The Second Incompleteness Theorem and Bounded Interpretations

Studia Logica (2012) 100: 399–418DOI: 10.1007/s11225-012-9385-z © Springer 2012

Albert Visser The Second IncompletenessTheorem and BoundedInterpretations

Dedicated to the memory of Leo Esakia

Abstract. In this paper we formulate a version of Second Incompleteness Theorem. The

idea is that a sequential sentence has ‘consistency power’ over a theory if it enables us to

construct a bounded interpretation of that theory. An interpretation of V in U is bounded

if, for some n, all translations of V -sentences are U -provably equivalent to sentences of

complexity less than n. We call a sequential sentence with consistency power over T a

pro-consistency statement for T . We study pro-consistency statements. We provide an

example of a pro-consistency statement for a sequential sentence A that is weaker than an

ordinary consistency statement for A. We show that, if A is S12, this sentence has some

further appealing properties, specifically that it is an Orey sentence for EA.

The basic ideas of the paper essentially involve sequential theories. We have a brief

look at the wider environment of the results, to wit the case of theories with pairing.

Keywords: Second Incompleteness Theorem, interpretability.

Introduction

In his posting

http://www.cs.nyu.edu/pipermail/fom/2010-January/014290.html

of January 7, 2010, Harvey Friedman introduces a new approach to theSecond Incompleteness Theorem.1

According to this approach the distinguishing feature of a consistencystatement is its power to support a proof of Godel’s Completeness Theoremfor the theory at hand.

We write U � V for: U interprets V . Friedman defines:

A is a sufficient formalization of con(V ) in U iff (U + A) interprets V ,(U + A) � V .

1See also the introduction to Friedman’s forthcoming book on Boolean Relation Theory.

Special issue dedicated to the memory of Leo EsakiaEdited by L. Beklemishev, G. Bezhanishvili, D. Mundici and Y. Venema

400 A. Visser

He notes that, for every RE theory T , there is a Π01 sufficient formalization

of con(T ) over a fixed theory, to wit a usual arithmetization of con(T ), in EA(aka EFA or IΔ0 + exp). In fact this also works when we replace EA by Buss’theory S1

2 or even by Robinson’s Arithmetic Q. Here are two of Friedman’sconcrete proposals for versions of the Second Incompleteness Theorem. LetEA� be IΔ0 + supexp.

a. Let K : V � EA�. Then there is no Π01 sufficient formalization P of con(V )

in EA such that V � PK .

b. Let K : V � EA. Then there is no Π01 sufficient formalization P of con(V )

in Q such that V � PK .

We can replace Q in (b) by S12 or by IΔ0 + Ω1. We rewrite (a) and (b) to

the following.

a . For no Π01-sentence P we have: (EA + P ) � T and T � (EA� + P ).

b . For no Π01-sentence P we have: (Q + P ) � T and T � (EA + P ).

The results follow immediately from:

a . For every Π01-sentence P : (EA + P ) �� (EA� + P ).

b . For every Π01-sentence P we have: (Q + P ) �� (EA + P ).

Here (a ) and (b ) follow from the application of the Second IncompletenessTheorem in the strong form due to Pudlak to the fact that EA� + P �con(EA + P ), resp. the fact that J : (EA + P ) � (Q + con(Q + P )), where Jis a superexponential cut (see [16], [7]).

These formulations of Godel’s Second Incompleteness Theorem have afew disadvantages. It is clear that the choice of the pairs EA, EA� and Q,EA does the trick. But why these pairs? Surely some further informallyrigourous explanation is due? Secondly, the choice of Π0

1-sentences is an apriori restriction on the form of the consistency statements that should bemotivated. Thirdly, these results do not seem to cover a perfectly reasonablespecial case like S1

2 � con(S12). Fourthly, taking EA or Q as the basis in

which we formulate sufficient formalizations of consistency statements makesconn(A) a sufficient formalization of A (for a sentence A and sufficientlylarge n). This does not seem to do justice to the fact that the SecondIncompleteness Theorem does not hold for restricted consistency statements.E.g. ACA0 � (S1

2 + conn(ACA0)). It seems to me that this important insightshould be reflected in one’s framework.

In this paper, I formulate and study a different proposal that is, how-ever, based on the underlying scheme of Friedman’s idea. A fundamental

Bounded Interpretations . . . 401

ingredient of both Friedman’s and my realisation is that we replace modelexistence with interpretation existence. However, by considering mere inter-pretation existence one looses some information about the construction: wedo not just construct an interpretation, but an interpretation equipped witha truth predicate given by the complete Henkin theory.

For this reason, I propose the following notion: a pro-consistency state-ment for a theory T is a sequential sentence A such that there is a boundedinterpretation of T in A. Here a sequential sentence is a sentence that pro-vides sufficient coding possibilities. A bounded interpretation is an interpre-tation for which we can reduce the complexity of any translated T -formulamodulo A-provable equivalence to an A-formula of complexity below a fixedstandard k. Of course, boundedness is a consequence of the presence of atruth predicate, but seems to be a bit weaker. It has the advantage that itsformulation demands no coding.

