Preservation theorems and restricted consistency statements in bounded arithmetic

Annals of Pure and Applied Logic 126 (2004) 255–280

www.elsevier.com/locate/apal

Preservation theorems and restricted consistencystatements in bounded arithmetic

Arnold Beckmann1

Institute of Algebra and Computational Mathematics, Vienna University of Technology,Wiedner Hauptstr. 8-10, A-1040 Vienna, Austria

Abstract

We de(ne and study a new restricted consistency notion RCon∗(T j2 ) for bounded arithmetic

theories T j2 . It is the strongest ∀�b

1-statement over S12 provable in T j

2 , similar to Con(Gi) inKraj012cek and Pudl0ak (Z. Math. Logik Grundl. Math. 36 (1990) 29) or RCon(T i

1) in Kraj012cekand Takeuti (Ann. Math. Arti(cial Intelligence 6 (1992) 107). The advantage of our notion overthe others is that RCon∗(T j

2 ) can directly be used to construct models of T j2 . We apply this by

proving preservation theorems for theories of bounded arithmetic of the following well-knownkind: The ∀�b

1-separation of bounded arithmetic theories Si2 from T j

2 (16 i6 j) is equivalentto the existence of a model of Si

2 which does not have a �b0-elementary extension to a model

of T j2 .

More speci(c, let M ��nst1 denote that there is a nonstandard element c in M such that the

function n �→ 2log(n)c is total in M . Let BL�b1 be the bounded collection schema for �b

1-formulas.We obtain the following preservation results: the ∀�b

1-separation of Si2 from T j

2 (16 i6 j) isequivalent to the existence of1. a model of Si

2 + �nst1 which is 1b-closed w.r.t. T j

2 ,2. a countable model of Si

2 + BL�b1 without weak end extensions to models of T j

2 .c© 2003 Elsevier B.V. All rights reserved.

MSC: 03F30; 03F25; 03H15

Keywords: Bounded arithmetic; Restricted consistency; Preservation theorems

1 Supported in part by the Deutschen Akademie der Naturforscher Leopoldina grant #BMBF-LPD 9801-7with funds from the Bundesministerium fGur Bildung, Wissenschaft, Forschung und Technologie, and by aMarie Curie Individual Fellowship #MCFI-2000-02005 from the European Commission.

E-mail address: [email protected] (A. Beckmann).

0168-0072/$ - see front matter c© 2003 Elsevier B.V. All rights reserved.doi:10.1016/j.apal.2003.11.003

mailto:[email protected]

256 A. Beckmann /Annals of Pure and Applied Logic 126 (2004) 255–280

0. Introduction

This article is a contribution to the investigation of the inJuence of consistency no-tions to the (nitely axiomatization question of bounded arithmetic. The usual notion ofconsistency is too strong to serve as a separating sentence for bounded arithmetic theo-ries because S2 0ConS−1

2, cf. [21], where S−1

2 is the induction-free fragment of boundedarithmetic S2. Also the weaker consistency statement BDCon, which refers to proofs thatuse only bounded formulas, still is too strong: Buss in [6] proved that Si+1

2 �BDConSi2

holds for at most one i, and later Pudl0ak showed in [18] that S2 0BDConS12, hence

only S2 �BDConS02

remains to be possible. On the other hand I have been able to showin [4] that S1

2 can prove the consistency of equational theories which base only on therecursive de(nition of the underlying function symbols. In particular S1

2 �Con(S−∞2 ),

where S−∞2 is the equational theory based on the recursive de(nition of the function

symbols of bounded arithmetic. This result disproves a plausible conjecture of Takeuti[11, p. 5, problem 9]. It gives hope that consistency statements can lead to a negativeanswer of the (nitely axiomatization question of bounded arithmetic.

The focus of this paper are new restricted consistency statements for theories ofbounded arithmetic and applications of them for proving preservation theorems fortheories of bounded arithmetic in the manner of the following well-known one. 2 LetL be a (rst order language, S ⊆T be L-theories and � a class of L-formulas whichis closed under conjunction and negation. With ∀� we denote the universal closure of(all formulas in) �.

Fact 1. S is ∀�-separated from T if and only if there is a model M of S whichcannot be extended �-elementarily to a model of T .

Proof ideas. The direction from left to right follows directly from the assumptionsusing the upwards persistency of ∃�-formulas w.r.t. �-elementary extensions.

For the direction from right to left let M be a model of S which cannot be extended�-elementarily to a model of T . Then T plus the �-diagram of M is inconsistent.Using compactness (and the closure of � under conjunction) we obtain some ’(a) inthe �-diagram of M such that T +’(a) is inconsistent, hence T �¬’(a). Applying thelemma of new constants we obtain T �∀x¬’(x). On the other hand M �’(a), henceM �∃x’(x). Thus S �=∀� T .

0.1. Introducing bounded arithmetic

Before we explain which restricted consistency statements we will consider andwhich preservation theorems will be proved by them let us brieJy introduce boundedarithmetic. Bounded arithmetic is intended to characterize low complexity computabil-ity, i.e. the polynomial hierarchy. Every primitive recursive function is provable total inI�1, hence I�1 is much stronger than bounded arithmetic. By Parikh’s Theorem ([16],or see [6, p. 83, Theorem 11]) the provable total functions of I�0 (in the language

2 I am grateful to Alex Wilkie for pointing out the simple proof of Fact 1 to me.

A. Beckmann /Annals of Pure and Applied Logic 126 (2004) 255–280 257

LPA of Peano arithmetic) are bounded by polynomials. Hence I�0(LPA) is weakerthan bounded arithmetic. Furthermore, only a constant number of elements 6n can becoded in a sequence s= nO(1): If we try to code l elements 6n in s we get

s =log n bits︷︸︸︷——— : : :

log n bits︷︸︸︷———-︸︷︷︸l times

hence s consists of l log n bits, hence s≈ nl. Thus, metamathematical arguments arein general not formalizable in I�0(LPA).

Allowing l=m many elements 6n would result in an exponential growth rate, againtoo strong.

As argued in [16] the right growth rate is obtained by allowing l= logm elements6n to be coded into one sequence. Then

s = nlog m ≈ 2|n|·|m| =: n#m;

where |m| is the number of bits in the binary representation of m. Bounded arithmeticcan be formulated now as I�0 in the language LBA of bounded arithmetic, that isLPA extended by |:|; #, or, equivalently, as I�0 +�1 (where �1≡∀x∃y(|x|2 = |y|)), thelatter being the original formulation of bounded arithmetic, see [21]. The provable totalfunctions of bounded arithmetic are the functions computable by a Turing machine inpolynomial time using oracles from �0(LBA), i.e. the polynomial time hierarchy.

A strati(cation of bounded arithmetic, which corresponds to the strati(cation ofthe polynomial time hierarchy, is obtained by putting restrictions on induction axioms;namely, allowing induction only for certain classes, �b

i , of bounded formulas, and usinglength induction (LIND) in place of successor induction (IND). The most importantsub-theories of bounded arithmetic are the theories Si

2, axiomatized by �bi -LIND, and

the theories T i2, axiomatized by �b

i -IND. The following is known for these theories:

S12 ⊆ T 1

2 ∀�b2S2

2 ⊆ T 22 ∀�b

3S3

2 : : :

and their union is the theory S2 =T2 = I�0(LBA) [6,9]. Here T ∀�biT ′ means that T ′

is a ∀�bi -conservative extension of T . Furthermore, the class of predicates de(nable by

�bi (or �b

i ) formulas is precisely the class of predicates in the ith level �pi (or �p

i ,resp.) of the polynomial time hierarchy. In addition, the �b

i -de(nable functions of Si2

are precisely the functions computable in polynomial time using an oracle for �pi−1

(cf. [6]).The main open problem for bounded arithmetic is the question if S2 is (nitely

axiomatizable. This question is connected to the open problem whether the polynomialtime hierarchy collapses, hence also to P = ?NP. The precise connection is that S2 is(nitely axiomatizable if and only if S2 can prove that the polynomial time hierarchycollapses [10,23]. As Si

2 and T i2 are (nitely axiomatizable, one can conclude from

this that S2 is (nitely axiomatizable iT there exists an i with T i2 = Si+1

2 . The commonconjecture is that the answer to all these questions is NO!


