ALBERT VISSER

THE PROVABILITY LOGICS OF RECURSIVELY

ENUMERABLE THEORIES EXTENDING PEANO

ARITHMETIC AT ARBITRARY THEORIES

EXTENDING PEANO ARITHMETIC

1. INTRODUCTION

Solovay’s proof of the arithmetical completeness theorem for provability logic shows more than is stated in the theorem. The idea is roughly this. Let cp be a modal propositional formula. Suppose cp is not derivable in L, i.e., Lab’s Logic (also known as G, for ‘Giidel’). There is a finite, transitive, irreflexive Kripke model K such that K F p. Solovay provides a specific arithmetical interpretation connected with K of the atoms of cp to show that not all arithmetical interpretations of p are derivable in PA, Peano Arithmetic. It turns out that Solovay’s interpretation does more: it connects the interpretation of cp with sentences of the form otAl, reflecting the way cp is connected with sentences of the form q k 1 in the specific model K. We employ this fact to state a lemma that captures the content of the proof better.

Consequences:

We fully characterize the provability logics of RE theories T extending PA at PA and at T.

We show that ‘nearly all’ provability logics of RE theories T extending PA at arbitrary U extending PA are between Lob’s logic L and Solovay’s logic 5’.

At the end of the paper we take a brief look at what goes on between L and S. We give examples of new phenomena there.

We only give definitions and facts where our way of doing things differs from the literature. For further details see [2] or [7].

2. DEFINITIONS AND ELEMENTARY FACTS

For technical convenience we do not employ finite, transitive, irreflexive Rripke models but tail models.

Journal ofPhilosophical Logic 13 (1984) 97-113. 0022-3611/84/0131-0097$01.70 0 1984 by D. Reidel Publishing Company.

98 ALBERT VISSER

2.1. DEFINITION. A tail model K is a triple (a,<, f> where

w is the set of natural numbers.

<is a binary, transitive, irreflexive relation on o.

f: w + the power set of 9, where @‘is the set of propositional atoms of PO, the language of propositional modal logic.

ifm#OthenO<m.

if m f 0 and m <n then m > n. Here > is the usual order on Cd.

for some N # 0: foreveryn>Nandeverym#Oifn>mthenn<m, for every n > N, f(n) = f(N), f(0) = f(N)-

Such an N will be called a tail element. kK is defined as usual, where we take f(n) as the set Of pi such that

n +K Pi*

Example with N = 5:

I 6 . p,r

I 7 l p,r

0 l p,r

PROVABILITY LOGICS 99

Clearly every finite transitive irreflexive Kripke model is the top part of some tail model modulo isomorphism and vice versa: proper top parts of tail models are finite, transitive, irreflexive Kripke models.

2.2. TAIL LEMMA. For every tail model K:

0 l==,cpiffforsomeM,foralln>M,n l=rr~p,

0 #Kqifffor someM,foralln>M,n I#K+x

Roof: Induction on cp, n

2.3. CONVENTION, For any set of formulas Y we will write L + Y for the closure under modus ponens of the theorems of L plus Y. Thus:

s = L+ {(%-v)lcp~~a).

2.4. COMPLETENESS THEOREMS FOR TAIL MODELS

(i)

(3

L k cp iff for all tail models K, for all n: n l=~ cp.

L l-- cp iff for all tail models K, for all tail elements N:Nk-,cp.

(iii) S l- up iff for all tail models K: 0 krr cp.

&oof: (i) and (ii) are trivial consequences of the corresponding completeness theorems for fmite, transitive, and n-reflexive Kripke models.

(iii) “=Q’. 0 satisfies the theorems of L because K is upwards well- founded. Closure under modus ponens is trivial. Suppose 0 l=~ q, then for all n > 0, n kK cp. Hence by the tail lemma 0 l=K 9.

