Faith & falsity

Annals of Pure and Applied Logic 131 (2005) 103–131

ANNALS OFPURE ANDAPPLIED LOGIC

www.elsevier.com/locate/apal

Faith & falsity

Albert Visser∗Department of Philosophy, Utrecht University, Heidelberglaan 8, 3584 CS Utrecht, The Netherlands

Received 1 November 2002; received in revised form 19 March 2004; accepted 28 April 2004

Available online 24 July 2004

Communicated by S.N. Artemov

Abstract

A theory T is trustworthy iff, whenever a theoryU is interpretable inT , then it is faithfullyinterpretable. In this paper we give a characterization of trustworthiness. We provide a simple proofof Friedman’s Theorem that finitely axiomatized, sequential, consistent theories are trustworthy. Weprovide an example of a theory whose schematic predicate logic is completeΠ 0

2 .© 2004 Elsevier B.V. All rights reserved.

MSC:03B52; 03F25; 03F30; 03F45; 03H13

Keywords:Rosser arguments; Faithful interpretations; Sequential theories;Σ -soundness

Motto: Quis comparabit comparatores ipsos?

1. Introduction

We begin with a definition.

Definition 1.1. A theory T is trustworthyif every U interpretable inT is also faithfullyinterpretable inT . Thus, our trustworthiness is the trustworthiness of someone who is, whenever he is able totell a story at all, able to tell it without false embellishments. Trustworthiness is a peculiarnotion that has nothing to do with strength. It has to do with the constraint a theory putson the available linguistic resources. InSection 6we will probe deeper into the true andproper nature of trustworthiness.

∗ Tel.: +31-30-253-21-73; fax: +31-30-253-28-16.E-mail address:[email protected] (A. Visser).

0168-0072/$ - see front matter © 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.apal.2004.04.008

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104 A. Visser / Annals of Pure and Applied Logic 131 (2005) 103–131

This paper is a study of trustworthiness. We aim to show that the notion oftrustworthiness is interesting both in its own right and by its connection to other notions.

1.1. Contents of the paper

Three central results form the core of the paper. The first is a characterization oftrustworthiness. This characterization is provided inSection 5.

As the second central result, we will reprove Friedman’s Theorem concerningtrustworthiness. The theorem is reported in [12, Theorem 3, on p. 224]. The theoremstates that finitely axiomatized, adequate (sequential)1, consistent theories are trustworthy.The proof of the result is provided inSection 5. Friedmans’ Theorem will be proved as aconsequence of our characterization and of a theorem that is proved inSection 4. In fact,the results ofSection 4make a strengthening of Friedman’s result possible.

Our third central result is the description of trustworthiness in terms of an adjunctionbetween the preorder of faithful interpretability and the preorder of interpretability. Thisresult is proved inSection 6.

An important method used in the paper is the use of the FGH Theorem, whichapproximately says that we can prove the following principle in Elementary Arithmetic.Let T be a theory into which a suitable fragment of Arithmetic can be interpreted. Then,for anyΣ0

1 -sentenceS, there is aΣ01 -sentenceR, such that(S∨ incon(T)) is equivalent to

T R. I.o.w., if T is consistent thenS is equivalent to aT-provability statement. Since theFGH Theorem plays such an important role, I devoteSection 3to an extensive discussionof it and its applications.

A side result with some independent interest is contained inAppendix C. We give anexample of a theory whose schematic logic is completeΠ 0

2 .

1.2. Prerequisites

Most of what is needed to understand the paper is contained in the textbook [4].

1.3. History of the paper

The present paper is a sequel of [19]. In that work a somewhat sharper version ofTheorem 4.1of the present paper was proved. The present proof is, however, considerablysimpler. The article [19], was the result of reflecting on the paper [6]. In that paper Kraj´ıcekstudies ViteslavSvejdar’s question “When is it consistent for inconsistency proofs to liebetween cuts?”. In other words, for which theoriesT and for whichT-cuts I and J isthe theoryT + conJ(T) + inconI (T) consistent? Kraj´ıcek proves that, for every finitelyaxiomatized, sequential and consistent theoryT , and for everyT-cut I , we can find aT-cutJ such thatSvejdar’s question has a positive answer forT , I andJ.

Neither Krajıcek nor I noted that Kraj´ıcek’s Theorem is an immediate consequenceof Friedman’s Theorem on trustworthiness.2 I only realized this recently after HarveyFriedman reminded me of his result in e-mail correspondence. It turns out that, in the other

1 We will usesequentialinstead ofadequatein this paper.2 SeeRemark 5.7of the present paper.

A. Visser / Annals of Pure and Applied Logic 131 (2005) 103–131 105

direction, the methods of [19] yield a proof of Friedman’s Theorem. This paper reports thisproof.

2. Arithmetization

In this section we introduce some basic notions and conventions.

2.1. Theories and interpretations

Theories in this paper are theories of first order predicate logic. Unless stated otherwise,we will assume that theories have an axiom set that is p-time decidable. Interpretationsbetween theories are relative interpretations. For a description of the notion of relativeinterpretation, see the classical [15], or e.g. [20]. We write:

• K : T U , for: K is an interpretation ofU in T .• T U , for ∃K K:T U .

We will be interested in theories in which a sufficiently large fragment of arithmetic isrelatively interpretable. Let us fix a weak, finitely axiomatized, arithmetical theoryF. Thistheory has as language, the arithmetical language with 0, S,+,×,≤. The theoryF isaxiomatized by Robinson’s ArithmeticQ plus axioms that≤, is linear, plus the axiomx ≤ Sy ↔ (x ≤ y∨ x = Sy).3 We useF instead ofQ, because it is pleasant to have someimportant properties of the Rosser ordering in one’s simplest theory.

The theoryF is interpretable inQ on a definable initial segmentI . See [4, pp. 366–371].We comment on some details in ourAppendix D.

To numerizea theoryT is to specify an interpretationN such thatN : T F. Thus,a theoryT is numerizableif T F. We will also need the notion ofnumerized theory. Anumerized theoryT is a pair〈T,N 〉, whereN : T F. The numerized theory〈T,N 〉 isa numerizationof the numerizable theoryT . In the context of numerized theoriesT , thevariablesx, y, z, . . . will range over the numbers provided byN . Thus, e.g.∀x . . . willmean∀x (δN (x)→ . . .). We will useξ, η, . . . for general variables. We will writeT + Afor 〈T + A,N 〉, etc.