One can show that, for an RE-theory T , the sentence S12+con(T ) is a pro-

consistency statement for T . Thus, our explication includes the usual cases.The Second Incompleteness Theorem, for pro-consistency statements,

now takes a particularly simple form:

No sequential sentence A has a bounded interpretation of itself.

It follows that, if there is a bounded interpretation of T in A, then T can-not interpret A, since the composition of a bounded interpretation and aninterpretation is a bounded interpretation.

An attractive feature of the Friedman style approach is the completeemancipation of the (pro-)consistency statement in the Second Incomplete-ness Theorem from its formulation in the theory we are thinking about. Theconsistency statement becomes a theory rather than a sentence. ‘Proving itsown consistency’ becomes a relation between theories.

If B is a sequential sentence, then S12 plus the cut-free (or: tableaux, or: Her-

brand, or: restricted) consistency of B is not a pro-consistency statement.We do have an interpretation of B in S1

2 plus the cut-free (etc.) consistencyof B, but not a bounded one. We also have an interpretation in the otherdirection: S1

2 plus the cut-free (etc.) consistency of B is interpretable inB. This illustrates that cut-free consistency need not be stronger than theoriginal theory and, thus, that the incompleteness theorem for cut-free (etc.)consistency fails.

However, as we will illustrate there are pro-consistency statements thatare weaker than the usual consistency statements and are, in a sense, be-tween consistency and cut-free (etc.) consistency. See Section 4 on super-logarithmically bounded consistency.

402 A. Visser

We note that we still have an a priori restriction on the form of our con-sistency statements: they should allow (sequence) coding. It seems to methat the presence of coding as a restriction does have some intuitive support.Still it would be very interesting to widen our net and consider lighter con-straints. E.g., if we replace sequential in our present formulation by theory-with-pairing, what happens then? The present paper leaves such questionsopen for further meditation. However, I submit that these questions point inthe right direction: after all is said and done, the point is not to give betterexplications of the second incompleteness theorem but to extend it in newdirections.

For a somewhat different approach to the Second Incompleteness Theorem,inspired by earlier FOM ideas of Harvey Friedman, see [14].

1. Preliminaries

We consider one-sorted RE theories with a finite signature. An interpre-tation between theories is a relative, more-dimensional interpretation. Weallow translation of identity by an equivalence relation. We write K : U �Vfor: K is an interpretation of V in U . An interpretation is direct iff it is one-dimensional, unrelativized and preserves identity. For precise definitions, seee.g. [13] or [12].

A theory is sequential iff it directly interprets adjunctive set theory AS. HereAS is the following theory in the language with only one binary relationsymbol.

AS1. � ∃x∀y y �∈ x,

AS2. � ∀x, y ∃z ∀u (u ∈ z ↔ (u ∈ x ∨ u = y)).

So the basic idea is that we can define a predicate ∈� in U such that ∈�

satisfies a very weak set-theory involving all the objects of U . Given thisweak set theory, we can develop a theory of sequences for all the objects inU , which again gives us partial truth-predicates, etc. In short, the notion ofsequentiality explicates the idea of a theory with coding.

A theory is a pair theory if it directly interprets a weak theory of non-surjective pairing. The theory of non-surjective pairing PAIR is given asfollows. It has, apart from identity, one ternary symbol pair. It has, apartfrom the axioms of identity, the following axioms.

po1 � ∀u, v ∃x pair(u, v, x),


po2 � ∀u, v, u′, v′, x ((pair(u, v, x) ∧ pair(u′, v′, x)) → (u = u′ ∧ v = v′)).po3 � ∃x ∀u, v ¬ pair(u, v, x).

The demand that our pairing is non-surjective is, modulo mutual directinterpretability, equivalent to the axiom that there are, provably, at leasttwo objects in the domain. For more information, see [13].

Restricted provability plays an important role in this paper. An n-proofis a proof from axioms smaller or equal than n only involving formulas ofcomplexity smaller or equal than n. To work conveniently with this notion,a good complexity measure is needed. This should satisfy three conditions.(i) Eliminating terms in favour of a relational formulation should raise thecomplexity only by a fixed standard number, (ii) Translation of a formulavia the translation corresponding to an interpretation K should raise thecomplexity of the formula by a fixed standard number depending only on K.(iii) The tower of exponents involved in cut-elimination should be of heightlinear in the complexity of the formulas involved in the proof. Such a goodmeasure of complexity —a form of nesting degree of quantifier alternations—is supplied in the work of Philipp Gerhardy. See [5] and [6]. In Figure 1,we give a nice formulation of Gerhardy’s measure that was formulated byVincent van Oostrom (in conversation). We say that C ∈ Γk iff ρ(C) ≤ k.