“e”. Suppose S l+ cp, then certainly L v M (a $i + Jli) + up, where the Jli are those subformulas of cp that have a box in front of them. Hence there are a tail model K and a tail element N such that N l=K M (n$r + J/t) and N #K y.~ That 0 & 9 follows easily from:

CXzim. For all subformulas x of cp:

ifNl==,xthenforallm<N,m l=Kx,

ifN#,xthenforallm<N,m &x.

l+oof of claim. By Induction on x. The cases of atoms and propositional

100 ALBERT VISSER

connectives are trivial. In case x = np we have: if N l=K up then, because p is a subformula of cp with a box in front, N l=K p. By Induction Hypothesis for all m< N, m l=:K p. On the other hand for all m >N, m F=K p. So for all k, k bK p. Conclude for m,<N: m l==K op. The second case is trivial. n

We will be interested in closed sentences, i.e., sentences in which no vari- ables pi occur and degrees of falsity.

2.5. DEFINITION

001 :=I

ok+* I:= q (Okl) q w I:= (-1).

Similarly we write $4 for the interpretation of ok1 in T and q !$1 for (0 = 0).

Note that our notation CP is perhaps a bit misleading for there is a perfectly natural interpretation in arithmetic for that expression different from ours.

2.6. DEFINITION. Consider a tail model K. Define the depth d(n) of a node n as follows:

d(n) := 1 + sup {d(m) I n -Cm}.

Note that if n is a top element of K then d(n) = 1 and that d(0) = w. Moreover:

d(n) = a iff n l= 0~1 and n # ofi1 for allo< a.

2.7. DEFINITION. A set X of formulas of 5?0 is standard closed if there is one element of the form 0~1 in X(a E w U {a}) and all other elements are of the form (nk+l l+okl)wherek+ l<a.

2.8. NORMAL FORM THEOREM FOR SETS OF CLOSED FORMULAS. For any set Y of closed formulas of -PO there is a unique standard closed XsuchthatL + Y=L +X.

Proof: Entirely routine. n

PROVABILITY LOGICS 101

We turn to arithmetical interpretations.

2.9. DEFINITION. (i) Let YpA be the set of sentences of the language of Arithmetic. A function g: 9+ YpA will be called an a-assignment.

We define for a-assignmentsg and RE theories T extending PA in the language of PA:

Wk T) := gh).

(* Ng, T) commutes with the propositional constants.

(“dk T) := %d(p)tg, 0.

As far as this paper is concerned we could as well have chosen to consider g from 9 to the formulas of Arithmetic under the appropriate conventions for handling free variables within the range of +. It could very well be that for further research this latter choice is the more natural.

(ii) Let T be an RE theory in the language of PA extending PA, let U be an arbitrary theory in the language of PA extending PA. Let I’,,, I’r be sets of formulas of 2,. Define:

FO; Fr l= cp(U, T) iff for all a-assignments f:

U + {($)(g, T) I J/ E F,,, g a-assignment}

+ ~(xW-,T)Ix~~,) I-W(f, 0.

ro; I= cp(u, T) iff PO; 0 k cptu, 0.

rl k du, T) iff 0; rl k cpV4 0.

I= CPUJ, T) iff 0; 0 I= VW, 0.

L(U, T) := {cp 1 l= cp(U, T)}. (L(U, T) is the provability logic of T at U.)

Our definition of l= is of a subtlety not really needed in the paper. Still it is nice to have the general notion around and to see how certain results can be formulated in terms of it.

2. lo. THEOREM. r l= (p(PA, PA) iff L + r l- cp. Proof: This is just a version of Solovay’s theorem. To show that F may

be infinite one uses the uniformized version of Solovay’s theorem as proved by Artyomov, Montagna and Boolos (see, e.g., [ 1, 3-S]). n

102 ALBERT VISSER

2.11. COROLLARY (Artyomov). Let X be a set of closed formulae of 4po then:

X,!=v iff L+Xl-v7.

Proof. Clearly X, I= up iff X l= cp. Use 2.10. l

3. A SHARPENED VERSION OF SOLOVAY’S THEOREM

Consider an RR theory T extending PA in the language of PA and a tail model K. We assume that the proof predicate Proof&, y) satisfies:

l-p* Proof&, JJ) A Proof&, 2) + y = 2,

I-PA iProof~(O, y).

Define for g: w + o: lim g = s iff for some m, g(m) = s and for all p, n if g(p) = s and n > p then g(n) = s.