0.2. Restricted consistency notions

We assume familiarity with [6]. From now on let L be LBA, the (rst order languageof bounded arithmetic. For convenience we assume that L contains some more symbolsfor polytime functions ((nitely many), e.g. for coding and decoding sequences (e.g.we could take the language L2 from [17]).

Several restricted consistency notions are known from the literature. Above we havedescribed some of them. The notion of restricted proof studied here will be similar tothe notions “i-regular proof” in [13, De(nition 10.5.2] and [15, De(nition 1.4], and“strictly i-normal proof” in [20, p. 81], but combined with a new idea. To explainthis let us (rst explain why usual approaches for proving consistency do not work inweak arithmetic. The reason for this is that in case of the usual feasible coding ofsyntax (cf. [6]) it is impossible to feasibly evaluate closed terms from the languageof bounded arithmetic—their values grow exponentially in their GGodel-numbers. Whathappens if we play with the growth rate of GGodel numbering? On the one hand,as mentioned above the usual “feasible coding” of syntax yields S2 0BDConS1

2. On

the other hand, if we take a “very unfeasible” sequence coding, e.g. one based onexponentiation like 〈n1; : : : ; nk〉= 2n1+1·3n2+1·: : :·pnk+1

k , pk being the k-th prime number,then soundness of S2-proofs can be proven in weak fragments of bounded arithmetic.

We have S12 �BDConS2 where in BDConS2 syntax is coded in the “very unfeasible” way.

Of course, in this setting we loose something, namely based on the very unfeasiblecoding GGodel’s incompleteness theorems are not provable, because substitution of termsgrows exponentially.

What we will do in this paper is that we will adjust the growth rate in a certainway which allows us to feasibly evaluate GGodel numbers of terms, with the cost thatGGodel’s incompleteness theorems will not be provable. But still there will be availableenough other properties of formalized provability RProv∗

T i2

(De(nition 9). We will have

that S12 proves

RProv∗T i

2(p�; ’q) and RProv∗

T i2(p�;¬’q) implies RProv∗

T i2(p�q)

(Lemma 10); that T i2-proofs can be normalized

RProv∗T i

2(p’q) if and only if T i

2 � ’

(Theorem 11); and that a certain ∃�b1-Completeness for S1

2 holds (Theorem 15), whichwill be a re(nement of

S12 � ’(u)→ RProv∗

T 02(’(Iu))

for ∃�b1-formulas ’.

0.3. Preservation theorems

The main application of our restricted provability notion in this paper will be that wewill prove certain preservation theorems. Other applications may lie in the construction


of models of bounded arithmetic with certain properties. In order to explain the preser-vation theorems we are heading for, let us (rst (x some notions. For L-structuresM;N we write M≺b

0 N iT N is a �b0-elementary extension of M . We write M≺b

1 NiT N is an ∃s�b

1-elementary extension of M . Here ∃s�b1 is the set of all formulas

(∃x)(∃y6t)’ with ’∈�b0. By s�b

i we always mean the prenex (or strict) version of�b

i , etc.For the following de(nitions let M;N be L-structures and T; T ′ be L-theories. Let

logM be {a∈M : (∃b∈M)(a6|b|)}. Let |t|3 be |||t||| and let log3 M := {a∈M : (∃b∈M)(a6|b|3)}. Let us call an extension M ⊆N log3-proper, if log3 M �= log3 N .

The notion “weak end extension” to be de(ned next is the natural adaption of thewell known notion “end extension” to the setting of bounded arithmetic, cf. [7].

De�nition 2 (Weak end extension). N is called a weak end extension of M (M ⊆we N ),

if N is an extension of M and logN is an end extension of logM , i.e. for all a∈ logM ,b∈ logN with N � b6a we have b∈ logM .

Weak end extensions for models of bounded arithmetic are in some aspects similarto end extensions for (general) models of arithmetic. For example, weak end extensionsare always �b

0-elementary.Let the function !1 be de(ned by !1(x) = 2|x|2

, i.e. !1(x) = x#x, and !(y)1 be the

y-fold iteration of !1. We want to de(ne sentences �nst1 and �∞

1 such that �nst1 ex-

presses that a nonstandard iteration of !1 (or, equivalently, of the smash function #)exists. �∞

1 should express that an upper bound to all (nite iterations of !1 exists,but not necessarily a nonstandard iteration. Now ∀x∃y(‖x‖ · c= ‖y‖) expresses that∀x !(|c|)

1 (x) exists. Hence we let the L!1!-sentences �∞1 ; �nst

1 be de(ned by

�∞1 := ∀x∃y

∧k ∈ !

(‖x‖ · k ¡ ‖y‖);

�nst1 := ∃c

[( ∧k ∈ !

(k ¡ c)

)∧ ∀x∃y(‖x‖ · c = ‖y‖)

]

where k is some canonical numeral associated with k.In the next de(nition we adopt the notion “1-closeness” (cf. [2]) to the setting of

bounded arithmetic. 1-closed models of Peano arithmetic PA satisfy ¬Con(PA) [2]. Wewill show something similar for 1b-closed models of bounded arithmetic and theoriesT i

2, namely that a model M of S12 +�nst

1 which is 1b-closed w.r.t. T i2 cannot be a model

of our restricted consistency notion for T i2.

De�nition 3 (1b-Closeness). M is called 1b-closed w.r.t. T , if for any model N of Tsuch that M≺b

0 N we have M≺b1 N .

With BL�b1 we denote the following bounded collection schema:

(∀x 6 |t|)(∃y)’(x; y)→ (∃z)(∀x 6 |t|)(∃y 6 z)’(x; y)


for ’∈�b1 which may contain parameters. Buss has shown in [8] that Si

2 and Si2 +BL�b

1have the same ∀�b

1-consequences.We are now ready to state the preservation theorems which we will prove.

Theorem 4. Let 16i6j. The following are equivalent:(1) Si

2 is ∀�b1-separated from T j

2 .(2) There is a model of Si

2 which does not have a log3-proper �b0-elementary extension

to a model of T j2 .

(3) There is a model of Si2 + �nst

1 which is 1b-closed w.r.t. T j2 .

(4) There is a countable model of Si2 +BL�b

1 without weak end extensions to modelsof T j

2 .

The equivalence between (1) and (2) is well-known, it is included just to complete thelist. One big impact of Theorem 4 is that solving the main open problem of boundedarithmetic with consistency statements is reduced to constructing models of boundedarithmetic with certain properties.

In the next section we will prove the easy directions of Theorem 4, i.e. the one from(1) to (2), (3), resp. (4). In Section 2 we will de(ne a restricted consistency notion ofT j

2 – RCon∗(T j2 ) – which will be the strongest consistency statement (over S1

2 ) provablein T j

2 . That is, every ∀�b1-consequence of T j

2 follows in S12 from RCon∗(T j

2 ). Suchconsistency notions are known from the literature, e.g. Con(Gi) in [14] or RCon(T i

1)in [15], or see [13] for a treatment of both. The advantage of our approach over theothers is that our consistency notion is for theories in the same language LBA, hencewe can use RCon∗(T j

2 ) to (directly) construct models of T j2 . This is needed in the

following 3 sections to complete the proof of Theorem 4. In Section 6 we extend ourresults by adding a ∀(�b

i ∪�bi )-sentence to the theories. In the last section we sketch

how the separation problem of bounded arithmetic can be connected to models withoutproper weak end extensions. This is stressed because there is a similar (open) questionfor models of I�0

3 : Are there models of I�0 + B�1 without proper end extensions tomodels of I�0? 4

1. Basic properties

From the de(nitions we directly obtain

Lemma 5. Models which do not have a �b0-elementary extension to a model of T are

1b-closed w.r.t. T and do not have weak end extensions to models of T , as weak endextension are �b

0-elementary.