We will be sloppy betweennumerizableand numerizedin the case of ‘explicitlyarithmetical’ theories, likePA. Officially, PA is a numerizable theory. However, we willconfuse it with the numerized theory〈PA, id〉, whereid is the identity interpretation.

We will fix an arithmetization of metamathematical notions in the language ofF. Thearithmetization is supposed to be efficient so that we can verify all relevant facts in Buss’S1

2. See e.g. [1] or [4].4 We will write U A (U A), for provU (#A) (provU (#A)). Theuse ofU (U ) will be only meaningful inside a numerized theoryT = 〈T,N 〉. Theformalization of an outer will always be in the designated numbers given byN . SoU A will be a different formula inside〈T,N 〉 than inside〈T,K〉, if N andK are different.

3 Our version of Robinson’s Arithmetic has≤ as an atomic symbol and includes the axiomy ≤ x↔ ∃z z+ y = x. SeeAppendix D.

4 As is well known, we can replaceS12 by a variant in the arithmetical language. We assume we are working

with this variant.


Boxes inside boxes will take their numerization from the numerized theory correspondingto the first box above in the parse tree. InAppendix Athis convention is made precise. Theconvention is best illustrated by some examples.

Example 2.1. SupposeT = 〈T,N 〉 andU = 〈T,K〉 are numerized theories.

• ‘T ∀ξ ∃y Q(ξ, y)’, whereQ is an atomic predicate, means:T ∀ξ ∃η (δN (η) ∧ Q(ξ, η)).

• ‘T U ∀ξ ∃y R(ξ, y)’, whereR is an atomic predicate, means:

T NU ∀ξ ∃η (δK(η) ∧ R(ξ, η)).

• ‘ T U A → U B’ is meaningless. There is nothing to tell us from which set ofnumbers to take the witnesses forU .

• ‘ T U A → U B’ is meaningless. The witnesses for outer’s must come from thenumerization ofT .

• ‘T U A → U B’ means:T NU A → N

U B (for A and B without boxes ornumerical variables).

• ‘T U A → U U A’ is meaningless. Where could the witnesses for the lastU

come from?• ‘T U A → UU A’ means:T N

U A → NU K

U A (for A without boxes ornumerical variables).

Schematic lettersA, B, range over the expanded language with boxes and two kinds ofvariables or over the original language. Schematic letters forΣ0

1 -formulas receive the sametreatment as boxed formulas: they range overΣ0

1 -formulas relativized to the stipulatednumbers.

Free variables in a formula inside a will be treated according to the usual conventionso that they are still free in the resulting formula. Thus,A(x) inside a box will really standfor a term that defines the following function: we map the numbern to Godelnumber ofthe result of substituting the (binary) numeraln of n for x in A.5

There are various orderings for interpretations ofF in a numerizable theoryT . The onethat is relevant for us is given as follows.

• E : K ≤T N iff E is aT-formula whichT-provably gives an initial embedding of theK-numbers into theN -numbers. We omit the subscript if the theory is clear from thecontext.

We give the clauses forE. To increase readability we usePlus for + andTimes for×.

(1) T ∀ξ ∀η (E(ξ, η)→ (δK(ξ) ∧ δN (η))),(2) T ∀ξ (δK(ξ) → ∃η (δN (η) ∧ E(ξ, η))),(3) T ∀ξ ∀η ((E(ξ, η) ∧ η′ ≤N η)→ ∃ξ ′ (E(ξ ′, η′) ∧ ξ ′ ≤K ξ)),(4) T ∀ξ ∀ξ ′ ∀η ∀η′ ((E(ξ, η) ∧ E(ξ ′, η′) ∧ ξ =K ξ ′)→ η =N η′),(5) T ∀ξ ∀ξ ′ ∀η ∀η′ ((E(ξ, η) ∧ E(ξ ′, η′) ∧ SK(ξ, ξ ′))→ SN (η, η′)),

5 The ‘term’ mentioned here need not be really a term, but can be given as a suitable formula of which thetheory proves that it behaves in the desired way.


(6) T ∀ξ ∀ξ ′ ∀ξ ′′ ∀η ∀η′ ∀η′′ ((E(ξ, η) ∧ E(ξ ′, η′) ∧ E(ξ ′′, η′′) ∧PlusK(ξ, ξ ′, ξ ′′))→ PlusN (η, η′, η′′)),

(7) T ∀ξ ∀ξ ′ ∀ξ ′′ ∀η ∀η′ ∀η′′ ((E(ξ, η) ∧ E(ξ ′, η′) ∧ E(ξ ′′, η′′) ∧TimesK(ξ, ξ ′, ξ ′′))→ TimesN (η, η′, η′′)).

Any provably initial embeddingE : K → N can be split into two parts:E0 : K → I ,andemb : I → N . HereE0 is a provable isomorphism andI is an initial segment of theN -numbers, satisfyingF. The embeddingemb is the identical embedding ofI intoN . Wewill call such an initial segment ofN satisfyingF a T -cut ofN . If we are considering anumerized theoryT = 〈T,N 〉, then we will call aT-cut ofN simplya T -cut.

A sequential theory is a theory with a good notion of sequence for all objects of thedomain of the theory. This notion is due to Pavel Pudl´ak. See e.g. [10], or [4, p. 151]. Thenotion ofsequential theoryis equivalent to Harvey Friedman’s notion ofadequate theory.See [12]. A sequential theory is always numerizable. Here are a few facts about≤ and cuts.

Fact 2.2. Consider a numerizable theory T . The variablesK, M, N will range overinterpretations ofF.

(1) For any numerized theoryT = 〈T,N 〉, there is aT -cut I , such that, for standardk, T ∀x∈I ∃y itexp(x, k) = y. Here,itexp(x, 0) := x and itexp(x, m+ 1) :=2itexp(x,m). This theorem is due to Robert Solovay (in an unpublished manuscript “OnInterpretability in Set Theories”). Later a sharper version was proved in [10]:S1

2 ∀z∃I T ∀x∈I ∃y itexp(x, |z|) = y. Here|n| = entier(2 log(n)). Thus|n| isthe binary length of n.

(2) For any numerized theoryT = 〈T,N 〉, there is aT -cut I , such that I: T (I ∆0 +Ω1). Since I is a cut,Π 0

1 -sentences are downwards preserved fromN to I andΣ01 -

sentences are upwards preserved from I toN . This theorem is due to Alex Wilkie.See [4, pp. 366–369]. See also our remarks inAppendixD.

(3) Suppose that T is sequential. Then, for allM, N , there is aK with K ≤ M andK ≤ N . This theorem is due to Pavel Pudlak [10].6 Note that, by(2), we can alwaysassume thatK : T I ∆0 + Ω1.