We can define or paraphrase in the language of arithmetic a term ‘Z(u) for ‘lim g’ where g is a total recursive function with index a.

By the recursion theorem we find a recursive function h with index e such that:

h(O) := 0,

n k(k + 1) :=

if for some n > h(k), Proof& + 1, r Z(e) f n’)

h(k) otherwise.

Write ‘I’ for ‘Z(e)‘. We have:

bA “II is weakly monotonic in 4,

bA “I exists”.

Because PA shows the existence of I scope problems are irrelevant. Define :

p{z = i I i !==K(P}

MK T) := if for only finitely many i, i !=pp,

m{l’f i Ii t#gq} if for only finitely many i, i #pp.

(We set w@:= (I), M@.:= (0 = 0)). By the tail lemma [cp](K, T) is always defined.

PROVABILITY LOGICS 103

Let gK,&J := [FJ(K, T) and 6&K, T) := (q)(grcT, T). We have:

3.1. MAIN LEMMA

PA I- W(K, T) * [PI& 2’).

Roof: By induction on cp. The only interesting case is: cp = 09. (a) In case there are only finitely many i s.t. i VR q JI by the tail lemma:

n i=K q $ for all n and hence n l=K IJJ for all n. By Induction Hypothesis:

l-pA <$ )(K, T) * (0 = 0). Hence

!-PA •T(($)(~9 T)), Or

t-PA(‘$)(K, T) * [OtiI(K, T).

(b) Suppose there are only finitely many i s.t. i kK q $. Let iO, . . . , j, be all j such that j kK q $, j vK J/. Note that if i vK o$, then there is a k such that i < jk. Clearly by Induction Hypothesis and the fact that T extends PA it is suffkient to show:

I-PA •~['kl(K T) * i"J/l(K 0

Argue in PA.

“+“. Suppose q T [ $](K, T). We have: +(I # j,) by the definition of [ ] and

the fact that jk titK JI. Suppose Proof & + 1, ‘I# j, 3 and h(p) = y. In case y< jk: h(p + 1) = jk, so certainly not: I< jk. In case not y <& clearly not I< jk. So not 1< jk. Conclude: M (not I<& I k = 0, . . . , s}. So by elementary reasoning: w(Z = i I i bK q J/}.

“Q’. Suppose I = i for an i l=K q $. By the definition of h and the fact that

i f 0 (for 0 #K q J/ ex hypothesi) we find: OT If i (“How else could h move up to i!“) Moreover 3x hx = i so by Z-completeness: 0,3x h.x = i. Combining: q T I>i. Hence +w{ I = j 1 j i==K J/}. m

3.2. DEFINITION. Let T be an BE theory extending PA in the language of PA, let U be an arbitrary theory extending PA in the language of PA.

104 ALBERTVISSER

Define :

a(U, T) := the smallest Q f w U (0) such that U I- q $l.

X(U, T) := (ok+1 l+@llk+l<a and

U I- q kT+ll + q “,l} U (nal}, where a = a(U, T).

Note that X(U, T) is standard closed.

3.3. CLOSED PART LEMMA. Let T, U be as in 3.2. We have: if X(U, T) is finite then L (U, T) = L + X(U, T).

proof: Set X := X(U, T), a := a(U, T). Trivially L + XC L(U, T). Suppose X is finite. Suppose further cp EL (II, T) and L + X p cp. We have L lj M X + cp, so there is a tail model K and a tail element N such that N kK M X and N kK cp. Clearly ( q dtN)l + •~(~-‘1) 4 X. Moreover d(N) < a. For any m: m kK I,P+ (odtN)l + gdcN)-’ I), since:

if d(m) < d(N): m kK Do-’ 1.

if d(m) = d(N) then N = m, N being a tail element, so m p=g -I Q.

if d(m) > d(N)m kK 1 •~(~~1.

So by our main lemma:

PA /- (Q + (od(N)l + q d(N)-1 l))(X, T).

Or PA k (Q)(K, T) + ($? + #N)-l 1).