ρ(P�x) := 0

ρ(¬A) := ρ(A)

ρ(A ∧ B) := max(ρ(A), ρ(B))

ρ(A ∨ B) := max(ρ(A), ρ(B))

ρ(A → B) := max(ρ(A), ρ(B))

ρ(∀x A) := ρ∀(A)

ρ(∃x A) := ρ∃(A)

ρ∀(P�x) := 0

ρ∀(¬A) := ρ∃(A)

ρ∀(A ∧ B) := max(ρ∀(A), ρ∀(B))

ρ∀(A ∨ B) := max(ρ(A), ρ(B)) + 1

ρ∀(A → B) := max(ρ(A), ρ(B)) + 1

ρ∀(∀x A) := ρ∀(A)

ρ∀(∃x A) := ρ∃(A) + 1

ρ∃(P�x) := 0

ρ∃(¬A) := ρ∀(A)

ρ∃(A ∧ B) := max(ρ(A), ρ(B)) + 1

ρ∃(A ∨ B) := max(ρ∃(A), ρ∃(B))

ρ∃(A → B) := max(ρ∀(A), ρ∃(B))

ρ∃(∀x A) := ρ∀(A) + 1

ρ∃(∃x A) := ρ∃(A)

Figure 1. Gerhardy’s complexity measure

404 A. Visser

2. Bounded Interpretations

In this section, we introduce the notion of bounded interpretation and provesome basic facts about it.

An n-dimensional interpretation K : U � V is bounded when, there is anatural number k, such that, for all V -formulas A(x0, . . . , xm−1), there is anU -formula B(�y0, . . . , �ym−1) in Γk, such that:

U � ∀�y0 ∈ δK , . . . , �ym−1 ∈ δK (B(�y0, . . . , �ym−1) ↔ AK(�y0, . . . , �ym−1)).

In our definition we assumed that our interpretation was one-piece andparameter-free. Moreover, we assumed that our theories are one-sorted.It is obvious how to adapt the definition to more complex cases. However,it is somewhat laborious to actually do it. We write U �bo V for: there is abounded K with K : U � V .

2.1. General Results

We give a number of results that hold for all RE theories.

Theorem 2.1. We have:

i. If U �bo V � W , then U �bo W .

ii. If U � V �bo W , then U �bo W .

Proof. Claim (i) is a triviality. Claim (ii) uses that ρ(AK) := ρ(A)+ c, forsome standard c that depends only on K. �

We remind the reader that �(W ) := S12+{coni(U) | i ∈ ω}, where coni(U) is

consistency of U for Γi-provability only involving axioms witnessed below i.2

See [14] for more information. The following theorem is a version of theFeferman-Henkin-Wang-Hilbert-Bernays-Godel theorem.

Theorem 2.2. �(U) �bo U .

See [3] for Feferman’s original paper. A sketch of the proof of the presentversion can be found in [14] (the proof of Theorem 3.9). For more detailssee e.g. [11]. The boundedness is a bonus from the fact that we constructour ‘internal model’ from a predicate H defining a complete Henkin theory.This predicate H functions as a satisfaction predicate and, thus, witnessesthe boundedness.

2We pronounce �(U) as: mho of U , where mho rhymes with Joe.


2.2. Sequential Theories

In this subsection, we collect a number of results that are specific for se-quential theories.

Theorem 2.3. Suppose U and V are sequential RE theories. Then, U �boViff U � �(V ).

Proof. Since we are considering sequential theories, we may assume thatall interpretations are one-dimensional.

From right to left. We have U � �(V ) �bo V . We apply Theorem 2.1 toobtain U �bo V .

From left to right. Suppose K : U �bo V and let k be the bound thatwitnesses this fact. We choose fixed interpretations M and N of S1

2 in U ,respectively V . These will be the numbers of U , respectively V .

Let I be the common cut of M and K ◦ N . This means that I is adefinable cut of M , such that there is a definable, U -verifiable, isomorphicembedding F of I in a cut I� of K ◦ N . By [9], such a common cut alwaysexists in sequential theories. Consider any formula B. We are interested inB viewed as a property of I�-numbers. Consider B’s back-projection, sayD, to I-numbers:

D�x := ∃y0, . . .∃ym−1 (F (x0, y0) ∧ . . . ∧ F (xm−1, ym−1) ∧ B(y0, . . . , yk−1))

We note that ρ(D) ≤ max(ρ(F ), ρ(B)) + 1. We will be interested in suchformulas D for B ∈ Γk. This motivates the following definition. Let k� :=max(ρ(F ), k) + 1.

We have, in U , a satisfaction predicate S for formulas of complexity k�,that works for formulas coded on some definable cut C of M . Let ε be thecode of the empty sequence. Let form0 be the virtual class of codes in C offormulas E with just x0 free in E. Let cut be the function that transformsthe Godelnumber of E in form0 to the Godelnumber of Cut(E), the formulaexpressing that E is a cut of M . Clearly, cut maps formulas of form0 tosentences in C. We define:

J(x) :↔ ∀e ∈ form0 (S(ε, cut(e)) → S(〈x〉, e)).We easily see that J is a definable cut of M that is initial in all definablek�-cuts of M . We note that I(x) :↔ ∃y Fxy. So I is a k�-cut and hence Jis U -verifiably a sub-cut of I.