(4) Suppose I is aT -cut. Then we have:S12 ∀x T x ∈ I . This theorem is theobis-

principle. It shows that numbers that arebig outsideare alwayssmall inside. The resultis proved e.g. in [22].

Remark 2.3. Consider a numerized theoryT = 〈T,N 〉. Let I , J range overT -cuts. Wecan assign an invariant toT as follows:

li(T ) := A | ∃I ∀J≤I T AJ.Hereli stands for ‘limes inferior’. It is easily seen that, ifT is consistent, thenli(T ) is alsoconsistent. One can show thatli(T ) extends the theoryI ∆0 + BΣ0

1 + conn(F) | n ∈ ω.HereBΣ0

1 is theΣ01 -collection principle:

∀x≤a∃y S0(y)→ ∃b∀x≤a∃y≤b S0(y),

6 Our statement is not precisely Pudl´ak’s, who considers a numerized theory and takesK to be a cut of thedesignated numbers. The two statements are easily seen to be equivalent.


where S0 ∈ ∆0. The formulaconn stands for consistency w.r.t.n-provability. (SeeSection 4, for further explanation.)

In caseT is sequential, byFact 2.2(3), li(T ) will be independent of the numerizationTof T . Thus, we may writeli(T), whenT is sequential. For sequential theoriesT andU , wefind the following.

(1) li(T) extendsI ∆0 + BΣ01 + conn(T) | n ∈ ω.

(2) If li(T) ⊆ li(U), thenT is locally interpretable inU .(3) If T is finitely axiomatized and consistent, thenli(T) is Σ0

1 -sound. (This follows fromTheorem 4.1.)

Open Question 2.4. Remark 2.3suggests the following questions. What are the possiblecomplexities of theli’s? Do we have, for sequentialT andU , that if T is interpretable inU , thenli(T) ⊆ li(U)?

2.2. Preliminaries to Rosser arguments

SupposeA = ∃x A0(x) andB = ∃x B0(x). HereA0 andB0 are arbitrary formulas ofthe language of some numerized theoryT = 〈T,N 〉. Remember thatx andy range overtheN -numbers. We write:

• A ≤ B :↔ ∃x (A0(x) ∧ ∀y < x ¬ B0(y)),• A < B :↔ ∃x (A0(x) ∧ ∀y ≤ x ¬ B0(y)),• If C = (A ≤ B), we write C⊥ for (B < A). If D = (A < B), we write D⊥ for

(B ≤ A).

Formulas of the formA ≤ B and A < B are calledwitness comparison formulas. Thewitness comparison notation was introduced by David Guaspari around 1976. They werefirst systematically studied in [3].

We present some facts about witness comparison formulas.

Fact 2.5. We have:

(1) T A ≤ B → A.(2) T A < B → A ≤ B.(3) T A ≤ B → ¬ (B < A).(4) T (A ≤ B ∧ B ≤ C)→ A ≤ C.(5) T A ≤ A→ (A ≤ B ∨ B < A).(6) T (A∧ ¬ B)→ A < B.(7) T ((A→ A ≤ A) ∧ B)→ (A ≤ B ∨ B < A).(8) T (A < B ∨ B ≤ A)↔ (A ≤ B ∨ B < A).

Proof. We prove (5). Reason inT . SupposeA ≤ A. This tells us thatx | A0(x) has asmallest element, sayx0. We have∀y<x0 ¬ B0(y) or ∃y<x0 B0(y). In the first case, wefind A ≤ B, in the second,B < A.

In I ∆0, we can prove the∆0-minimum principle. So,I ∆0 S→ S≤ S, for S∈ ∃∆0. Infact,∆0-induction is equivalent to this principle, assuming we allow free parameters inS.


Similarly, Buss’ theoryT12 proves theΣb

1 -minimum principle.7 So, T12 S → S ≤ S,

for S ∈ ∃Σb1 . In fact, Σb

1 -IND is equivalent to this principle, assuming we allow freeparameters inS. (See [1, p. 61, Theorem 24].) Thus, we can draw the following corollaryfrom Fact 2.5(5, 7).

Corollary 2.6. Let S be∃∆0 [∃Σb1 ]. Suppose thatN : T I ∆0 [N : T T1

2]. Then,T (S∨ A)→ (S≤ A∨ A < S).

Note that it follows, from the conclusion ofCorollary 2.6, by substitutingS for A, thatT S→ S ≤ S, which expresses the∆0-minimum principle [Σb

1 -minimum principle],and hence∆0-induction [Σb

1 -IND]. If, in the Σb1 -case, we could prove our corollary using

S12, it would follow thatT1

2 = S12, deciding an open problem. However, we can prove a

related fact forS12, which is sufficient for some important applications.

Fact 2.7. Let T be a numerized theory. Let := T . Suppose that S is∃Σb1 and that

A = ∃x A0(x). We haveS12 S→ (S≤ S), and, hence,S1

2 S→ (S≤ A∨A < S).

Proof. Reason inS12. SupposeS. By Σb

1 -completeness, we findS0(x), for somex.By the obis-principle, we findSI , for any T -definable cutI . By Fact 2.2(2), we canpick I such that it satisfiesI ∆0 + Ω1.8 It follows that in (S ≤ S)I and, thus,(S≤ S).

The following fact is, modulo some insignificant differences, verified in [16].

Fact 2.8 (Small Reflection Principle).LetT = 〈T,N 〉 be a sequential numerized theory.Suppose that T is either finitely axiomatized or an extension by finitely many axioms ofI ∆0 + Ω1 (relativized toN ). Let := T . Let S be∃Σb

1 . Let A be any sentence in thelanguage of T . We have:

S12 S→ (A ≤ S→ A).

We finish this section by providing a verification of Rosser’s Theorem inS12 for theories

finitely axiomatized over eitherS12 or I ∆0+Ω1. The idea of this argument is due to Viteslav

Svejdar [14].

Theorem 2.9 (Fast Rosser Theorem).Let T be a sequential numerized theory. Supposethat T is either finitely axiomatized or an extension by finitely many axioms of I∆0 + Ω1(relativized toN ). Let := T . Let R be such thatS1

2 R ↔ ¬R ≤ R. We have:S1

2 (R∨¬R) → ⊥.

Proof. Reason inS12. Suppose (a)R. By Fact 2.8, we have (b)((¬R≤ R) → ¬R).