Ex hypo thesi U I- (Q) (K, T) and U extends PA, so

Now (odtN)l + 0~(~)-‘1) $ X so not d(N) < a. We saw before: d(N) < a. Conclude: d(N) = a. But then U t- 0 $tN)-’ 1, contradicting the minimahty of cy. n

3.4. THEOREM. Let U, T be as before and a := a(U, T). Suppose U C T, we have:

L(U, T) = L + 0~1.

PROVABILITY LOGICS 105

proof: If 0 ‘+’ I+ ok1 were in X(U, T) we would have:

u t q p1 + q $l, so Tt”, k+ll+ q $, hence

PA t o&+1l + +), so by Liib’s Theorem:

PA t q !++rl, so

utoy1 9 conclude :

u t q k,l. This contradicts the definition of X(U, T). Hence only q l is in X(U, T).

So X(U, T) is fmite and L(U, T) = L + 0~1. m

3.5. CONSEQUENCE

(9 L(PA, PA) = L.

(3 L(PA, T) = L + q Q(P*.T)l = L + q l+W,T)l. (iii) L(T, T) = L + IP(~*~)I.

3.6. THEOREM. Let T, Ube as in 3.2. Suppose L(U, T) $ S, then a(U, T) E o, and so L(U, T) = L + X(U, T).

Proof: Let (9 E L(U, T), cp 4 S. There is a tail model K with 0 kcx 9. BythetaillemmaforsomedEwandallm:ml=Kcp+~dl.Byourmain lemma: U tbp)(K, T) -+ $1. Conclude U l- q d,l. n

3.7. CONSEQUENCE. L(Th(IN), PA) = L(PA + RFN(PA), PA) = S.

3.8. TWO EXAMPLES (i) Let T, U be as before. Suppose (o(p vq) + (Op v q q)) E L(U, T). Clearly L(U, T) $ S. Inspection of the proof shows: 01 E L(U, T) or L(U, T) = L + 01. Of course this can also be seen more directly by substituting the Z! Rosser Sentence R - satisfying PA tR *(+ 1R < q TR) - for p and -R for q.

On the other hand:

L+{o(~vIL)-,(O(PVOJ/)I(P,JI~~~)tool

as can be seen by substituting 01 for cp and 101 for J/, but:

106 ALBERT VISSER

This can easily be shown by considering a linear tail model: the two highest nodes will satisfy ~(9 v $) + (09 v q $).

Facts like the above are often more naturally formulated in terms of I==; for example we have :

(ii) Let R be the Zt Rosser Sentence as in (i). Then L(PA + -R, PA) = L Proof. It is clearly sufficient to show that for no k: PA + -rR I- q Ei1l +

okAl, 1R being true. Suppose PA l--R + (oil11 + okpAl).

In case k = 0: PA t 0pAl-f R, SO PA I- •p~np~l+ q p~R, hence

PA t •~A+A~ + q p~l. @od non.

In case k > 0, we have from PA I- lnff~l--* 1R :

PA t lt$Al+ (OS:’ I+ c$~l), hence

PA t’=#;l.‘+ookpAi. Quad non. l

3.9. REMARK. Using 3.3 and 3.6 we can state weak local versions of Solovay’s theorem in terms of b, for example:

(9 L tfcp*forsomekEwoIp; l=o&l(PA,PA),

(ii) S lfg*forsomekEwrp; l=n’l(PA,PA).

4. BETWEEN L AND S

What L(U, T)‘s are there between L and S? We show that the question is not trivial by treating some examples, proving that the most obvious attempts at axiomatization are not complete.

PROVABILITY LOGICS 107

Clearly (9 I J/i I= 9VJ’, T)} is an L(U, T), and we will use this fact with- out mentioning it.

Define: D := b I (VP v q 4 + (a=‘~ v q q)); I= 9(pA, PA)).

E := (9 I 0 up + 0~; I= 9(PA, PA)}.

LD :=L+(o(ocpvoJ/)-*(oocpvooJ/)IIp,J/E~~}.

L&” :=L+{oocp+ocpIcpE 2%).

The following charming theorem is a useful tool.

4.1. THEOREM (Goldfarb). Let T be an RE theory extending PA in the language of PA. We have:

(i) For every IZ! sentence A there is a Z! sentence S such that:

PA t-A v +I* q.S.