Consider any n. Since V is sequential, there is a V -definable cut L ofN , such that V � conL

n(V ). Consider the U -cut D := {x | ∃y (Fxy ∧LKy)}.

406 A. Visser

We can find an E in Γk such that D := {x | ∃y (Fxy∧Ey)}. So D is in Γk� .We may conclude that J is U -verifiably a sub-cut of D. Hence, U � conJ

n(V ).Since, n was arbitrary, it follows that J : U � �(V ). �

The following theorem is due to Pavel Pudlak (see [9]).

Theorem 2.4. Let A be a sequential sentence. Then, A �� (A).

As a corollary, we have the following theorem.

Theorem 2.5. Let A be a sequential sentence. Then, A ��bo A.

Open Question 2.6. Is there a pair sentence A with A �bo A?

Using the fact that � is the right adjoint of the projection functor of thedegrees of interpretability of sequential RE theories into the degrees of lo-cal interpretability of sequential RE theories3, we also have the followingcorollary.

Corollary 2.7. Suppose U and V are sequential RE theories. We have:U �bo V iff, for all sequential RE theories W , if V �loc W , then U � W .

2.3. Excursion: Pairing and First-Order Comprehension

In this subsection, we have a brief look at the case of pair-theories. LetFOC(V ) be the functor that adds a new sort of classes to the signature of Vand adds the scheme of first order comprehension. The idea is that FOC isfor pair theories what � is for sequential theories. Regrettably, the analogy isstill incomplete. (We are diverging here from our policy to consider only one-sorted theories. However, in the presence of pairing, we may replace FOC(U)by its one-sorted flattening replacing the sorts by one-place predicates thatpartition the domain, etcetera.)

A theory V is locally interpretable in U iff, for every finitely axiomatizedsubtheory V0 of V , we have U � V0. We write U �loc V for: V is locallyinterpretable in U .

Theorem 2.8. For any pair theory U , we have U �loc FOC(U).

Proof. Consider formulas Ai(x, �y), for i = 0, . . . , n − 1, in the language ofU . We assume that the sequence x, �y is part of the data for specifying theformula. The free variables of Ai are among the x, �y. Let the length of thesequence �y of Ai be �i.

3See [14].


We define formulas seqj(y0, . . . , yj−1, x), as follows:

seq0(x) :↔ ∀y, z ¬ pair(y, z, x),

seqj+1(y, �z, x) :↔ ∃u (seqj(�z, u) ∧ pair(y, u, x)).

Here, pair is the formula that represents pairing in U . It is easily verified inU that seqj indeed defines sequences of length j with the right projections.Moreover sequences of length i are disjoint from sequences of length j, fori �= j.

We define the second order domain of classes by:

δc(u) :↔ ∨i<n ∃v, w, �y, �z (seqi(�z, v) ∧ seq�i

(�y, w) ∧ pair(v, w, u)).

Here the first component of the pair u just represents the index i. Thesecond component represents the parameters in the definition of the class.The definition of elementhood is as follows:

x ∈ u :↔ ∨i<n ∃v, w, �y, �z (seqi(�z, v) ∧ seq�i

(�y, w) ∧pair(v, w, u) ∧ Ai(x, �y)).

Finally, we define identity on classes as extensional identity w.r.t. ∈. It iseasy to verify that our present definition validates first-order comprehensionfor the formulas A0, . . . , An−1. �

We write U ≡ V for: U � V and V � U .

Theorem 2.9. If V is sequential �(V ) ≡ FOC(V ).

Proof. From left to right. The Feferman-Henkin interpretation of V in�(V ) comes equipped with a satisfaction predicate. We use that to definethe desired classes. See the proof of Theorem 3.9 of [14] for a sketch of theFeferman-Henkin interpretation.

Alternatively, we may reason as follows. Since V is sequential, it is afortiori a pair theory. Hence, by our previous theorem, V �loc FOC(V ). ByTheorem 3.11 of [14], we may conclude �(V ) � FOC(V ).

From right to left. In FOC(V ), we can take the intersection, say J , of allV -definable cuts (for a given set of V -numbers N). For every n, we canfind a V -definable cut I (of N) such that V � conI

n(V ). This follows fromthe existence of partial satisfaction predicates in V . Since Π0

1-sentences aredownwards preserved to initial sub-cuts, we have �(V ) on J . �

Theorem 2.10. Suppose U is a pair theory. We have FOC(V ) �bo V .

408 A. Visser

Proof. We show that the identity interpretation of V in V is bounded inFOC(V ). We define formulas seqn as in the proof of Theorem 2.8.

