppose we have already assigned Jtk\ok. We will have insured that Mk satisfies PA + -i vk + (vk + 1). Let I be a direct successor of k. In case v? = v^, we put Jtt := Jtk. In case v? < v*;, we proceed as follows. Note that -i Vk is PA-provably

equivalent to con (PA + -

(v^ ?

1)). Since v? < Vk ?

1, we have:

J?k H PA + con(PA + -.v?).

By L?b's Theorem, it follows that:

Jtk h PA + con(PA + -nv? + (v? + 1)).

By the Feferman-Henkin construction, we can find an interpretation

&vt : (PA + con(PA + -.v? + (v? + 1))) > (PA + -.v? + (v? + 1)).

We take ^ := MOD(^Ve)(Jtk), i.e., the internal model of Mk given by %fn. Clearly: Jtt |= PA + ^v? + (v? + 1).

Since the cone above each node is internally definable in the model associated to that node, we find, by Smory?ski's reasoning, that ET* I h HA. See [8]. Moreover, since, for ̂-sentences S, we have k I h S <^> Jtk |= S, we find: A: I h n <^ ̂ |= n.

By our construction, it follows that fl is equal to ET* if we restrict ?T'* to the degrees of falsity (modulo embs). We may conclude that fJ"* ? c?. We can easily adapt our argument to establish a similar result for HA -f- MP and

HA + MPpR. The proof of the finite model property for Stable is word for word the same. It follows that Stable is decent.

Let's say that a finite Kripke model for Basic is upwards reflecting if, for every node k there is a top-node k' > k. such that k II?> vk, i.o.w., such that the atomic

forcing of k' is the same as the atomic forcing of k. By a simple argument one can show that a finite Kripke model for Basic is a finite Kripke model for Stable iff it is

upwards reflecting. We may conclude:

Theorem 4.3. We have completeness for Stable in the class of finite upwards re

flecting Kripke models for Basic.

Also the argument for the multi-exponential computability of astabie is the same.

Theorem 4.4. We have: astable is total and computable by a multi-exponential algorithm.

We clearly have that Stable is a subtheory of TDFHa+mppr and TDFha+mp- Sup pose Stable h (?. We show that HA + MPh </>. Let ET be an upwards reflecting finite Kripke tree such that the root of ̂ does not force c?. Consider the model SA* constructed above. Note that, by the construction, it has the property that above each node k there is a top node k' such that J?k> = Jtk. We claim that ?T* forces Markov's Principle. Suppose k IF Vx (Aox V -Âox)

and k II??-i 3x A^x. Let k' be a top node above k such that J?ki ? J?k. Clearly, k' IF 3xAqx. Let a be a domain element such that k' IF Aâ. We find that k ? Â?a, and so, by decidability, k IF Aoa. Thus, k IF 3x Aqx.

It follows that Stable provides a complete axiomatization for TDFHa+mp and

TDFha+MPpr

4.2. Formalization of Kripke models. Our next step is to verify the completeness side of the two above proofs inside HA. The result gets the form: HA h UTc? ?>

UTap((?), where 7 is one of our three target theories. By Theorem 2.3, this is

precisely what we need to show that ap is left adjoint to up. Thus, we obtain the desired characterization.

To realize our program, we will have to eliminate the reference to models and

replace it fully by talk about interpretations. In fact we will view k IF A as an arithmetical formula B. Let y be any finite tree model of Basic which forces some finite N. Let the ordering of iT be -<. We write -<\ for: direct successor in the

^-relation. We define a translation as follows.

Suppose/: -<\ t. Then, %?u := ID, iff v^ = ve an?%fke ' =

%fy,t?fvi < vk. Here

%fVi is as defined in Subsection 4.1. We write Fk? for (a formula representing) the Dedekind embedding of ID into %fkt> I.o.w., Fk? is an arithmetical formula that represents the initial embedding of the numbers of the given extension of PA into the numbers of the interpretation XVf.

1092 ALBERT VISSER

Suppose k = ?o -<\ ?\ -<\ -<\ tn-\

= k'. Then:

We write ?kk> for 6&kk,. It is convenient to give our definitions by restricting the given Kripke model to the cone above a given node k. Thus, our definitions are for 'the model as it looks according to k\

Suppose k ^ k' = ?0 -<\ i\ -<\ - into ffkk<>.

We define a mapping ( ) lh(.) ( ) from pairs of nodes and arithmetical sentences to arithmetical sentences.

t (k'?kPx) :+* (Px)*""'. t k' ?k ( ) commutes with _L, T, conjunction and disjunction. t (k'?k(Ax -> Bx)) :<

Nk??k> Vy((?Fkk>k>>yAk" ?k Ay) -> k" \Vk By). t (k' ?k VuAux) :<->

l\kn^k,Vu\?kknVy(xFkkfkny^> k' ?k Auf). y (k' \\-k 3u Aux) :<-> 3u:?kk> k' \\-k Aux.