By Fact 2.7, we have (c)((¬R ≤ R) ∨ (R < ¬R)). Combining (b) and (c), wefind: (d) ¬R. Combining (a) and (d), we get⊥. The proof from the assumption¬Ris similar.

7 For a description ofT12, see [1] or [4].

8 In fact, we need only a sufficiently large finite fragment ofI ∆0 +Ω1 here.


Note that it follows, by Buss’s results, that there is a p-time transformation of a proof ofRto a proof of⊥, and, similarly, for proofs of¬R.

Open Question 2.10. The restriction on the theories ofTheorem 2.9 is somewhatunsatisfactory. So one might ask whether the theorem also holds for non-sequential theoriesor for sequential theories that are not either finitely axiomatized or finitely axiomatized asextensions ofI ∆0+ Ω1.

It is well known that, ifS12 did prove “NP=co-NP”, then the usual formalization of

Rosser’s Theorem would work. Thus, a negative answer to our question would entail:S1

2 NP=co-NP.

3. A miraculous argument

Sometimes, in mathematics, we meet an argument that is utterly simple, and, yet, hasmany surprising consequences. The reasoning leading to the FGH Theorem surely qualifiesas an example of such an argument. It is a Rosser type argument and, thus, it inherits theinherent mystery of such arguments. It is a simple Rosser type argument, not much morecomplicated in terms of number of steps than Rosser’s original argument, even simpler interms of the definition of the fixed point. However, the formalization of the FGH Theoremseemsto ask for more resources than the formalization of Rosser’s, as will be explainedbelow.

3.1. The FGH Theorem

Let us first state the FGH Theorem. LetEA be Elementary Arithmetic, i.e.I ∆0+ exp.This theory is calledEFA in [12].

Theorem 3.1 (FGH). Consider any numerized theoryT = 〈T,N 〉. Let := T . Let SbeΣ0

1 and let R be such thatQ R↔ S≤ R. We have:

EA (S∨⊥) ↔ (R∨ ⊥)

↔ R

or, equivalently,EA+ con(T) (S↔ R) ∧ (S↔ R).

‘FGH’ stands for Friedman–Goldfarb–Harrington. The history is as follows. Around 1976or very early 1977, Harrington proved a principle very close to the FGH principle. Themain difference was that Harrington’s sentenceR wasΠ 0

1 and notΣ01 . Harvey Friedman

saw Harrington’s result and realized that one can also get the result forR in Σ01 . He wrote

down his result in a manuscript “Proof Theoretic Degrees”, dated February 1977. An earlypaper reporting the result is [11, p. 366]. Smory´nski refers to Friedman’s unpublishedmanuscript.

Warren Goldfarb rediscovered the principle independently in November 1980. Hecommunicated the result to George Boolos. Subsequently, Boolos promulgated it to thelogic of provability community. Via this channel I learned of it. So I called itGoldfarb’sPrinciple. I guess everyone gets due credit in my new name for it:The FGH Theorem. Hereis the proof.


Proof. Reason inEA.

Step 1. SupposeS∨⊥. We want to deriveR∨⊥. If we have⊥, we are done. SupposeS. It follows that R∨ R⊥. In the first case, we are again done. In case we haveR⊥, wefind (a)R, sinceR⊥ = (R < S). Moreover, byΣ0

1 -completeness, we have (b)R⊥.Combining (a) and (b), we obtain⊥.

Step 2. SupposeR∨ ⊥. By Σ01 -completeness, we findR∨⊥, hence,R.

Step 3. SupposeR. We want to deriveR∨⊥. We find:R∨ R⊥. Now we may proceedas in step 1.

Step 4. SupposeR∨ ⊥. We may immediately conclude thatS∨ ⊥.

Remark 3.2. We can also proveEA ¬R⊥ ↔ R. Right-to-left is trivial. In the otherdirection, letI be aT -cut satisfyingI ∆0+ Ω1. We have:

EA ¬R⊥ → (¬R⊥)I

→ (R→ R)I

→ (I R→ RI )

→ (I R→ R)

→ R.

The last step is an application of L¨ob’s Theorem for〈T, I 〉. For a discussion of L¨ob’sTheorem with shifting interpretations, see [19, Section 4].

An immediate generalization of the FGH Theorem is due essentially to Franco Montagna.

Theorem 3.3. Consider any numerized theoryT = 〈T,N 〉. Let := T . Let S(x) beΣ0

1 and let R be such thatQ R↔ S(#R) ≤ R. We have:

EA (S(#R) ∨⊥) ↔ (R∨ ⊥)

↔ R

or, equivalently,EA + con(T) (S(#R)↔ R) ∧ (S(#R)↔ R).

It is easy to see that Rosser’s Theorem is an immediate consequence of Montagna’sTheorem. We end this subsection, by proving a variant of a part of the FGH Theoremthat will be used inSection 4.

Theorem 3.4. Consider any numerized theoryT . Let := T . Let A be∃∀∆0 andlet R be such thatQ R ↔ A ≤ R. Let BΣ0

1 be theΣ01 -collection principle:

∀x≤a∃y S0(y)→ ∃b∀x≤a∃y≤b S0(y), where S0 ∈ ∆0. We have:

EA+ BΣ1 R→ (A∨⊥),

or, equivalently,EA + BΣ1+ con(T) R→ A.

Proof. Reason inEA + BΣ1. SupposeR. We haveA ≤ R or R < A. In thefirst case, we may concludeA, and we are done. SupposeR < A. This has the form


∃p (proof(p, #R) ∧ ∀y≤p ∃z ¬ A0(y, z)), where A0 is in ∆0. By Σ01 -collection, our

formula is equivalent to:

C := ∃p∃x (proof(p, #R) ∧ ∀y≤p ∃z≤ x ¬ A0(y, z)).

Thus, we find:C, and, hence,(R < A). I.o.w., R⊥. Combining this with ourassumptionR, we find⊥ and we are done.

3.2. The FGH Theorem andS12

It is an open problem whether the FGH Theorem can be formalized inS12, even for

S∈ ∃Σb1 . However, for a restricted range of theories, we can prove a salient consequence

of FGH Theorem.

Theorem 3.5. LetT = 〈T,N 〉 be a sequential numerized theory. Suppose that T is eitherfinitely axiomatized or an extension by finitely many axioms of I∆0+Ω1 (relativized toN ).We write := T . Let A be any T -sentence. Let R be such thatQ R ↔ A ≤ R.We have:S1

2 A↔ R.

Proof. Reason inS12.

SupposeA. By the small reflection principle (Fact 2.8), we have (a):

((R < A)→ R).