Consider any V -formula A. For each subformula B of A we introduce aclass variable XB. This class is intended to contain the sequences of objectssatisfying the formula. These sequences correspond precisely to the freevariables of the formula, say in a fixed standard ordering of the variables.We define a formula αB as follows.

Suppose B is P�x. We set:

αB :↔ ∀�x, u (seqn(�x, u) → (u ∈ XB ↔ B(�x))).

Suppose B is C ∧ D and the free variables of B are �x, the free variables ofC are �x0 and the free variables of D are �x1. We set:

αB :↔ ∀�x, u, v, w ((seqn(�x, u) ∧ seqm(�x0, v) ∧ seqp(�x1, w)) →(u ∈ XB ↔ (v ∈ XC ∧ w ∈ XD))).

Suppose B is ∃y C. We assume that y occurs in C. The case that it doesnot is easier. Let the free variables of C be �x0, y, �x1. We set:

αB :↔ ∀�x0, �x1, u (seqn(�x0, �x1, u) →(u ∈ XB ↔ ∃y, v (seqn+1(�x0, y, �x1, v) ∧ v ∈ XC))).

The other cases are similar. We have:

A�x ↔ ∃XA, XB, . . .∃v (αA ∧ αB ∧ . . . ∧ seqn(�x, v) ∧ v ∈ XA).

The formulas of the form seqn(�w, u) have depth of quantifier alternations atmost two, where the outer quantifier is existential. It follows that the depthof quantifier alternations of the α is at most three. Hence the quantifieralternation depth of our equivalent of A is four. �

Theorem 2.11. Suppose U is RE and sequential and V is an RE pair theory.We have: U �bo V iff U � FOC(V ).

Proof. From left to right. Suppose our interpretation is K with witnessingbound n. We choose a set of numbers M on U satisfying S1

2, such that wehave a satisfaction predicate S for formulas of complexity n. We use that todefine our classes in the obvious way.

From right to left. Suppose U � FOC(V ). We have U � FOC(V ) �bo V , and,hence, U �bo V . �


We note that FOC(PAIR) � VS � R, where VS is Vaught Set Theory4 andwhere R is the Robinson-Mostowski-Tarski theory R. So, FOC(PAIR) is es-sentially undecidable. We know that PAIR has extensions U with quantifierelimination. See e.g. [4] and [2]. For such extensions we have: U �bo U . Sowe have U �bo U , but U �� FOC(U) (since U is decidable and FOC(U) isessentially undecidable). This example leaves the matter of pair sentencesuntouched. I do not know whether there are finitely axiomatizable pairtheories with quantifier elimination.

Open Question 2.12. Are there finitely axiomatizable pair theories withquantifier elimination?

3. A Version of the Second Incompleteness Theorem

Harvey Friedman’s idea is that the defining characteristic of a consistencystatement is the fact that a Feferman-Henkin-style interpretation can bebased on it. In our version we zoom in on the fact that such interpretationsare bounded.

Definition 3.1. A sequential sentence A is a pro-consistency statement forU iff A �bo U .

We note that ordinary consistency statements which we identify with S12 +

con(U) are pro-consistency statements. The second incompleteness theoremfor pro-consistency statements is now a triviality.

Theorem 3.2. Suppose U is an RE theory. Then U does not interpret anypro-consistency statement for U .

Proof. Suppose A is a pro-consistency statement for U . Suppose U � A.Then A �bo U � A, and hence A �bo A, quod non. Hence U �� A. �

Suppose that A is a pro-consistency statement for U and B is sequentialwith B � A. Then, clearly, B is a pro-consistency statement for U . Thissuggests that we may consider pro-consistency statements as sequential sen-tences modulo mutual interpretability, or, i.o.w., as degrees of interpretabil-ity. The weakest finitely axiomatized sequential theory is, modulo mutualinterpretability, S1

2.

Example 3.3. Let INF be the theory in the language of pure equality with,for every n, an axiom stating that there are at least n elements. Since this

4See [10, 13]. In the appendix we prove that FOC(PAIR) � VS.

410 A. Visser

theory has quantifier elimination, we have INF�bo INF and, hence, S12�bo INF.

So S12 is a pro-consistency statement for INF

In fact S12 interprets the ordinary consistency statement for INF, to wit

S12 + con(INF). This can be seen as follows. By a result of Wilkie and Paris

(see [16]) it is sufficient to show that EA � con(INF). Using the constructionfrom [8], we can build a Δ0-truth predicate for the theory of a number inEA. Reason in EA. Suppose that we have a INF-proof p of ⊥. Then, theaxioms employed in p are satisfied in some number n ≤ p. Using the Δ0-truth predicate, we can prove by induction on the sub-proofs of p, that everysub-conclusion of p is true in n. A contradiction.

Reflection on the above argument shows that a small adaptation showsa bit more, to wit: S1

2 �(S12 +con(R)), for the theory R introduced by Tarski,

Mostowski and Robinson.