Given an L-structure M there is always an elementary extension to an L-structureN such that N ��nst

1 . This can be seen by a simple compactness argument applied to

eldiag(M) ∪ {n ¡ c : n∈!} ∪ {∀x∃y(‖x‖ · c = ‖y‖)};3 See [1] for a summary of the most important partial results on the end extension problem.4 To the authors best knowledge this question was raised by Kirby and Paris in 1977.


where eldiag(M) denotes the elementary diagram of M . Hence T , T + �∞1 , T + �nst

1all prove the same (rst order sentences.

A similar argumentation as in [2] Remark 1.2 using these observations yields

Lemma 6. Let T ⊆T ′ be L-theories, where T ⊆∀∃s�b1, then there exists a countable

model of T + �∞1 which is 1b-closed w.r.t. T ′.

Proof. We repeat the argument from [2] Remark 1.2 in an adapted form. Let t0; t1; : : :be an enumeration of all triples

〈m; 〈k1; : : : ; kn〉; ’〉;where m; n; k1; : : : ; kn ∈N, ’∈∃s�b

1 and the number of free variables in ’ is n. W.l.o.g.tl = 〈m; 〈k1; : : : ; kn〉; ’〉 implies m6l.

We de(ne a tower M0≺b0 M1≺b

0 : : : of countable models of T + �∞1 . Let M0 be a

countable model of T +�nst1 . Assume that we have de(ned M0; : : : ; Ml and enumerations

{xmk : k ∈N} of Mm for m6l. Let tl = 〈m; 〈k1; : : : ; kn〉; ’〉.If there is a �b

0-elementary extension M ′ of Ml which is a model of T ′ + ’(xmk1; : : : ;

xmkn), then we can (nd an elementary extension M ′′ of M ′ satisfying �nst1 (we can restrict

ourselves to countable M ′; M ′′). Let Ml+1 =M ′′. Otherwise let Ml+1 =Ml. We (x anyenumeration {xl+1

k : k ∈N} of Ml+1. Now let M =⋃{Ml : l∈N}. Then M �T + �∞

1which is 1b-closed w.r.t. T ′.

This construction does not produce a model of T + �nst1 which is 1b-closed w.r.t.

T ′, because �nst1 is “�3”, where �∞

1 is only “�2”, and the model is constructed as aunion of a chain of models, which, in general, does not preserve �3-sentences.

One direction of our desired results is easy, that the ∀�b1-separation yields certain

models. The argumentation follows the one from Fact 1.

Theorem 7. Let S12 ⊆T ⊆T ′, T; T ′ L-theories. If T is ∀�b

1-separated from T ′, thenthere exists a countable model M of T + �nst

1 which does not have a �b0-elementary

extension to a model of T ′. Hence M is also 1b-closed w.r.t. T ′, and also M doesnot have weak end extensions to models of T ′.

Proof. Under the assumption there is a ’∈∀s�b1 such that T ′ �’ and T 0’ as S1

2knows ∀�b

1 =∀s�b1. The above remark shows T +�nst

1 0’, hence there is a countablemodel M of T+�nst

1 +¬’. Now M does not have a �b0-elementary extension to a model

of T ′, because if M≺b0 M

′, then M ′ �¬’ using upwards persistency of ∃s�b1-formulas,

and therefore M ′ 2T ′ as T ′ �’.By Lemma 5 every model which does not have a �b

0-elementary extension to amodel of T ′ is already 1b-closed w.r.t. T ′, and also does not have weak end extensionsto models of T ′.

This proves directions (1)⇒ (2) resp. (1)⇒ (3) of Theorem 4. For (1)⇒ (4) observethat for ’∈∀s�b

1 with Si2 0’ and T j

2 �’ we also have Si2 + BL�b

1 0’ as Si2 + BL�b

1is ∀�b

1-conservative over Si2, hence there is a countable model M of Si

2 + BL�b1 +¬’.


The same argument as in the proof of Theorem 7 shows that there are no weak endextensions of M to models of T j

2 .We are going to prove converses of Theorem 7. That is, from the existence of

certain models we will derive the ∀�b1-separation of bounded arithmetic theories. The

separating sentence will always be a restricted consistency notion of the “stronger”theory which will be de(ned in the next section.

2. Restricted proofs

Our notion of restricted proof will be similar to the notions “i-regular proof” in [13,De(nition 10.5.2] and [15, De(nition 1.4], and “strictly i-normal proof” in [20, p. 81].The main diTerence between these notions will be that we will vary the coding offormal terms in order to make the coding more “unfeasible” so that values of codesof closed terms are bounded by the codes themselves, where e.g. in [20] restrictionsof the provability notion to proofs of small and very small sizes are considered.

We assume that we have constants cn for each n∈! in our language L, and asuitable axiomatization of them in our theories, for example Sicn = cSin for i = 0; 1.Hence in this setting Iu from [6] can simply be de(ned by Iu := cu.

Let p:q be a usual feasible GGodelisation as in [6], then sub(w; x•; n) – the result ofreplacing the variable 5 x• in the string w by the numeral In – is a �b

1-de(nable functionin S1

2 and S12 can prove all necessary properties. We further have l(sub(w; x•; n)) = l(w),

where l(w) denotes the length of the sequence w, i.e. the number of elements formingw. For formal terms t let l(t) denote its formal length, i.e. the number of symbolsforming t. Obviously we have l(t)6l(ptq).

One problem with feasible GGodelisation is that the values of terms have a biggergrowth rate that their GGodel numbers, e.g.

c := p2# : : : #2︸︷︷︸|n| times

q = 2O(|n|) = nO(1) but val(c) = X(2n):

Here we use the common O- and X-notion to compare the growths of functions:let f; g be functions from the natural numbers into the reals, then f(n) = O(g(n))(f(n) = X(g(n)) respectively) iT there is some c¿0 such that eventually f(n)6c ·g(n)(f(n)¿c · g(n) respectively).

Of course there cannot be a GGodelisation G of closed L-terms such that both

(1) val(c)6cO(1) for c∈G, and(2) sub(w; x•; n)6(w + n)O(1),

because then f(w; n) := val(sub(w; x•; n)) would be a universal bounding function forall polynomially growth rate functions, which itself grows polynomially. By diagonal-isation this cannot be possible.

Let us review the growth rates which are achieved via feasible GGodelisation. Assumethat we have a function symbol !1 in L with !1(n) = n#n= 2|n|2

. With !(k)1 (n) we

5 We use x• to speak about codes indicating variables.


denote the k-fold iteration of !1. Then !(|m|)1 (n) = 2|n|m ≈ 22(‖n‖ ·m) with 22(n) = 22n

.Hence a natural candidate for a GGodelisation which has property (1) but fails for(2) is

ptq∗ := 22(‖ptq‖ · 2l(ptq)) + ptq:

(In this de(nition we can think of l(ptq) as the number of symbols l(t) in the formalterm t – we have that l(ptq) is an upper bound to l(t).) Let rem(n) be n without itsleading bit, then ptq= rem(ptq∗). We de(ne val∗(c) and sub∗(w; x•; n) to be the valueof the closed term c resp. the result of substituting for x• in w the numeral In, thistime with respect to p:q∗. I.e., with w′ = rem(w) we can write

val∗(c) := val(rem(c));

sub∗(w; x•; n) := 22(‖sub(w′; x•; n)‖ · 2l(w′)) + sub(w′; x•; n);

l∗(w) := l(rem(w)):

As argued above sub∗ cannot be polynomially bounded as sub∗ and val∗ togetherdiagonalise 2|n|k . But we observe that for a (xed w with l∗(w) standard the functionn �→ sub∗(w; x•; n) is polytime because sub∗(w; x•; n) can be bounded by !(l∗(w)+O(1))

1 (n).Now we can prove property (1) for p:q∗

Lemma 8. val∗(c)6c.