By A andFact 2.7, we have (b)(A ≤ R∨ R < A). Combining (a) and (b), wefind R.

Conversely, supposeR. By the small reflection principleFact 2.8, we have:

((A ≤ R) → A),

i.e.(R→ A). Ergo,A.

3.3. Some consequences of the FGH Theorem

The proof ofTheorem 4.1is our central application of the FGH Theorem in this paper.We also use it in the proof ofTheorem C.7. In this subsection we spell out some moreimmediate consequences ofTheorem 3.1. These consequences are not strictly needed forthe rest of the paper. They have, however, heuristic value. Moreover, they are interesting intheir own right. For some further information, the reader is referred to [13, Chapter 7].

3.3.1. 1-ReducibilityWe give a quick proof of a well-known fact.

Theorem 3.6. Suppose T can be extended to a consistent numerizable theory W. Then,any RE set is 1-reducible to T . A fortiori, T is of Turing degree0′.

Proof. Clearly, we may assume thatW is a finite extension ofT , sayW = T + A. LetW = 〈W,N 〉 be a numerization ofW. Consider any RE setX with indexe. Let Rn be theFGH sentence for the theoryW corresponding to the sentenceSn := (en 0). Clearly,the mappingn → (A → RN

n ) is recursive. By the FGH Theorem, formulated externally,


we have:n ∈ X ⇔ T A→ RN

n .

3.3.2. Closure under disjunctionWe show that provabilities are closed under disjunction.

Theorem 3.7. LetT be a numerized theory. Let := T . For any sentences A and B ofthe language of T , there is aΣ1-sentence C such thatEA C ↔ (A∨B).

Proof. TakeS := (A∨B) in Theorem 3.1.

Note thatC can in fact be taken to be∃Π b1 .

3.3.3. Degrees of provably deductive consequenceLet T be numerized. LetA andB be sentences of the language ofT . Let := T . We

define:

• A T B :⇔ T A→ B.

• A ≡T B :⇔ A T B andB T A.

We call T provably deductive consequenceand we call≡T provably deductiveequivalence. Clearly, these notions yield a degree structure on the sentences ofT .

Theorem 3.8. LetT = 〈T,N 〉 be a numerized theory. SupposeT provesEA (relativizedtoN ). Then, each degree of provably deductive equivalence ofT contains a∃Π b

1 -sentence.

Proof. Let γ be such a degree. SupposeC ∈ γ . TakeS := C in Theorem 3.1.

We can extendTheorem 3.8to certain weaker theories by usingTheorem 3.5.

Theorem 3.9. LetT = 〈T,N 〉 be a sequential numerized theory. Suppose that T is eitherfinitely axiomatized or an extension by finitely many axioms of I∆0 + Ω1 (relativized toN ). Suppose thatT provesS1

2 (relativized toN ). Then, each degree of provably deductiveequivalence ofT contains a∃Π b

1 -sentence.

3.3.4. Smorynski’s TheoremThe following application is due to Smory´nski. See [11, p. 366] or [13, p. 312].

Theorem 3.10 (Smorynski’s Theorem).Let T = 〈T,N 〉 be numerized theory. SupposeT EA. Then, we have, verifiably inEA, that T is Σ0

1 -sound iffT is consistent andT + con(T) is Σ0

1 -conservative overT .

Proof. We write := T . Reason inEA.SupposeT is Σ0

1 -sound. LetS be in Σ01 . Suppose(con(T) → S). Then, we find

(S∨⊥). By Σ01 -soundness, it follows that(S∨⊥). Hence, byΣ0

1 -completeness,S.Suppose thatT is consistent andT + con(T) is Σ0

1 -conservative overT . SupposeS.Applying the first equivalence of the FGH Theoreminside the, we obtain(R∨ ⊥).Ergo,(con(T)→ R). By Σ0

1 -conservativity, it follows thatR. We may conclude, nowapplying the FGH Theorem outside the, thatS.


Note that the assumption thatT EA, was only used in the second part of the proof in the‘internal’ application of the FGH Theorem. We can extend the result to theoriesT suchthat everyT -cut I has a subcutJ with J : T T . Examples of such theories areS1

2,I ∆0+ Ω14+ con(F), I ∆0+ Ωn+1 | n ∈ ω andPA + incon(PA).

Theorem 3.11. Let T = 〈T,N 〉 be numerized theory. Suppose that everyT -cut I has asubcut J with J: T T . Then, we have, verifiably inEA, that T is Σ0

1 -sound iffT isconsistent andT + con(T) is Σ0

1 -conservative overT .

Proof. We replace the second part of the previous proof by the following variation.Suppose thatT is consistent and thatT + con(T) is Σ0

1 -conservative overT . SupposeS. UsingFact 2.2(2), (1), we can find aT -cut J such that (a)J : T (I ∆0+ T) and (b)(∀x∈J ∃y 2x = y). By (a), we findSJ . Ergo(R∨R⊥)J , Hence,(R∨(R⊥)J). Also(c) ((R⊥)J → R). Since, in the proof ofΣ0

1 -completeness forT , the transformationof the witnessx of aΣ0

1 -sentenceS′ to a proofp of S′ is of order 2xm, for standardm, we

get by (b):((R⊥)J → R⊥). Ergo (d)((R⊥)J → ⊥). We may conclude from (c)and (d):(R∨ ⊥).

Hence,(con(T)→ R). By Σ01 -conservativity, it follows thatR. We may conclude,

by the FGH Theorem, thatS.

Here is a corollary fromTheorem 3.10. SupposeT = 〈T,N 〉 EA. We say thatT = 〈T,N 〉 is reflexiveif it proves for everyn the statementcon(Tn). HereTn is thetheory axiomatized byEAN plus theT-axioms with Godelnumber less than or equal ton.

Corollary 3.12. SupposeT is a consistent, numerized, reflexive theory such thatT EA.Suppose there is an n, such that, for allΣ0

1 -sentences S, wheneverT S, we haveTn S.ThenT is Σ0

1 -sound.

Proof. Let := T andn := Tn . Supposen(con(Tn) → S), then, by reflexivity,S. Hence,nS. Applying Theorem 3.10to Tn, we findS. SoTn is Σ0

1 -sound. Hence,Tis alsoΣ0

1 -sound.

The above theorem tells us that, if a theory that is consistent, numerized, reflexive andverifiesEA, proves a falseΣ0

1 -sentence, then it is forced to tell more and more complexlies, i.e. it will prove falseΣ0

1 -sentences the proofs of which need more and more axioms.