(ii) For all sentences B and C there is a Zy sentence S such that:

PA I-- q-B v +C * ~7~s.

R-oojI (i) By the arithmetical fued point theorem pick S such that:

We have in PA:

[oTS< A]@

OT@TS<A)

[A < +S]@ OT1s ‘Ts

q Ts A q Tl

AGoTSVOTS<A Av+l AvoTl AvoTI

1,

[A < +S]@

EAl@ S [oTS< A]@

[“Tllo A<DTSVOTS<A q Ts q Ts AvnTl q TS ‘Ts

2

q Ts 3

(it) q TB v +Cis ct so by i for some 2’: sentence S:

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Hence PA I- +B v oTC * oTS.

Note that we could as well have proved 4.1 for formulas under the appropriate conventions for handling free variables.

4.2. FACT. We have:

(O(EOpi)+E opi)EE (kEti)a

Proof: L&B,,,..., Bk +l be any sentences of Arithmetic. By 4.1 there is an S such that:

k+l PA t y. q pABi ++ 0~~s.

Hence k+l

PA + q PAOPAS + q PAS I- q PA iyo ‘PABi + q PAOPAS

So certainly:

k+l * ,yo ‘PABi

(PA PA)*

4.3. FACT

LE ~"(o~O~O~l)~(o~O~O~,).

Proo$ Consider the transitive irreflexive Kripke model K:

*PO l Pl I I

As is easily seen

.Po

I

.Pl I

ko I=K oov+ov

*PO .Pl and hence ko l=g LE.

* \*/ * Moreover k. kK q (opo v Opl),

ko but ko #K GPO v 0~1).

PROVABILITY LOGICS 109

4.4. FACT

Proof L 5 D: Inclusion is trivial. Consider the irreflexive transitive Kripke

model K :

.Po I

.Pl Clearly k. l==K L and

\

I l 0 l P1 ko I=K q (OPO v 0~11 but

/ ko . ko I& 00~0 VOOPI.

D 5 E: Inclusion follows by 4.2. We claim 0 q l+ q 16! D. For:

~A+OPAOPA~ FOPAPPAA VOPA@

+C"pADp~A V"~~o~~B-

So if 0 0 I+ 0 1 were in D, it would follow that:

PA+op~op~l~op~op~I~opAl.

Hence PA t q p~ ‘JPA I+ 0p~ 1. Quad non.

E : S: Inclusion is trivial. We claim: 01--f 1 BE. For:

PA + ‘Jp~l I- •p~op~A + q p~A.

Hence if ml+ 1 were in E:

PA+“pAll-opAl+I, SO

PA t ‘pAl+l. Quodnon.

4.5. FACT. q ool+oolED. Proof: Consider the tail model K:

1 *PO I

2 Pl We have: for every n:

3* 0 4 .Pl

\/

T n ~K~n~o~O"o~~~~OO~O"OO~~~

sop0 I

6.~0

+(oool+ool).

So by our main lemma we can find A and B such that:

110 ALBERT VISSER

. PA l-("p~(n~~A VOPA@+

Hence OiPo

VP0 v UPI) + q OPO v ‘Jnpl; ~o~~l+~~l(PA,PA). n

4.6. FACT. LD /+~nnl+onl. Proof: Any linear, finite, transitive, irreflexive Kripke model satisfies LD.

l

4.7. FACT

Proof: Induction on k. The cases k = 0, 1 are trivial. Suppose k > 1. For any sentences Al, . . . , Ak we can find by 4.1 a ZZtS such that

PA t i% ‘PAAi +) q pAs. Hence

PA+"~A(n~A~o~opA~+o~An~~~~~npAoPA~

k +'PA iy 'PAAi + i$l 'PAnPAAi t

k "PA iFo ='e~At --f i i 0 i-o PAnPAAi-

By IH we may conclude: k

4.8. FACT

LD Hni+o oPi+,$o “Pi*

Proof: Consider the transitive irreflexive Kripke model K:

PROVABILITY LOGICS 111

ko

Clearly k. satisfies L. Suppose k. kK q (O cp v q $). Then one of UP, q $ is true at at least two of kl, k2, k3, e.g., •~ is true at kl, k2. But in that case cp is true at k4, ks, kg, k,. Moreover k5 and k8, k, and kg satisfy the same formulas so cp is true at k,, kg. Hence k. FK 0 0~.