Open Question 3.4. Do we have S12 � PresA, where PresA is Presburger

Arithmetic? Do we have S12 � (S1

2 + con(PresA))? I think we should have yesto both questions.

Do we have S12 �PAIR? Do we have S1

2 �(S12 +con(PAIR))? I think we should

have no to both questions.

Pro-consistency statements can be both stronger and weaker than consis-tency statements. In the next section we study an example of a weaker typeof statement: the super-logarithmically bounded consistency statements.

4. Super-logarithmically Bounded Consistency

In this section, we study an interesting pro-consistency statement that isweaker than the ordinary consistency statement. We define:

itexp(m, 0) := m, itexp(m, n + 1) := 2itexp(m,n).

supexp(n) := itexp(1, n).

We note that we can write down Δ01-graphs for both functions5 so that

there is no problem coding them. We can do this in such a way that theobvious properties for the functions can be verified in S1

2. E.g., we haveitexp(itexp(x, y), z) is defined iff itexp(x, y + z) is defined, and, if both aredefined, they are equal.

Definition 4.1. We define the super-logarithmically bounded consistency ofU , or slb-consistency, by: slb-con(U) :↔ ∀x ( supexp(x)↓ → conx(U) ).

5See e.g. Chapter V of [7] for the basic ideas for constructing such definitions.


We note that slb-con(U) is Π01.

Theorem 4.2. Let U be any RE theory. Then S12 + slb-con(U) is a pro-

consistency statement for U .

Proof. It is easy to see that (S12 + slb-con(U)) � �(U) and, hence, we find

that (S12 + slb-con(U)) �bo U . �

Theorem 4.3. Let A be a consistent sequential sentence. Then, we have:(S1

2 + slb-con(A)) �� (S12 + con(A)).

Proof. Suppose (S12 + slb-con(A)) � (S1

2 + con(A)). Then, by a theorem ofWilkie & Paris ([16]), we have: ( ) (EA + slb-con(A)) � con(A). We nowreason as follows. (Each of the steps is explained below.)

EA + conn(A) � EA + slb-con(A) (1)� EA + con(A) (2)� EA + con(S1

2 + conn(A)) (3)� EA + conk(EA + conn(A)) (4)� S1

2 + con(EA + conn(A)) (5)

Here n and k are supposed to be larger than the quantifier alternation com-plexities of A, resp. EA = conn(A). Step (1) is by Lemma 4.1 below. Step(2) is by ( ). Step (3) is because A � (S1

2 + conn(A)), for sufficiently large n,since A is sequential. Step (4) is by Lemma 4.2 below. Step (5) is as follows.Let J be the superexponential cut of EA. We find by cut-elimination:

EA + conk(EA + conn(A)) � (S12 + con(EA + conn(A)))J .

Composing all steps, we find (EA + conn(A)) � (S12 + con(EA + conn(A))).

This is impossible by the Second Incompleteness Theorem. �

It would be nice to have a less elaborate and more direct proof. Here is ourfirst promised lemma.

Lemma 4.1. We have: (EA + conn(A)) � (EA + slb-con(A)).

Proof. We can construct an EA-cut J such that:

EA � ∀x∈J ∀y itexp(y, x)↓.We work in EA. Consider the class N of all x such that x ≤ supexp(z), forsome z ∈ J . Our interpretation is relativization to N . It is easy to see thatwe have EA in N .

412 A. Visser

Consider any x in N . Suppose that supexp(x) exists in N . It followsthat supexp(x) ≤ supexp(u), for some u ∈ J , and, hence, that x ∈ J .

Let z be in N . We claim that itexp(z, x) exists and is in N . For some y inJ , we have z ≤ supexp(y). We note that itexp(z, x) ≤ itexp(supexp(y), x) =supexp(x + y) (if defined). Since both x and y are in J , we find that x + yis in J . Hence supexp(x + y) exists and is in N . Ergo, itexp(z, x) is in N .

It follows that ∀x∈N (supexp(x)↓ → ∀y itexp(y, x)↓ )N . Using this, wecan, in N , apply cut-elimination of x proofs, whenever supexp(x) exists.Thus, inside N , we may conclude, from conn(A), to conx(A), wheneversupexp(x) exists. �

Here is our second promised lemma.

Lemma 4.2. Consider a sentence ∃x∀y Axy, where A is Δ0. We have:

(†) EA � conn(EA + ∃x∀y Axy) ↔ ∀I∈S12-cuts con(S1

2 + ∃x∈I ∀y Axy).

Proof. We note that both sides of the equivalence in ( ) can be written asΠ0

1-sentences. By the result of Wilkie & Paris ([16]), ( ) is equivalent to:

(‡) (S12+conn(EA+∃x∀y Axy)) ≡ (S1

2+∀I∈S12-cuts con(S1

2+∃x∈I ∀y Axy)).

Now ( ) in its turn is equivalent to:

($) (EA + ∃x∀y Axy) ≡ (S12 + ∀I∈S1

2-cuts con(S12 + ∃x∈I ∀y Axy)).