4. Σ01 -soundness in potentia

In this section, we prove a theorem that will be the main lemma to our proof thatconsistent, finitely axiomatized, sequential theories are trustworthy. LetEA+ be I ∆0 +supexp, wheresupexp is the axiom stating that the superexponentiation function is total.

Theorem 4.1. Let T := 〈T,N 〉 be a finitely axiomatized, sequential theory. We write := T . There is aT -cut I such that, for allΣ0

1 -sentences S,EA+ (S∨⊥)↔ SI ,or equivalently,EA+ + con(T) S↔ SI .

Before proving our theorem we formulate and prove an immediate corollary.


Corollary 4.2. Let T := 〈T,N 〉 be a finitely axiomatized, sequential theory. There is aT -cut I such that〈T, I 〉 is (EA+ + con(T))-verifiablyΣ0

1 -sound.

Proof. Let I be the cut promised inTheorem 4.1. We have, for anyΣ01 -sentenceS,

EA+ + con(T) T SI → S, and hence,EA+ + con(T) 〈T,I 〉S→ S.

To get the proof ofTheorem 4.1going, we need a few preparatory steps. We will applythe FGH Theorem to a restricted proof predicate, where the formulas in the proof arerestricted to formulas of a certain complexity. We take as measure of complexityρ, whereρ(A) is the depth of quantifier changes. This measure is discussed in some detail in [19].We takeΓn to be the set of formulas of complexity at mostn andΓ cl

n the set of sentencesof Γn. m-provability will be provability from axioms with G¨odelnumber belowm, wherethe formulas occurring in the proof are all inΓm.

The notation A(k) is somewhat misleading. In general we are working in someinterpretationof number theory. So the termk occurs in unwound relational form. Ourmeasureρ is designed to be insensitive to such fine points.

Lemma 4.3. ρ(A(k)) is independent of k.

Proof. Suppose, for simplicity, that we are working with tally-numerals.A(k) in T couldlook like this:

∃x0 . . . ∃xk (0N (x0) ∧ SN (x0, x1) ∧ . . . ∧ SN (xk−1, xk) ∧ A(xk)).

The complexity of this formula is max(ρ(0N (x)), ρ(SN (x, y)), ρ(A(x))) + 1. Thisformula is clearly estimated byρ(A(x)) + c, for a fixed standardc. Similar reasoningworks for efficient numerals based e.g. on binary notations.

Here is a fundamental lemma aboutn.

Lemma 4.4. Suppose thatT := 〈T,N 〉 is a finitely axiomatized theory. Let andm bethe provability and the m-provability predicates ofT . We have, for any T -sentence A andk > ρ(A) and k larger than the complexities of the axioms of T ,EA+ k A↔ A.

Proof. The left-to-right direction is obvious. To prove the right-to-left direction, reasonin EA+. SupposeA. We can, usingsupexp, find a cutfree proof in predicate logic ofC → A, whereC is the conjunction of theT-axioms. See [4, Part V, Chapter 5], fordetails. By the subformula property, this proof is also ak-proof.

Note that we used the fact thatT is finitely axiomatized in an essential way in the proof.

Remark 4.5. It has been shown by Philipp Gerhardy that it is possible to replace, inthe usual superexponential estimate of the growth involved in cut elimination, the usualmeasure of complexity (depth of connectives) by a measure that is a minor modification ofour measureρ of depth of quantifier changes. See [2]. For Gerhardy’s more recent results,see: 〈http://www.daimi.au.dk/∼peegee〉.

Lemma 4.6. Let T := 〈T,N 〉 be finitely axiomatized. Let andm be the provabilityand the m-provability predicates ofT . Consider aΣ0

1 -sentence S. We can find Rm such thatQ Rm ↔ S≤ mRm, by the Godel Fixed Point Lemma. Note thatρ(Rm) := ρ(S)+ c,

http://www.daimi.au.dk/~peegee


for a standard c which is independent of m. Choose n> ρ(S) + c. We have:EA+ (S∨⊥)↔ Rn.

Proof. We want to apply the FGH Theorem. To do this we must verify that the stepsin the proof go through for ourn-provability. Note e.g. thatn is large enough to have:EA Rn → n Rn andEA R⊥n → nR⊥n . Thus, we have:EA (S∨ n⊥)↔ n Rn.Now applyLemma 4.4.

Our proof strategy will be to provide a cutI , such that,EA+-verifiably, we haveRn ↔SI . Then we may applyLemma 4.6. To get the desired result, we need a reflectionprinciple.

Lemma 4.7. Let U := 〈U,M〉 be any sequential theory. Let beU-provability and letn beU-n-provability. For any n, we can find aU-cut J such thatEA ∀A∈Γ cl

n (Jn

A→ A).

Proof. This is Fact 2.4.5(ii) of [19]. The idea is that, inU , we can define a satisfactionpredicate forΓn and proveΓn-reflection by replacing induction over proof length by theuse of a definable cut.

The next lemma is nearly the theorem we are aiming to prove. The only defect is thatI isstill dependent onρ(S).

Lemma 4.8. Let T := 〈T,N 〉 be a finitely axiomatized, sequential theory. We write := T . For anyΣ0

1 -sentence S, there is aT -cut I such that,EA+ (S∨⊥)↔ SI ,or equivalently,EA+ + con(T) S↔ SI . The cut I depends only onρ(S).

Proof. Take n and Rn as in Lemma 4.6. Let R := Rn. We have, byLemma 4.6, (a)EA+ (S∨ ⊥) ↔ R. Choose a reflectingT -cut I for n as in Lemma 4.7. ByFact 2.2(2), we can chooseI in such a way that it verifies∆0-induction. Note thatI willonly depend onρ(S).

The left-to-right direction is immediate by theobis-principle. We treat the otherdirection. By (a), it is sufficient to show thatEA+ SI → R.

Reason inEA+. SupposeSI . Since we have∆0-induction in I , it follows that(S ≤ nR∨ n R < S)I and so((S ≤ n R)I ∨ (n R < S)I ). The first disjunct isequivalent toRI , which impliesR. To the second disjunct we apply the reflection principlefrom Lemma 4.7to infer R. Thus, we obtainR.

We want to make the cutI independent of theΣ01 -sentenceS. The problem is thatΣ0

1 -sentences may have arbitrarily largeρ-complexities. If we would haveN : T EA,there would be no problem, since we haveEA S ↔ trueΣ (#S), wheretrueΣ is theordinaryΣ0

1 -truth predicate, which is itself given by aΣ01 -formula. All sentences of the

form trueΣ (#S) have some complexity below a fixed finiten. We can use the idea even inthe absence ofEA by making our cut smaller. Here is another lemma.