Clearly: k. FKo(opO vop, vop2)+(o~po voop, voop2). m

4.9. FACT

Proofi We show: For any A E Et-sentences:

Our claimed result then follows by noting that for any

Step 1. For any RE theory T extending PA in the language of PA we have :

q po VOPl) + (“UP0 v q op,)~ooolt-*nl(PA,T).

This follows by more or less copying the proof of 4.5. Step 2. Let A be a I$-sentence; B1, C, E YpA. We have:

PA + (op,&p~B V 0 PAC)'@PA~PABV~PA~PAC) 1

B,CE ~A~t~PA+~A~nPA+~~~~~npA+~~~~)~

(0 PA+~A~PA+-ABIV~PA+~A~PA+~ACI)-

Reason in

PA + (npA(npAB v q PA~)~~~PA~PA~~~PA~~A~)~~~~~~~A}~

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Suppose %A+ 4%+ 431 v UPA+ TA Cd. We have:

q PA@ v nPA(‘A + BJ v q pA(1A + cr)). Because

AEZ1? we have o PA@ +n~~A), SO

•~~(A + ‘JPA(~A + El)) and q pA(A + q pA(lA + Cl)).

Hence we find:

so:

‘PA@PA~A +BI) V q PAW + cd).

Conclude: q PA+ TAOPA+ -ABI V SPA+ TAOPA+ TACI-

Step 3. For k > 0: PA I- q kpAA * mEA+ -Al, where A is a Z$sentence. By induction on k. The induction step is:

PA t q $A:AA * ~pA~kp;;:~Al

* “p& v %i -Al)

f, q PA+ &i&&

The middle step is because: PA t ‘J~A(A + J$,: 7 Al). Step 4. Take T of Step 1: PA + ‘A, where A is a Zf-sentence. We have:

PA + t”~A(n~~B VOPAC)+(OPAOPAB V”PAOPAC) I & CE SPA)

t PA + h+ -x4(’ PA+ YAB v ‘PA+ -AC) -+

(n~~+~~n~~+7~B Vn~~+y~npA+~~C) 1 B, CE SPA}

l-0 PA+ lAOPA+ -AnPA+ TA ~+DPA+~AnP.4+~Al

t ‘PA’PA~PAA + q PA’PAA. 8

4.9 is not surprising for those familiar with Friedman’s proof that BE theories extending Heyting’s Arithmetic satisfying the Disjunction Property also satisfy the Existence Property or with applications of Shepherdson’s Fixed Points. Our argument in 4.5 is closely related to uses of Friedman’s and Shepherdson’s Fixed Points. For a survey of various applications of Fixed Points see [6 1.

PROVABILITY LOGICS 113

REFERENCES

[l] S. N. Artyomov, ‘Arithmetically complete modal theories’, (Russian), Semiotics and Information Science, No. 14 (Russian) (1980), pp. 115-133, Akad. Nauk SSSR, Vsesojuz. Inst. NauEn. i Tehn. Informacii, Moscow.

[2] G. Boolos, The Unprovability of Consistency, Cambridge University Press (1979). [3] G. Boolos, ‘Extremely undecidable sentences’, JSL 47 (1982), 191-196. [4] F. Montagna, ‘On the diagonabizable algebra of Peano Arithmetic’, BoZZettino

UnioneMatematicaZtaliana (5) 166 (1979), 795-812. [S] F. Montagna, ‘Relatively precomplete numerations and arithmetic’, Journal of

Philosophical Logic 11, No. 4 (1982) 419-430. [6] C. Smorynski, ‘Calculating self-referential statements’, Fund. Math. 109 (1980)

189-210. [7 ] R. M. Solovay, ‘Provability interpretations of modal logic’, Israel .f. Math. 25

(1976), 287-304.

Department of Philosophy, University of Utrecht, Utrech t, T&e Netherlands.