Let Tk := PA + -.v* + (v* + 1).

Fact 4.5. Suppose k^k' <k" <?. We have in Tk:

?. %fkkn =

%?kk, O <%V?:" .

ii. 4*" =? iii. F^/?

= Fkk?? o Fkk>k?.

iv. Fkk?? =

F^kk;,?.

Proof. We treat (iv). Suppose k" = mo - k'k"i ^ mN_2m?_i ^ u ^ momi /

- (F^k'mn-2 \^kk/ (F^k'>?0)^kk,

_ ^o^?,n_ xH.,o^?,?0 ? L mn-2^n-\ * mom]

= Fkkn?. H

Fact 4.6. Suppose k < k' -< k". We have: Tk h (k" ?k A) <- (k" hk, A)*"'.

Proof. Suppose k -<k' -< k". The proof is by induction on A. For the base case, we reason in Tk. We have: k" lh Px iff (Px)^kk". But

= Xkk'O&k'k?- So, (Pj?)***" iff ((p*)**'*")***' iff (?" lh^ Pj?)***'. The cases of J_, T, conjunction, disjunction and existential quantification are

easy.

We treat the case of implication. The case of the universal quantifier is similar. Reason in Tk. We have:

3Note the reversal of order here. This is because the MOD-functor is contravariant.

k" IF* {Bx -+ Cx) ~ /\ \Jy((xFkkl/iyAi?k By) -* t \Vk Cy) l>k"

~ A V^: &w ̂Fk>t'?iy^ tt H' By)***')

^k" -^(k?tyM\H> By) - ? IF*/ Cy))*"' ?>Z.k"

^(k"\Vk, (Bx-> Cx))*"' H

Fact 4.7. Persistence: Suppose k -< k' < k". We have:

Tk V- ((xFkklk,yAk' IF* Ax) -> k" IF* Ay).

Proof. The proof is by a simple induction on A. The atomic case uses the fact that the Dedekind embedding is an embedding of structures. H

Fact 4.8. We have, verifiably in HA, that, for any finite Yx, ifYx Hqc Ax* then'.

Tk h a v^: ?k> (( A k'lh*fij?) ->k'lh^A*)' k'hk BxeTx

Proof. The proof is by induction on the number of steps in the IQC-proof. H

Fact 4.9. Let a finite tree for Basic be given. We have, verifiably in HA, that, for any k' >k and for any axiom C of HA, we have Tk \- (kf \\-k C).

Proof. We treat, for example, the axiom h Vx Sx ^ 0. Reason in Tk. We have to show that /\k,yk Vx : ?k> ̂ (kf \\-k Sx =

0). In other words, we want to show that:

A Vx:ok,^(Sx=0)*"'. k'yk

But this is trivial, since ?^*,*/ is an interpretation of PA. We turn to the case of induction. We reason in HA. Suppose we have verified

7*/ h (k" IF*/ C), for all k" >z k' >- k and for all instances of induction C. Fix k and k'. Consider any Dxy. We define:

pxogx(Dxy) :<-? DOy A Vz (Dzy ?> DSzy). indx(Dx/) :<-+ (progx(Dxy)

? MxDxy).

We want to prove: Tk \- y: ?kk> ?> (k' IF* \x\?x(Dxy)). We have, for k" y k,

Tk" H (k" ?k? mdx{Dxy)).

Since, %fkk>> is an interpretation of Tk>> in Tk, it follows that:

Tk h /: <W - (*" H" ?ndx(?x/))^''. (f) We reason in 7*. Suppose that k" >z k' and /: Skkn and /c7' IF* progx(Dxy).

Case 1: Suppose first that k" y k. It follows that (k" IF*// progx(7)xjT))^,/. By (f), we have (k" IF*// Vx Dxy)*"". Hence, k" IF* Vx 7>x/.

Case 2: Suppose k" = k. It follows that kf = k. We have to show /: IF* Vx Z)x/. Note that by persistence, we have, for any ? y k, and any z such that yFkk?z, that I IF* progx(Dxz). Hence, by Case 1, we find ? IF* Vx Dxz. So it is sufficient to show that Mxk IF* Dxy. From k IF* progx(7>x/), we can derive: progx(k IF*

Dxy). Hence the desired result follows by the induction scheme of the classical

theory 7*. H

1094 ALBERT VISSER

Fact 4.10. Suppose Sx is a Sj-formula. Then

Tk h x: ?kk, -+ (S^kk'x^ (kf ?k Sy)).

This fact is verifiable in HA.