Proof. We identify function symbols with the functions they represent. Then the as-sertion follows from

|t|¡ |ptq∗|for closed L-terms t. All function symbols f in L represent polytime functions, hencethere are constants cf such that

|f(n)|¡ max(|n|; 2)cf :

Assume t =fu1 : : : uk , then ‖ptq‖¿max(‖puq‖) and, w.l.o.g., l(ptq)¿max(l(puq)) +|cf|, hence we obtain inductively

|t|¡ max(|u|; 2)cf 6 2max(‖u‖;1)·cf

i:h:6 2maxi=1;:::k (‖puiq‖·2l(puiq))·cf

6 2max(‖ ˜puq‖)·max(2l( ˜puq))·cf

6 2‖ptq‖·2l(ptq)= |ptq∗|:

The form of restricted proofs we will consider here is similar to the notions “i-regularproof” in [13, De(nition 10.5.2] and [15, De(nition 1.4], and “strictly i-normal proof”in [20, p. 81].


De�nition 9. RProof ∗T i

2(〈p; b; t; d; T ; D〉; �) holds if the following conditions are satis-

(ed:(1) p; b; � are coded via p:q; t; d; T ; D via p:q∗.(2) p is a Tait-style 6 derivation of the set � using the s�b

i -IND-rule, and all formulasin p are in �b

∞.(3) All cut formulas in p are in s�b

i ∪ s�bi .

(4) p is in free variable normal form.(5) If a are all parameters (i.e. free variables in �) and b= (b0; : : : ; bk−1) all other

variables in p, then(a) if the elimination inference of bi is below the elimination inference of bj, then

i¡j.(b) t is a k-tuple of monotone terms with variables among a; d is a k-tuple of

proofs.(c) the elimination inference of bi is one of

(s�bi -IND)

�;¬A(bi); A(bi + 1)�;¬A(0); A(r(a; b0; : : : ; bi−1))

(∀6)�; bi � r(a; b0; : : : ; bi−1); A(bi)�;∀x6r(a; b0; : : : ; bi−1)A(x)

and di is a proof of

b0 � t0(a); : : : ; bi−1 � ti−1(a); r(a; b0; : : : ; bi−1) 6 ti(a)

that is without the IND-rule, is quanti(er-free and contains only the variablesb0; : : : ; bi−1. (E.g., ti = 1[r]b0 ;:::;bi−1 (t0; : : : ; ti−1) with 1 the metafunction de(nedin [6].)

(d) We de(ne the set Bdb;t’ (T ) for bounded ’ via sets A’ and B’, which are

de(ned by recursion on ’:

’ A’ B’

Pu atomic ∅ {u} ◦ 3 A ∪ A3 B ∪ B3

Qx 6 u (x) (A (x))x(a’) ∪ {a’ � u} (B (x))x(a’) ∪ {u}Then

Bdb;t’ (T ) := {bi � ti : i 6 k} ∪ A’ ∪

∧u∈B’

(u6T )

expresses, that T is an upper bound to the “world” of ’. By this we meanthat all values which are considered by ’ are bounded by T .

6 With “Tait-style” we mean that sets (or formally: sequences) of formulas are derived, and that negationis a syntactic operation, not a symbol of our formal language.


T is a list of monotone terms with variables among a, D is a list of proofssimilar to (5c), such that for every in p there is some T in T and a proofd of Bdb;t

’ (T ) in D, which contains only the same free variables as Bdb;t’ (T ).

This means that max(T ) bounds the “world” of p, which allows us tocompute all values occurring in a soundness proof of a RProof ∗

T i2-derivation,

if we are able to compute val(max(T )).Now we de(ne

RProv∗T i

2(�) := ∃P RProof ∗

T i2(P; �);

RCon∗(T i2) := ¬RProv∗

T i2(p∅q):

If RProof ∗T i

2(P; p�q) then we say that P is a restricted-T i

2-proof of �.

Lemma 10. (S12 ) If ’∈ s�b

i , RProv∗T i

2(p�; ’q) and RProv∗

T i2(p�;¬’q), then RProv∗

T i2

(p�q).

Theorem 11 (Normalizing). RProv∗T i

2(p’q)⇔T i

2 �’.

Proof. ⇒: is clear.⇐: We give two proofs. We will sketch a prooftheoretic one, and give a precise

modeltheoretic one because we will extend the model-theoretic method later on.

Prooftheoretic proof. Assume T i2 �’. Then there is a derivation d1 of ’ which has as

axioms instances of BASIC and which uses the s�bi -IND-rule. By partial cutelimination

and further normalisations (see [6] or [3,5]) we obtain a derivation d2 of ’, in whichcutformulas are in s�b

i ∪ s�bi and the only free variables in d2 are those occurring free

in ’ (the parameters) and the eigenvariables of d2. By renaming eigenvariables andcollecting data we obtain a restricted-T i

2-proof of ’.

Modeltheoretic proof. Assume ¬RProv∗T i

2(p’q). Via modeltheoretic forcing (cf. [12])

we construct a countable model N of T i2 + ¬’. Hence T i

2 0’.We start adding new constants (dn)n∈! (the witnesses of our forcing construction)

to our language (call it L+). Again we call the formalized restricted proof predicatesin this extended language RProof ∗

T i2

and RProv∗T i

2. Then again ¬RProv∗

T i2(p’q), because

the new constants can be replaced by 0.We shall de(ne an increasing chain (Tn)n∈! of (nite sets of L+-sentences, such

that for every n

¬RProv∗T i

2(p¬Tnq): (1)

In the end their union T+ :=⋃

n Tn will be a Hintikka set for L+ (cf. Section 2.3in [12]), and the canonical model of the atomic sentences in T+ will be a model of


T i2 + ¬’. To ensure that T+ will have these properties, we carry out several tasks as

we build the chain. We consider the following tasks:

(T1) ¬’∈T+.(T2) For all s�b

i (L+)-sentences , ∈T+ or ¬ ∈T+.(T3) (x);t (for a s�b

i (L+)-formula in one free variable and t ∈L+ closed) (0);¬ (t)∈T+⇒ (s);¬ (s + 1)∈T+ for some closed s∈L+.

The following tasks (together with (T2) will insure that T+ is a Hintikka set:

(H1) ; 3 ∧ 3∈T+⇒ ; 3∈T+.(H2) ; 3 ∨ 3∈T+⇒ ∈T+ or 3∈T+.(H3)∀x (x) ∀x (x)∈T+⇒ (t)∈T+ for every closed t ∈L+.(H4)∃x (x) ∃x (x)∈T+⇒ (t)∈T+ for some closed t ∈L+.

If these tasks are all carried out, then T+ is a Hintikka set. E.g. if ∈ s�bi and ∈T+,

then ¬ =∈T+, because otherwise ;¬ ∈Tn for some n and obviously RProv∗T i

2(p¬ ; q)

contradicting condition (1) for n. Or, if s= t ∈T+ for some closed terms s; t ∈L+ thenalso t = s∈T+ using (T2) and RProv∗

T i2(ps �= t; t = sq).