Finally, ( ) is Lemma 4.1 of [11]. �

We have shown, in Lemma 4.1, that (EA + conn(A)) � (EA + slb-con(A)).The next theorem shows that we can also move in the other direction andinterpret EA + conn(A) + slb-incon(A) in EA + conn(A).

Theorem 4.4. Suppose A is sequential. We have, for sufficiently large n:

(EA + conn(A)) � (EA + conn(A) + slb-incon(A)).

Proof. We have, by Lemma 4.2, for sufficiently large k,

EA � �EA,k(conn(A) → slb-con(A)) → ∃I∈S12-cuts

�S12(conn(A) → slb-conI(A)).

It follows that:

EA � �EA,k(conn(A) → slb-con(A)) → A � (S12 + slb-con(A)).


By the Second Incompleteness Theorem for superlogarithmetically boundedconsistency, we find:

EA � �EA,k(conn(A) → slb-con(A)) → �A⊥.

So, by the Wilkie-Paris result, for some S12-cut J , we have:

(♥) S12 � �J

EA,k(conn(A) → slb-con(A)) → �A⊥.

Let J be the superexponential cut. We find EA+conn(A) � conJ (A). Thus,by (♥), we may conclude that:

EA + conn(A) � conJJk (EA + conn(A) + slb-incon(A)).

It follows that: (EA + conn(A)) � (EA + conn(A) + slb-incon(A)). �

Open Question 4.5. Consider the proof predicate:

�slbS12A :↔ ∃x (supexp(x)↓ ∧ �S1

2,xA).

It is easy to see that �slb satisfies Lob’s Logic over S12.

Is Lob’s Logic complete for provability interpretations for �slb over S12.

(The answer for this question is open for ordinary S12-provability. See [1].)

What is the closed fragment of the provability logic of S12 for �slb with a

constant for Exp? (See [11], for an analogous case for ordinary provability.)

5. Orey Sentences

Consider any theory U . A sentence A is an Orey-sentence for U iff we haveboth U �(U +A) and U �(U +¬A). Both V = L and CH are Orey-sentencesof ZF. The Axiom of Foundation Fo is an Orey-sentence of ZF minus Fo. Forarithmetic, it is much more difficult to find examples of Orey-sentences. Theonly known examples in the case of PA are metamathematical artifacts. Thissuggests to look for analogues of the Paris-Harrington Theorem for indepen-dent sentences that are very ‘close’ to PA. If we consider weak theories wedo have a mathematically natural example. We remind the reader that thefunction ω2(x) := 22(log(log(x)))2

and that Ω2 is the statement that ω2 is total.

Theorem 5.1. The sentence Ω2 is an Orey-sentence of S12

We only give a proof sketch.

414 A. Visser

Proof. It is well known that S12 interprets Ω2 by Solovay’s method of short-

ening cuts (see e.g. [16]). In the other direction, we note that S12 interprets

the theory S12 + conk(S1

2 + Ω2), for a sufficiently large k. Let

U := S12 + Ω2 + {ω2(c) > ωn

1 (c) | n ∈ ω}.By a compactness argument, we find that S1

2 + conk(S12 + Ω2) � conk(U).

Using this last consistency statement we find an interpretation H : S12 � U .

Let J be the common cut of ID and H. We define N as those y in δH suchthat, for some j ∈ J , y ≤ ωj

1(c). As is easily seen, restricting δH to N givesus an interpretation of S1

2 + ¬Ω1 in S12. �

In a similar way, Ω1 is an Orey-sentence over IΔ0. The materials in Section 4yield an Orey-sentence of EA that is rather natural. First, we have thatEA� (EA+ slb-con(S1

2)), by Theorem 4.1, combined with () EA � conn(S12),

which is a consequence of a result of Wilkie & Paris in [16]. Secondly, wehave that EA � (EA + ¬ slb-con(S1

2)), by Theorem 4.4, combined with ().Thus, we have proved:

Theorem 5.2. slb-con(S12) is an Orey-sentence for EA.

The example of slb-con(S12) leads to another Orey-sentence. Reflecting on

the proof of Lemma 4.1, we find:

Theorem 5.3. ∀x (supexp(x)↓ → ∀y itexp(y, x)↓ ) is an Orey-sentence of EA.

Proof. Let C := ∀x (supexp(x)↓ → ∀y itexp(y, x)↓ ). The proof of Lemma4.1 tells us that EA � (EA + C). In the other direction, we note that:

EA + C � slb-con(S12).