Lemma 4.9. Let S= ∃x S0(x), where S0 ∈ ∆0. Let the truth predicate be of the form∃y trueΣ ,0(y, z), where(trueΣ ,0(y, z)) ∈ ∆0. There is a fixed standard k, such that

S12 (S0(x) ∧ 2xk ↓)→ ∃y≤2xk

trueΣ ,0(y, #S).


Proof. The proof is by inspecting the usualEA-proof of S → trueΣ (#S). See e.g. [4,Part C, Chapter 5(b)], for a detailed presentation.

Here is the proof ofTheorem 4.1.

Proof. Let J be the cut provided byLemma 4.8for the complexity of theΣ -truthpredicate.Let I be a shorter cut, such thatT ∀x∈I 2x ∈ J.

Let aΣ01 -sentenceS be given. The left-to-right direction is immediate, using theobis-

principle. We treat the direction from right-to-left. TakeS∗ := trueΣ (S). By Lemma 4.8,we getEA+ (S∗ ∨ ⊥) ↔ SJ∗ . By Lemma 4.9, we findT SI → SJ∗ . Thus, wehave:

EA+ SI → SJ∗→ S∗ ∨ ⊥→ S∨ ⊥.

So we are done.

Open Question 4.10. Can one find a numerized, non-sequential, finitely axiomatizedtheory for which there is a falseΣ0

1 -sentence which is provable on every definable cut?

We draw an obvious corollary.

Corollary 4.11. SupposeT is consistent, finitely axiomatized and sequential. Then thereare aT -cut I and a modelM of T such that, inM, witnesses ofΣ0

1 -sentences are eitherin the initial segment of theT -numbers isomorphic toω or not in I .

Proof. ChooseI as inTheorem 4.1. Clearly,U := T +¬SI | N #|= S is consistent. TakeM a model ofU .

Note thatCorollary 4.11, in its turn, directly impliesTheorem 4.1. Another immediatecorollary is as follows. This corollary is about the limes inferior of a sequential theoryT .The notion of limes inferior of a sequential theoryT or li(T) was introduced inRemark 2.3.

Corollary 4.12. Let T be a consistent, sequential, finitely axiomatized theory. Thenli(T)

is Σ01 -sound.

We can extendLemma 4.8partly to a wider formula class.

Definition 4.13. Consider any numerized theoryT . Let B := ∃x B0(x) be a formulaof the language ofT . Let I be aT -cut. We write B[I ] for ∃x∈I B0(x) (or: B < ∃xx #∈ I ).

Theorem 4.14. Let T := 〈T,N 〉 be a finitely axiomatized, sequential theory. We write := T . For any∃∀∆0-sentence A, there is aT -cut I such that,EA++BΣ1 A[I ] →(A∨⊥), or equivalently,

EA+ + BΣ1+ con(T) A[I ] → A.

The cut I depends only onρ(A).


Proof. Take R as inTheorem 3.4, with n, for a suitably largen, substituted for. Wefind, using cut elimination, fromTheorem 3.4:

EA+ + BΣ1 R→ (A∨ ⊥).

Let I be ann-reflectingT -cut satisfyingI ∆0. It is sufficient to show inEA+ + BΣ1 thatA[I ] impliesR.

Reason inEA+ + BΣ1. SupposeA[I ]. Since

(n R→ n R≤ n R)I ,

it follows, byFact 2.5(7), that(A[I ] ≤ InR∨I

n R < A[I ]).9 Clearly, the first disjunct isT -equivalent toR[I ], and, thus, implies inT that R. Moreover, the second disjunct impliesin T thatI

n R. Hence, sinceI is n-reflecting, the second disjunct impliesR in T . Thus,we find (outside ofT ): R. We can extendTheorem 4.1to a larger class of theories.

Theorem 4.15. Let T be a consistent, sequential, finitely axiomatized theory. Suppose thatT and U are mutually interpretable. Then there is aΣ0

1 -sound numerizationU = 〈U,P〉of U.

Note thatU need not be sequential! Before proving the theorem we need a lemma, whichis a strengthening of L¨ob’s Theorem.

Lemma 4.16. LetT = 〈T,N 〉 be a numerized, consistent, sequential, finitely axiomatizedtheory. Let I be aT -cut and let A be a sentence of the language of T . Then there is a ksuch that

I ∆0+ Ω1 T (IT ,k A→ A)→ T A.

The number k depends only on the complexities of the axioms of T , the complexity ofN ,the complexity of I and the complexity of A. Our complexity measure here isρ, i.e. depthof quantifier changes.

The lemma is a special case of Theorem 4.2 of [19]. We turn to the proof ofTheorem 4.15.

Proof. SupposeK : T U andM : U T . Note thatN ′ := N M K is an interpretationof F in T . (We write composition in the order of application here.) ByFact 2.2(3), there isa T -cut J that isT -provably isomorphic with aT-cut J ′ of N ′. By Fact 2.2(2), we mayassume thatJ satisfiesI ∆0 + Ω1. Let K be theρ-complexity of theΣ0

1 -truth predicate.By the external form ofLemma 4.16, we can find ak such that, for anyA ∈ ΓK+n, ifT J

T ,k A→ A, thenT A. Heren is a sufficiently large number.

By Lemma 4.7, we can find aT -cut I ∗ such thatT I ∗T ,kB → B, for any B ∈ Γk.

Let I be a subcut ofI ∗ such thatT ∀x∈I 2x ∈ I ∗. By Fact 2.2(2), we may chooseI ∗andI such that they satisfyI ∆0+Ω1. Consider anyΣ0

1 -sentenceS. Let S0 := trueΣ (#S).We have, byLemma 4.9, T SI → SI ∗

0 .

9 Note that, to apply the verbatim statement ofFact 2.5(7) we have to shift to the theory〈T, I 〉 first and, then,shift back toT . Alternatively, we can just run through the proof again for the modified statement.


Let R be such thatF R↔ S0 ≤ T ,k R. We have:

T SI → SI ∗0

→ (R∨ (T ,k R < S0))I ∗

→ R∨I ∗T ,k R

→ R.

We takeP := IM. Suppose, for anyΣ01 -sentenceS, that U S. This tells us that

M : U (T + SI ). ErgoM : U (T + RN ). We may conclude thatT RN M K ,i.o.w. T RN ′

. It now follows that

T JT ,k R→ RN ′ ∧ J ′

T ,k R

→ RJ ′

→ RJ

→ R.