Write N+ for the canonical model of the atomic sentences in T+. Then N+ isa model of T+. Using (T2) N+ is a model of the s�b

i ∪ s�bi -consequences of T i

2.Also N+ ful(lls s�b

i -IND, because if ∈ s�bi and N+ � (0)∧∃x¬ (x), then there

is some closed term t such that N+ � (0)∧¬ (t) as N+ is a term model. With(T2) we get (0);¬ (t)∈T+. By (T3) (x);t there is a closed L+-term s such that (s);¬ (s + 1)∈T+. Hence N+ �∃x( (x)∧¬ (x + 1)). Altogether this shows

N+ � (0) ∧ ∃x¬ (x)→ ∃x( (x) ∧ ¬ (x + 1));

thus

N+ � (0) ∧ ∀x( (x)→ (x + 1))→ ∀x (x):

By (T1) we further get N+ �¬’. Hence N+ is the desired model.It remains to show that all the countable tasks are enforceable. This means that for

any task T and any in(nite set X ⊂! such that !\X is also in(nite and 0 =∈X , weconsider the following game G(T; X ). The player, ∀ and ∃, pick the sets Tn in turn;player ∃ makes the choice of Tn if and only if n∈X . Player ∃ wins if T+ has propertyT, otherwise ∀ wins. Now “enforceable” means that in all these games player ∃ hasa winning strategy.

Consider a task. We describe an expert (feminine by convention) handling this task.Let X ⊂! be her subset and assume n∈X .

(T1) Let T0 := {¬’} (one expert is allowed to have 0 in her set X ).(T2) Let her list as ( n : n∈X ) all sentences of s�b

i (L+).If ¬RProv∗

T i2(p¬Tn−1;¬ nq), let Tn :=Tn−1 ∪{ n}. Otherwise she sets Tn :=Tn−1 ∪

{¬ n}. Thus she ful(lls her task.We have to check that condition (1) is not violated. Suppose for the sake of con-

tradiction that RProv∗T i

2(p¬Tnq) holds, then by construction RProv∗

T i2(p¬Tn−1;¬ nq)


holds and Tn =Tn−1 ∪{¬ n}. Now we obtain RProv∗T i

2(p¬Tn−1q) using Lemma 10

contradicting condition (1) for n− 1.(T3) (x);t If (0);¬ (t)∈Tn−1, then she chooses some witness d not occurring in Tn−1

and sets Tn :=Tn−1 ∪{ (d);¬ (d + 1)}. Otherwise she does nothing. This strategyworks, because her subset X contains arbitrarily large numbers.

Condition (1) is ful(lled, because if we have (0);¬ (t)∈Tn−1 and RProv∗T i

2

(p¬Tnq), then by construction we can replace d in Tn by a fresh variable a obtainingRProv∗

T i2(p¬Tn−1;¬ (a); (a+1)q), hence one application of formalized s�b

i -IND (inRProv∗

T i2) yields RProv∗

T i2(p¬Tn−1q).

Tasks (H1)–(H4) are treated in a similar way.

Theorem 12. (i¿0)

T i2 � RCon∗(T i

2):

Proof. The argument is similar to [13] Theorem 10.5.3.Suppose for the sake of contradiction P = 〈p; b; t; d; T ; D〉 is a restricted-T i

2-proof ofthe empty set, then by the sub formula property all formulas in p are in s�b

i ∪ s�bi .

Now S12 can evaluate all t; T (cf. Lemma 8). Consider a partial truth de(nition Tri(x; y)

for s�bi -formulas, (Tri ∈ s�b

i ), then S12 can prove Tarski’s conditions for all formulas

in p with b6t using T and D. Extend Tri to STri for sets of s�bi ∪ s�b

i -formulas(STri ∈�b

i+1), and consider the s�bi+1-formula ’(s)

∀r 6 s∀b0 6 t0 : : :∀bk 6 tkSTri(Sr; b) (2)

where p= (S0; : : : ; Sl). Local correctness, i.e. ’(s)→’(s + 1), is provable in T i2. With

s�bi+1-LIND we can show ’(l) which implies Tri(∅) – a contradiction. Hence Si+1

2

proves RCon∗(T i2), and as Si+1

2 is ∀�bi+1-conservative over T i

2, the same holds for T i2.

Remark 13. The GGodel sentences as proved in [6] as well as the following results inthis article admit another kind of restricting proof predicates. By adding weights wecan force the depth dp(p) of proofs p to be bounded by ‖p‖,

dp(p) 6 ‖p‖:Then assertion (2) in the proof of Theorem 12 can be reformulated as

’(s) := ∀r1; r2 6 l ∀b0 6 t0 : : :∀bk 6 tk [r1 6 r2 6 r1 + 2s &

∀h6 r1∀j 6 lhSTri(Sj;h; b)→ ∀h6 r2∀j 6 lhSTri(Sj;h; b)]

if p is written as (S0;0; S1;0; : : : ; Sl0 ;0; S0;1; : : : ; Sll; l) with j = depth of Si; j in p. Thus�b

i+1-L3IND su[ces to prove ’(|l|) from ’(0), as ’(s)∈�bi+1 and l6‖p‖. Hence for

any theory T =�bi -Lk IND we obtain

T + �bi+1-L3IND � RCon∗(T ):


Remember that we have n �→ sub∗(p8q∗; x•; n) as a band of polytime functions indexedby 8. De(ne

l∗(〈p; b; t; d; T ; D〉) := max(l∗(t); l∗(d); l∗(T ); l∗(D));

where, e.g., l∗(t1; : : : ; tk) means l∗(t1); : : : ; l∗(tk). An inspection of [6, Lemma 7.4.5]and [6, Theorem 7.4.4] shows

Lemma 14. For all terms t with free variables among x1; : : : ; xl there is a constantkt ∈! and a term st with free variables among x1; : : : ; xl such that

S12 � ∃P 6 st(u)[l∗(P)6kt ∧ RProof ∗

T 02(P; pt(Iu) = It(u)q)]:

Proof. Inspecting Lemma 7.4.5 in [6] shows that the lengths of all terms in the gen-erated proof of t(Iu) = It(u) is bounded by l(t) + O(1). Here it is important that in oursetting we have Iu := cu and hence l(Iu) = 1 for arbitrary u (even if u is nonstandard).

In the proof p of t(Iu) = It(u) generated in Lemma 7.4.5 of [6] only equations occur.Thus we may take T as the sequence of all v + w for v=w occurring in p, and D asthe sequence of simple proofs of v6v+w∧w6v+w. Then 〈ppq; p∅q; p∅q∗; p∅q∗; pTq∗;pDq∗〉 is a restricted proof of pt(Iu) = It(u)q.

Hence we can choose kt = l(ptq) + O(1).

Theorem 15 (∃�b1-Completeness for S1

2 w.r.t. p:q∗).(1) For ’∈�b

1 with free variables among x1; : : : ; xl there exists a constant k’ ∈!and a term s’ with free variables among x1; : : : ; xl such that

S12 � ’(u)→ ∃P 6 s’(u)[l∗(P) 6 k’ ∧ RProof ∗

T 02(P; p’(Iu)q)]:

(2) For ’∈∃�b1 with free variables among x1; : : : ; xl there exists a constant k’ ∈!

such that

S12 � ’(u)→ ∃P[l∗(P) 6 k’ ∧ RProof ∗

T 02(P; p’(Iu)q)]:

Proof. An inspection of Theorem 7.4.4 in [6] shows that the lengths of terms ingenerated proofs does not exceed those generated in Lemma 14. Hence we can choosek’ = l(p’q) + O(1).

Furthermore, proofs are generated by induction on the complexity of ’, they do notcontain applications of induction rules, and the structure of (∀6)-inferences and theirbounds reJect the structure of the bounded ∀-quanti(ers and their bounds in ’. Thuswe can de(ne “from the outside” lists b; t and d (all standard objects), such that if’(u) holds, then

〈ppq; pb q; pt q∗; pd q∗; pT q∗; pD q∗〉(where p; T ; D are collected similar as in the proof of Lemma 14) is a restricted proofof p’(Iu)q.


As a corollary we obtain that RCon∗(T i2) is the strongest ∀�b

1-statement (over S12 )

provable in T i2, similar to Con(Gi) in [14] or RCon(T i

1) in [15], or see [13] for atreatment of both.

Theorem 16 (i¿1). For all ’∈∀�b1 such that T i

2 �’ we have that S12 �

RCon∗(T i2)→’.