Proof. The main part of the proof is by induction on the complexity of Ao formulas A. The addition of the initial block of existential quantifiers is easy. We use the fact that we already know that HA is forced in our finite model, that HA

proves the decidability of A0-formulas and the fact that, for decidable A,

HA h ^\fx<t Ax <-? 3x<tÂx.

We treat the case of the bounded universal quantifier. Suppose A is A0 and satisfies the induction hypothesis. We reason in Tk. Suppose /: ?kk> and x is not among the y.

k' ?k Vx<tyAxy->\/x:Skk, ((x < tyYkk' -> k' ?k Axy) -> \/x:Skk> ((* < tyYkkl -? A^kk'xy) -> (\/x<tyAxyYkk',

k' ?k Vx<tyAxy ?> k' \\-k ^x<tyAxy ? k' \Vk 3x<tyÂxy -+ 3x:?kk, ((x < tyYkk' A k' ?k Âxy) -+ 3x:?kk, ((x < tyYkk' A k' ?k Axy) -> 3x:?kk, ((x < tyYkk' A ̂ A*kk'xy) ->

(3x < ty-ÂxyYkk' - -?

(Vx < tyAxyYkk'. H

Remark AA. There is an alternative proof of Theorem 4.10 following an idea of Smory?ski which avoids most of the inductive cases: one uses the fact that

Matijacevic's theorem can be verified in HA. This proof has the advantage of being simpler. It has the disadvantage of using a major theorem. Moreover, it does not downwards generalize?as far as we know?to e.g., z7A0.

Fact 4.11. Let a finite Kripke tree for Basic be given. We have, for k < k',

verifiably in Tk, thatk' ?k vk> andk' ?k (vk> + 1). Proof. Reasoning. We have already seen that a:7 lh^ HA. We also have/:7 lh^ v'k

iff vk,kk'. Since, %fkki interprets Tk>, we have ->

vk,kk'. Ergo, k' ?k v'k. The other case

is similar. H

Fact 4.12. Let a finite Kripke tree for Basic be given. Suppose k -< k'. Suppose k' Ih 0. We have Tk h (kf \\-k 0). Similarly, ifk' ? 0, then, Tk h (k' ?k 0).

Proof. The proof is by induction on 0 using Fact 4.11. H

Theorem 4.13. Consider any 0 in 2?. We have: HA h ?ha0 ?> DHAo;Bas?c(0) Proof. In case a?asic(0)

= oo, this is trivial. We assume that aBasic(0) is finite.

By Kripke completeness, we can find a finite Kripke tree for Basic with root b, such that and b ? 0 and vb = aBas?c(0). Reason in HA. Suppose nHA0. It follows that

D:rb(b Ihb 0). On the other hand, since b? 0, we have: ^rb(^ ^b 0). Ergo, ?^l. Hence, DPA((v{, -f- 1) ?> v&). By L?b's Theorem, in combination with the verifiable

n^-conservativity of PA over HA, we find that DpAVb and, thus, ?HAVb- In other

Words, ?HAU?BasicW. ^

To adapt our proof to HA + MPpr and HA + MP. The only extra step is to verify in that, for a finite upwards reflecting tree ST, we have, in the theory HA, that k' IF* (HA + MP). The argument needed to do that closely follows the argument at the end of Subsection 4.1. So we have:

Theorem 4.14. Consider any c? in 5?0- Let U be either HA + MPPR or HA + MP. We have: HA h Uu(? -> astable (<?)

We have proved the conditions of Theorem 2.3 for the pairs CFHa> Basic and

CFha+mPpR, Stable and CFHa+mp> Stable. This completes our characterization of the closed fragments of HA, HA + MPPR, and HA + MP. We have:

CFha =

AL-Basic,

CFha+MP =

CFhA+MPpr =

ALstable

Example 4.2. We provide an example of a theory that does not conform to our

pattern. Define 7 := HA + {(n -> (1 V ->1)) \n eco}. Since 7 is between HA and

PA, we find that 7 is a ha-theory. Suppose that 7 h Dr(l v -.1). It follows by the ̂-soundness of 7 that 7 h

1 V -ni. By a simple Kripke model argument we can show that 7 has the disjunction property. So 7 h 1 or 7 I?il. Quod non.

Suppose that 7 h ?r(l v -il) <-> n. It follows that 7 h (n + 1) ? n, and hence

7 f- ?. Quod non.

We may conclude that, for no ?, we have 7 h Dr(l v ->1) <-> ?. Note that TDF^ is not decent, so the question remains for a decent example that does not conform to the pattern.

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[12] J. F. A. K. van Benthem, Doctoral Thesis, 1974, Manuscript, Mathematical Institute, University of Amsterdam.

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DEPARTMENT OF PHILOSOPHY

FACULTY OF HUMANITIES

UTRECHT UNIVERSITY

HEIDELBERGLAAN 8

3584 CS UTRECHT THE NETHERLANDS

E-mail: [email protected]



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