By Theorem 5.2, EA � (EA + ¬ slb-con(S12)). Ergo, EA � (EA + ¬C). �

If we define suplog(x) as the largest y such that supexp(y) exists and issmaller or equal to x, then we can rewrite the formula

∀x (supexp(x)↓ → ∀y itexp(y, x)↓ ),

modulo EA-provable equivalence, as ∀x∀y itexp(suplog(x), y)↓. Thus, thefunction f(x, y) := itexp(suplog(x), y) is an example of a function that is notprovably total in EA, where EA � (EA + tot(f)) and EA � (EA + ¬ tot(f)).It seems that f if for EA, what ω1, or more precisely Buss’ smash function #,is for IΔ0. The relationship between functions like f and # has been ana-lyzed in great generality by Alex Wilkie in his beautiful paper [15]. We hope


to analyze, in a later paper, the relationship between consistency statementsand the functions of the Wilkie Hierarchy, that was introduced by Wilkiein [15].

We end this section by comparing conn(S12), slb-con(S1

2) en con(S12). It is

pleasing to see how these three consistency-like statements differ in behaviorover EA. We have:

EA � conn(S12), hence:

EA � (EA + conn(S12)) and EA �� (EA + inconn(S1

2));

EA � (EA + slb-con(S12)) and EA � (EA + slb-incon(S1

2));

EA �� (EA + con(S12)) and EA � (EA + incon(S1

2)), by the results of [11].

In a schema:

A EA � (EA + A) EA � (EA + ¬A)

conn(S12) + −

slb-con(S12) + +

con(S12) − +

6. Postscriptum

In this paper we have introduced the notion of pro-consistency statement asone way of formulating a coding-free version of the Second IncompletenessTheorem. The aim of providing such coding-free versions is not just thatthey avoid the unsatisfactory fact that the usual formulations of the SecondIncompleteness Theorem contain an unanalyzed quantification over ‘goodcodings’. The aim lies also in the hope that such coding-free versions willcreate a new playground for finding new generalizations. For example, weposed the question of generalization of the theorem to pair-theories.

We did provide one specific example slb-con(U) of a pro-consistencystatement that is weaker than an ordinary consistency statement. We thinkthat this is a convincing example of a sentence falling under the Second In-completeness Theorem that is not an ordinary consistency statement. Theexample points to interesting further questions like the connection with theWilkie Hierarchy. Moreover, the example leads to a different example of anatural Orey sentence over EA: to wit the statement of the totality of thefunction λx, y · itexp(suplog(x), y). We feel that such examples take Kreisel’schallenge to find natural examples of independent sentences in a new direc-tion, to wit: finding examples of natural sentences that are very close to

416 A. Visser

the theories they are independent of. In the case of set theories, we can findsuch examples in abundance, like the continuum hypothesis. However, forarithmetical theories, we have far less examples. Such a natural example isunknown for Peano Arithmetic.

Acknowledgements. I thank Vincent van Oostrom for enlightening con-versations and for his gracious permission to present his recursive charac-terization of Philipp Gerhardy’s complexity measure. I am grateful to theanonymous referee for his/her suggestions and corrections.

References

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A. Pairing and Vaught Set Theory

We prove that FOC(PAIR)�VS. Vaught Set Theory VS is defined as follows.

VS For each n ∈ ω, we have:� ∀x0, . . . , xn−1 ∃y ∀u (u ∈ y ↔ ∨

i<n u = xi).

The idea is that we code a finite set {a0, . . . , an} as 〈a0, 〈a1, . . . 〈an−1, b〉〉〉,where b is a non-pair. Of course, we do not have extensional pairing, so theabove representation is only heuristic. We define in FOC(PAIR):

DC(X) :↔ ∀z, u, v ((z ∈ X ∧ pair(u, v, z)) → v ∈ Z),

x ∈ y :↔ ∃z (∀X ((y ∈ X ∧ DC(X)) → z ∈ X) ∧ ∃v pair(x, v, z)).

Note that there could be y that are not in any X with DC(X). These wouldrepresent the universe.

To represent the empty set, we simply take y to be a non-pair. We have aminimal X with y ∈ X and DC(X), to wit: {y}. Note that {y} correspondsto the formula x = y. Suppose we have produced an y′ that represents theset containing x0, . . . , xn−1, with a mimimal class X ′ such that y′ ∈ X ′ andDC(X ′), where X ′ is defined by the formula A′. Then we take y such thatpair(xn, y′, y), and X := X ′ ∪ {y}, and A :↔ x = y ∨ A′. It is easy to seethat y represents the set consisting of x0, . . . , xn.

We note that this argument also works when we replace First Order Com-prehension by the much weaker analogue for classes of Vaught’s axioms:

VC For each n ∈ ω, we have:� ∀x0, . . . , xn−1 ∃Y ∀u (u ∈ Y ↔ ∨

i<n u = xi).

Moreover, if we replace First Order Comprehension by Adjunctive ClassTheory, with the axioms that an empty class exists and that we may adjoinany element to any class, we can use a variant of the above argument tointerpret adjunctive set theory.

418 A. Visser

Albert VisserDepartment of PhilosophyUtrecht UniversityJanskerkhof 13A,3512BL Utrecht, The [email protected]

The Second Incompleteness Theorem and Bounded Interpretations

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