Proof. Let S12 �’↔∀x (x) with ∈ s�b

1, then we have T i2 � (x), hence RProof ∗

T i2

(P; p (x•)q) for some standard P using Normalizing (Theorem 11). S12 can check that

RProof ∗T i

2(IP; p (x•)q) holds. Now l∗(P) is standard, hence we can substitute values

for x• in P, because u �→ sub∗(P; x•; u) is polytime. Thus we obtain S12 �∀uRProv∗

T i2

(p (Iu)q). The ∃�b1-completeness of S1

2 shows S12 �¬ (u)→RProv∗

T i2(p¬ (Iu)q). Hence

S12 �¬ (u)→¬RCon∗(T i

2).

Corollary 17. Let i¿1.(1) S1

2 + RCon∗(T i2) =∀�b

1T i

2.

(2) Si2 �=∀�b

1T j

2 ⇔ Si2 0RCon∗(T j

2 ).

3. �b0-elementary extension

As shown in Fact 1 it is well-known that the ∀�b1-separation of Si

2 from T j2 (16i6j)

is equivalent to the existence of a model of Si2 which does not have a �b

0-elementaryextension to a model of T j

2 .We repeat this argument in an adapted form to prove the next theorem. We do this

also because the proof of Theorem 18 is the bases of following proofs.

Theorem 18 (i¿1). Assume M is a model of S12 which does not have a log3-proper

�b0-elementary extension to a model of T i

2. Then M �¬RCon∗(T i2).

Proof. Suppose for the sake of contradiction M �RCon∗(T i2). Let LM be the extension

of the constants cn : n∈! of our language L to cd :d∈M , plus some new constant b.So Id = cd for all d∈M . Consider

T := T i2 + �b

0-diag(M) + {|Id|3 ¡ |b|3 : d ∈ M};where �b

0-diag(M) is the �b0-diagram of M

�b0-diag(M) := {8(Ia1 ; : : : ; Ian) : 8 ∈ �b

0; a1; : : : ; an ∈ M;M � 8(a)}:We will show that T is consistent. Suppose for the sake of contradiction T is incon-sistent. Then by compactness there is a (nite subset D⊂�b

0-diag(M) and some (niteN ⊂M such that T i

2 + D + {|Id|3¡|b|3 :d∈N} is inconsistent. Let d0 = max(N ) andd1 =d0#d0 ∈M . As b has been new, we can replace it by Id1 . Now |d1|3¿|d0|3¿|d|3for any d∈N , hence we can assume w.l.o.g. that T i

2 + D is inconsistent. Using


that �b0 is closed under conjunction, we even can assume w.l.o.g. that there is one

8(Ia1 ; : : : ; Ian)∈�b0-diag(M) such that T i

2 + 8(Ia1 ; : : : ; Ian) is inconsistent. Hence

T i2 � ¬8(Ia1 ; : : : ; Ian):

Using Normalizing (Theorem 11) there is a standard restricted-T i2-proof with parameters

from M deriving this formula, hence

M � RProv∗T i

2(p¬8(Ia1 ; : : : ; Ian)q):

By �b1-Completeness (Theorem 15) we have M �RProv∗

T i2(p8(Ia1 ; : : : ; Ian)q). Hence us-

ing a bounded cut (i.e. a re(nement of Lemma 10) we obtain M �RProv∗T i

2(∅) contra-

dicting our assumption M �RCon∗(T i2). Thus T is consistent.

Let N be a model of T , then (up to isomorphism) M≺b0 N because T includes the

�b0-diagram of M . Furthermore N �T i

2 +{|Id|3¡|b|3 :d∈M}, hence N is a log3-properextension of M and N is a model of T i

2, contradicting our assumption that M does nothave a log3-proper �b

0-elementary extension to a model of T i2.

Thus our assumption was wrong, and we have M �¬RCon∗(T i2).

This theorem can be used to reobtain Fact 1 (and, therefore, (2)⇒ (1) of Theorem4), i.e. that if there is a model of Si

2 which does not have a log3-proper �b0-elementary

extension to a model of T j2 then Si

2 is ∀�b1-separated from T j

2 (16i6j). Because, underthe assumption the last Theorem shows that Si

2 does not prove RCon∗(T j2 ), while T j

2does prove it (cf. Theorem 12).

4. 1b-closed models

1-closed models of PA satisfy ¬Con(PA) [2]. Here we show something similar for1b-closed models of bounded arithmetic and theories T i

2.

Theorem 19. (i¿1) Assume M is a model of S12 + �nst

1 which is 1b-closed w.r.t. T i2.

Then M �¬RCon∗(T i2).

Proof. The proof is similar to that of Theorem 1.1 in [2].Suppose for the sake of contradiction M �RCon∗(T i

2). Fix a nonstandard b∈M suchthat M �∀x∃y(‖x‖ · b= ‖y‖). Let (z; p’q; x) be the formula

∀P[l∗(P) ¡ |z| → ¬RProof ∗T i

2(P; p¬’(Iz; Ix; y•)q)];

where ’∈ s�b1.

We assert that is universal for (standard) ∃s�b1-formulas with one free variable

and parameter b, i.e.

M � (b; p’q; a)↔ ∃y’(b; a; y) (3)

for ’∈ s�b1 and a∈M .


Fix ’∈ s�b1 and a∈M .

⇐: Assume there exists some d∈M such that M �’(b; a; d). By �b1-Completeness

(Theorem 15) there exists P ∈M with

M � RProof ∗T i

2(P; p’(Ib; Ia; Id)q): (4)

Suppose for the sake of contradiction M 2 (b; p’q; a), i.e. there is a Q∈M such thatM � l∗(Q)¡|b| and M �RProof ∗

T i2(Q; p¬’(Ib; Ia; y•)q). As M � l∗(Q)¡|b| and M �∀x

∃y(‖x‖ · b= ‖y‖), we can substitute y• in Q by Id obtaining some Q′ ∈M such that

M � RProof ∗T i

2(Q′; p¬’(Ib; Ia; Id)q):

Now a bounded cut with (4) (cf. Lemma 10) yields M �RProv∗T i

2(∅) which contradicts

our assumption M �RCon∗(T i2).

⇒: Assume M � (b; p’q; a). Let LM be the extension of the constants cn : n∈! ofour language L to cd :d∈M . So Id = cd for all d∈M .

Consider

T := T i2 + �b

0-diag(M) + ∃y’(Ib; Ia; y):

We will show that T is consistent. Suppose for the sake of contradiction T is in-consistent. By compactness and the closure of �b

0 under conjunction there is some8(Ia1 ; : : : ; Ian)∈�b

0-diag(M) such that T i2 + 8(Ia1 ; : : : ; Ian) + ∃y’(Ib; Ia; y) is inconsistent,

hence

T i2 � ¬8(Ia1 ; : : : ; Ian);¬’(Ib; Ia; y):

Using Normalizing (Theorem 11) there is a standard restricted-T i2-proof with parameters

from M for this set of formulas, hence there is some P ∈M such that M � l∗(P)¡|b|and

M � RProof ∗T i

2(P; p¬8(Ia1 ; : : : ; Ian);¬’(Ib; Ia; y•)q):

By �b1-Completeness (Theorem 15) there exist P′ ∈M such that l∗(P′) is standard and

M � RProof ∗T i

2(P′; p8(Ia1 ; : : : ; Ian)q):

Hence using a bounded cut (i.e. a re(nement of Lemma 10) yields a Q∈M such thatM � l∗(Q)¡|b| and

M � RProof ∗T i

2(Q; p¬’(Ib; Ia; y•)q)

contradicting our assumption M � (b; p’q; a). Thus T is consistent.Let N be a model of T , then (up to isomorphism) M≺b

0 N because T includes the�b

0-diagram of M . Furthermore N �T i2 + ∃y’(b; a; y), hence M �∃y’(b; a; y) by our

assumption that M is 1b-closed w.r.t. T i2. This (nishes the proof of (3).

Considering ¬ (z; x; x) there is some ’∈ s�b1 such that S1

2 �∃y’(z; x; y)↔¬ (z; x; x).By (3) we have

M � (b; p’q; a)↔ ∃y’(b; a; y)↔ ¬ (b; a; a)


for all a∈M . Instantiating this with a := p’q we get

M � (b; p’q; p’q)↔ ¬ (b; p’q; p’q);

a contradiction.Thus our assumption was wrong, and we have M �¬RCon∗(T i

2).

Corollary 20 (16i6j). Assume that there is a model of Si2 +�nst

1 which is 1b-closedw.r.t. T j

2 . Then Si2 �=∀�b

1T j

2 .

Proof. Under the assumption the last Theorem shows that Si2 does not prove

RCon∗(T j2 ), while T j

2 does prove it (cf. Theorem 12).

This proves (3)⇒ (1) of Theorem 4.

Corollary 21. There is no model M of T i2 + �nst

1 which is 1b-closed w.r.t. T i2.

Proof. Otherwise we would get M �¬RCon∗(T i2) by Theorem 19, which contradicts

M �T i2 as T i

2 �RCon∗(T i2).

The author conjectures that there also is no model M of Si2 + �nst

1 which is 1b-closedw.r.t. Si

2.

5. Weak end extensions

Up to now our restricted consistency notion yielded �b0-elementary extensions. We

will change this so that weak end extensions are obtained. To this end we extend ourformal language to Lex adding uniform small conjunctions and disjunctions:

’(x) ∈Lex ⇒∨

b6|a|’(Ib);

∧b6|a|

’(Ib) ∈Lex:

The formal Tait-style rules for deriving them are given by(∨)�; ’(Ib) for some b6 |a| ⇒ �;

∨b6|a|

’(Ib);(∧)�; ’(Ib) for all b6 |a| ⇒ �;

∧b6|a|

’(Ib):

We de(ne �b; ex0 , s�b; ex

i etc. analogous to �b0, s�b

i etc. counting small conjunctions andsmall disjunctions as sharply bounded quanti(ers. Now we de(ne RProof ∗

T i2+WEnd similar

to RProof ∗T i

2in the language Lex with additional rules (

∨), (∧

) and an additionalaxiom schema

∀x 6 |Ia|∨

b6|a|x = Ib


for all a. Furthermore cuts are extended to s�b; exi ∪ s�b; ex

i -formulas. Induction need notto be extended to s�b; ex

i .We have

Theorem 22. T i2 �RCon∗(T i

2 + WEnd).

Proof. There is a polytime transformation elex :Lex→L given by

≡∨

b6|a|’(Ib) �→ ∃x 6 |Ia|’(x );

≡∧

b6|a|’(Ib) �→ ∀x 6 |Ia|’(x )

which can be formalized in S12 . We have

elex(�b;ex0 ) = �b

0; elex(s�b;exi ) = s�b

i ; etc:

Furthermore

elex

∀x 6 |Ia| ∨b6|a|

x = Ib

≡ ∀x 6 |Ia|∃y 6 |Ia|x = y

and

x �= I0 ∧ : : : ∧ x �= I|a| → x � I|a|

have simple proofs in T 02 of size linear in |a| resp. |a|2. Hence

S12 � RProv∗

T i2+WEnd(p�q)→ RProv∗

T i2(pelex(�)q)

and therefore

S12 � RCon∗(T i

2)→ RCon∗(T i2 + WEnd):

Now Theorem 12, T i2 �RCon∗(T i

2), yields the assertion.

Theorem 23. Assume M is a countable model of S12 +BL�b

1 without weak end exten-sions to models of T i

2. Then M �¬RCon∗(T i2 + WEnd).

Proof. Suppose for the sake of contradiction that M �RCon∗(T i2 +WEnd). We extend

our forcing construction from the proof of Theorem 11. Again we start extending ourlanguages L;Lex to include witnesses (dn)n∈! (we call them L+ resp. Lex+). We callthe formalized restricted proof predicate in the language Lex+ again RProof ∗

T i2+WEnd,

obtaining

M � ¬RProv∗T i

2+WEnd(p∅q):


We de(ne an increasing chain (Tn)n∈! of (nite sets of L+-sentences, such that forevery n

M � ¬RProv∗T i

2+WEnd(p¬Tnq): (5)

In the end their union T+ :=⋃

n Tn will be a Hintikka set for L+, and the canonicalmodel N of the atomic sentences in T+ will be a weak end extension of M and amodel of T i

2, contradicting our assumption.To ensure that T+ will have these properties, we carry out several tasks as we build

the chain. Beside tasks (T2)–(T3), (H1)–(H4) described in the proof of Theorem 11we need also the following task:

(T1′)a;t (for a∈M; t ∈L+ closed) If (t6|Ia|)∈T+, then there is some b∈M suchthat (t = Ib)∈T+.

If (T1′) is ful(lled, then N is a weak end extension of M . To see this, let a∈M andb∈N such that N � b6|a|, then we have to show b∈M . As N is the canonical model,there is some closed t ∈L+ such that N � b= t, hence N � t6|a|. T+ is s�b

i -complete(by (T2)), hence (t6|Ia|)∈T+. Now (T1′)a;t yields the existence of some e∈M suchthat (t = Ie)∈T+, hence N � t = e and therefore b= e∈M .

It remains to show that (T1′)a;t is enforceable. Therefore we need to describe anexpert handling this task. Let X ⊂! be her subset and assume n∈X . Let her list{(a; t) : a∈M; t ∈L+ closed} using her set X . Of course here we need that M iscountable. Assume that (a; t) is the element indexed by n in her list.

If (t6|Ia|) =∈Tn−1 let Tn :=Tn−1. Otherwise (t6|Ia|)∈Tn−1. Suppose for the sakeof contradiction that

M � ∀b6|a|∃pRProof ∗T i

2+WEnd(p; p¬Tn−1; t �= Ibq):

As M is a model of S12 + BL�b

1 we obtain

M �∃P∀b6|a|RProof ∗T i

2+WEnd((P)b; p¬Tn−1; t �= Ibq);

hence by (∧

)

M � RProv∗T i

2+WEnd

p¬Tn−1;∧

b6|a|t �= Ibq

:

As (t6|Ia|)∈Tn−1 we get

M � RProv∗T i

2+WEnd(p¬Tn−1; t 6 |Ia|q);

hence

M � RProv∗T i

2+WEnd

p¬Tn−1;∃x 6 |Ia|∧

b6|a|x �= Ibq


thus M �RProv∗T i

2+WEnd(p¬Tn−1q) by a �b; ex0 -cut contradicting (5) for n−1. Hence there

is some b6|a| in M such that

M � ¬RProv∗T i

2+WEnd(p¬Tn−1; t �= Ibq):

Let Tn :=Tn−1 ∪{t �= Ib}.Using this strategy she ful(lls her task as X is in(nite.

Corollary 24 (16i6j). Assume that there is a countable model of Si2 +BL�b

1 withoutweak end extensions to models of T j

2 . Then Si2 �=∀�b

1T j

2 .

Proof. Under the assumption the last Theorem shows that Si2 does not prove

RCon∗(T j2 + WEnd), while T j

2 does prove it (cf. Theorem 22).

This proves (4)⇒ (1) of Theorem 4.

6. Extending results

First let us remark that if we modify our restricted provability notion to include trues�b

k ∪ s�bk -formulas as axioms we would characterize the ∀�b

k -separation of Si2 from

T j2 . Also it is obvious how to adjust the coding p:q∗ to include functions with stronger

but sub-exponential growth rates like #3; #4; : : :.Now let < be an ∀(�b

i ∪ �bi )-sentence. We are going to argue that the previously

obtained results can be extended by adding < to the theories.W.l.o.g. let <=∀y<′(y) for some <′ ∈ s�b

i ∪ s�bi , because we are adding < to T i

2,and T i

2 knows �bi = s�b

i . We extend RProof ∗T i

2to RProof ∗

T i2+< by allowing all instances

of ∀y<′(y) as additional axioms, i.e. axioms of the form <′(t) for any L-term t. ThenRProv∗

T i2+< and RCon∗(T i

2 + <) are de(ned in the obvious way.

Theorem 25. T i2 + <�RCon∗(T i

2 + <).

Proof. See also Theorem 12.S1

2 proves Tarski’s conditions for all formulas in a restricted-T i2 + <-proof P of the

emptyset, therefore we have for all axioms <′(t) occurring in P = 〈p; b; t; d; T ; D〉∀y<′(y)→ ∀b0 6 t0 : : :∀bk 6 tk Tri(p<′(t)q; b):

Thus the argument runs the same way as in the proof of Theorem 12, i.e. we can show

T i2 � ∀y<′(y)→ RCon∗(T i

2 + <):

Again we have that RCon∗(T i2 + <) is the strongest ∀�b

1-statement (over S12 ) provable

in T i2 + <.


Corollary 26. For all ’∈∀�b1 such that T i

2 + <�’ we have S12 �RCon∗(T i

2 +<)→’.

Theorem 27. Assume M is a countable model of S12 +BL�b

1 without weak end exten-sions to models of T i

2 + <. Then M �¬RCon∗(T i2 + < + WEnd).

Proof. Assuming M �RCon∗(T i2 + < + WEnd) we modify our forcing construction

obtaining a Hintikka set T+ such that

<′(t) ∈ T+ for all closed L+-terms t:

Then the canonical model N generated from T+ ful(lls N �∀y<′(y), thus N �T i2 + <

and N is a weak end extension of M contradicting our assumption.

Similarly we can prove the following theorems.

Theorem 28. Assume M is a model of S12 which does not have a log3-proper �b

0-elementary extension to a model of T i

2 + <. Then M �¬RCon∗(T i2 + <).

Theorem 29. Assume M is a model of S12 +�nst

1 which is 1b-closed w.r.t. T i2 +<. Then

M �¬RCon∗(T i2 + <).

Hence we obtain

Corollary 30. Let 16i6j. The following are equivalent:(1) Si

2 + < is ∀�b1-separated from T j

2 + <.(2) Si

2 + < has a model which does not have a log3-proper �b0-elementary extension

to a model of T j2 + <.

(3) Si2 + < + �nst

1 has a model which is 1b-closed w.r.t. T j2 + <.

(4) Si2 + < + BL�b

1 has a countable model without weak end extensions to models ofT j

2 + <.

Remark 31. Considering �nst1 we can extend the results allowing < to be an ∃∀(�b

i ∪�b

i )-sentences which has as a parameter the nonstandard element given by �nst1 . I.e.,

assume that our language L is extended by a new constant c. Let �nst1 (c) be the

following L!1!-sentence:

�nst1 (c) :=

(∧k∈!

(k ¡ c)

)∧ ∀x∃y(‖x‖ · c = ‖y‖):

Thus �nst1 =∃z�nst

1 (z). Then we can consider <(c) instead of < in the previously obtained

results where �nst1 is replaced by �

nst1 (c).


7. Towards proper weak end extensions

In Section 5 we have connected the ∀�b1-separation of Si

2 from T j2 with models of

Si2 +BL�b

1 which do not have weak end extensions to models of T j2 . But we are really

interested in a connection to models without proper weak end extensions, because thereis a similar (open) question for models of I�0: are there models of I�0 +B�1 withoutproper end extensions to models of I�0? Furthermore, there exists a �1-sentence < (aversion of the Tableau consistency of I�0) and a model of I�0 + �1 + < + B�1 whichhas no proper end extensions to models of I�0 + �1 + < (cf. [1]).

Up to now we have not achieved a connection to proper weak end extensions. Wewill describe two possible ways in this direction now.

Wilkie and Paris in [22] de(ned a �3-sentence such that

I�0 + B�1 I�0 + B�1 + I�0 + B�1 + ¬ all have the same �1-consequences. This can be improved to

Proposition 32. There is a ∀∃�b1-sentence 3 such that for S1

2 ⊆T ⊆ S2

T T + 3 T + ¬3all have the same �1-consequences.

Proof. Let Loga(z) := =u:z62|a|u , then Loga(z) = u has a �b1-description in S1

2 . Let 3be the sentence equivalent to

∀a∀z(2|a|2Loga(z)

exists):

Then

T T + 3 T + ¬3all have the same �1-consequences. To see this, suppose

T + ¬3 � ∀x8(x); 8 ∈ �0

but

T 0 ∀x8(x):

Then we (nd a model K �T +¬8(a) such that 2|a|?t exists in K for some N¡?; t¡a.Let

LK (a; ?) := {x ∈ K : x 6 2|a|?n some n ∈ N} ⊆e K

then we have LK (a; ?) �T + ¬8(a) (this is true for arbitrary ?). Furthermore,LK (a; ?) �¬3, because b := 2|a|? ∈LK (a; ?) and Loga(b) = ?, but 2|a|2?�2|a|?n for alln∈N.

The case for 3 is similar, taking 2 in place of ? and observing LK (a; 2) � 3.


Remark 33. The same 3 from the proof of the last proposition also ful(lls that

T + BL�b1 T + BL�b

1 + 3 T + BL�b1 + ¬3

have the same �1-consequences. This can be seen by adapting Theorem 1 from [22]in the form

M ⊂e K � S12 ⇒ M � BL�b

1:

Open Problem 1. Can this be improved to (nding a sentence @ such that Proposition32 holds for @ instead of 3 and such that the formalized proof predicate can be extendedto @ and ¬@ ful(lling

T j2 + @ � RCon∗(T j

2 + @) T j2 + ¬@ � RCon∗(T j

2 + ¬@)?

Having this, we would get

T j2 � RCon∗(T j

2 + @) ∧ RCon∗(T j2 + ¬@)

using Proposition 32.On the other hand we would be able to show that if M is a model of S1

2 + BL�b1

without proper weak end extensions to models of T j2 , then

M � ¬RCon∗(T j2 + @) ∨ ¬RCon∗(T j

2 + ¬@):

To see this observe M �@ ∨ ¬@. W.l.o.g. we may assume M �@. Now assumingM �RCon∗(T j

2 +¬@) would produce a weak end extension N of M which is a modelof T j

2 + ¬@. But then M �=N as M �@ and N �¬@, hence the extension is a properone contradicting our assumption. Hence

Si2 0 RCon∗(T j

2 + @) ∧ RCon∗(T j2 + ¬@):

The second possibility bases on an ultrapower construction described by Buss in [7].Suppose

M � Si2 + BL�b

i + BB�bi

where BB�bi is the sharply bounded replacement schema

(∀x 6 |a|)(∃y 6 b)’(x; y)→ (∃w)(∀x 6 |a|)’(x; (w)x)

for ’∈�bi . Note that by results of Buss [8] and Ressayre [19] the theory Si

2 +BL�bi +

BB�bi is ∀�b

i+1-conservative over Si2. Buss shows in [7] that there is a proper (�b

i ∩�bi )-

elementary weak end extension N of M such that logN = logM and N �T i−12 .

Open Problem 2. Can this be improved such that N �BL�b1 and N is a ∀�b

1-elementaryextension of M?


Then we could argue as follows: If M does not have proper weak end extensions tomodels of T j

2 , then M �¬RCon∗(T j2 ). Because assuming M �RCon∗(T j

2 ) would implyN �RCon∗(T j

2 ), and our construction from Section 5 would yield a weak end extensionsof N to a model N ′ of T j

2 , but N ′ would now be a proper weak end extension ofM – contradiction.

Acknowledgements

I would like to thank Sam Buss for his hospitality during my stay at the Departmentof Mathematics of the University of California, San Diego, and for a lot of discussionsand remarks.

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Preservation theorems and restricted consistency statements in bounded arithmetic

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