Applying Lob’s rule, we haveT R. By cutelimination, we findT k R. Hence, by theexternal version of the proof of the FGH Theorem, we find thatS0 is true and, thus, thatSis true.

5. On the manufacture of faith

We repeat the definition of trustworthiness here.

Definition 5.1. A theory V is trustworthyif every U interpretable inV is also faithfullyinterpretable inV .

In this section, we will provide a characterization of trustworthy theories. Friedman’sresult that consistent, finitely axiomatized, sequential theories are trustworthy, will followfrom this characterization in combination withTheorem 4.1. Our treatment in this sectioncan be viewed as generalizing some of Per Lindstr¨om’s work on faithful interpretability.See [7, Chapter 6, Section 2]. The methods used are for a great part those developed by PerLindstrom and ViteslavSvejdar.

5.1. An upper bound

In this subsection we prove an upper bound result. We need two lemmas.

Lemma 5.2. Let T = 〈T,N 〉 be a numerized theory. LetΓ be any class of T -sentencesfor which T contains a definable truth predicate, sayTRUE. We only need thatTRUEsatisfies Tarski’s convention. Suppose that the set of codes of elements ofΓ has a fixedbinumeration inT . Then, there is a unary predicate of numbers A(x), such thatT (A(x) ∧ A(y)) → x = y, and such that, for any n,T + A(n) is Γ -conservative overT .We may consider A as representing a closed partial numerical termτ , writing ‘τ x’ for‘A(x)’.

We give the proof inAppendix B.


Lemma 5.3. Let T = 〈T,N 〉 be a numerized theory. LetL be a language of finitesignatureσ for predicate logic. We call predicate logic of signatureσ : FOLσ . Let α(x)

be any formula in the language of T such thatT proves that all elements ofx | α(x) arecodes ofL-sentences. We writeα for provability from the sentences coded by the elementsof x | α(x). We writecon(α) for ¬α⊥.

There is an interpretationH : (T + con(α)) FOLσ such that, for anyL-sentence A,we haveT + con(α)+α A AH. We say thatH is a Henkin interpretation ofα.

Proof. We can see this by inspection of the usual proof of the Interpretation ExistenceLemma. The basic idea is that we formalize the Henkin construction, employing definablecuts whenever we would have used induction inPA. See e.g. [17] or [18].

We proceed with our, somewhat technical, upperbound result. The bit with the sentenceAis present, because we want our result to be applicable also to some theories that arenotnumerizable.

Lemma 5.4. Let T be any theory. SupposeK : T U. Let A be any T -sentence. SupposeW = 〈T + A,N 〉 is numerized. Then there is an interpretationM : T U such that, forany U-sentence B, T BM ⇒W U B.

Proof. ConsiderW . We can, byFact 2.2(1) andLemma 4.9, shortenN to aW-definablecut J such thatZ := 〈T + A, J〉 contains a truth predicate for theΣ0

1 -sentencesof Z. (Remember that the meaning of ‘Σ0

1 ’ shifts with the numerization.) Note thatZ U B ⇒W U B. It follows that it is sufficient to prove our theorem forZ. Thus,we may, without loss of generality, assume thatW contains a truth predicate, saytrue, fortheΣ0

1 -sentences. Moreover, we may, byFact 2.2(2), assume thatW provesI ∆0+ Ω1.Let τ be the partial closed term promised byLemma 5.2for W andΣ0

1 . We fix somestandard enumerationCx of theU -sentences in such a way thatW verifies its elementaryproperties. We specifyM, in T , by cases. In case we have¬A, we takeM equal toK.Suppose we haveA. We may now work inW . Let U∗ := U + Cx | τ x. Note that(i) U∗ is not∆b

1-axiomatized, and that (ii) in talking aboutU∗ we are really talking aboutthe formula defining the axiom set and that (iii) the definition ofU∗ only makes sense inthe presence ofA. In caseincon(U∗), we takeM again equal toK. If con(U∗), we takeM equal to the Henkin-interpretationH of U∗. We give the clauses forM, for the casesof the domain of the interpretation and the translation of a binary predicate:

• δM(x) :↔ ((¬A∨ (A∧ inconN (U∗))) ∧ δK(x)) ∨ (A∧ conN (U∗) ∧ δH(x)),• PM(x, y) :↔ ((¬A∨ (A∧ inconN (U∗))) ∧ PK(x, y)) ∨

(A∧ conN (U∗) ∧ PH(x, y)).

(In writing e.g. ‘inconN (U∗)’, we intend no relativization of the formula defining theaxiom set.)

Clearly,M : T U . SupposeT BM. Let¬ B = Cn. We have:

W + τ n “ (U +¬B) = U∗” .

Hence,W + (τ = n) + con(U + ¬ B) ¬ BM. Thus,W (τ n) → U B. By theΣ0

1 -conservativity ofτ n, we findW U B.


5.2. The characterization

In this subsection, we provide the promised characterization of trustworthiness andprove Friedman’s result as a corollary.

Theorem 5.5. Let T be any∆b1-axiomatized theory. The following are equivalent.

(1) T is trustworthy.(2) T has a (finite) extension which has aΣ0

1 -sound numerization.(3) T has a (finite) extension on which there isΣ0

1 -sound interpretation ofQ.(4) There is a faithful interpretation of predicate logic with one binary relation symbol

into T .10

Proof. “ (1) ⇒ (2)”. SupposeT is trustworthy. Say the (relational) signature of numbertheory isσ . Trivially, the predicate logicFOLσ is interpretable inT . Hence, there is afaithful interpretation, sayK, of FOLσ in T . It is easily seen that〈T + (

∧F)K,K〉 is a

Σ01 -sound numerization of an extension ofT .“ (2) ⇒ (1)”. SupposeT has a (finite) extension which has aΣ0

1 -sound numerization,sayW . It follows, byΣ0

1 -soundness, thatW U B impliesU B.SupposeK : T U . By Lemma 5.4, we may conclude that there is a faithful

interpretationM : T U .“ (1)⇒ (4)”. This is immediate.“ (4) ⇒ (2)”. SupposeP is a faithful interpretation of predicate logic with one binary

relation symbol intoT . There is a finitely axiomatized set theory, sayS, in the languagewith just one binary relation symbol into whichF is faithfully interpretable, say viaQ. Seee.g. [9]. Hence,〈T + (

∧S)P ,QP〉 is aΣ0

1 -sound numerization of an extension ofT .“ (3) ⇔ (2)”. This is immediate, by the fact thatF can be interpreted inQ on a cutI .

Cuts are downwards closed under≤. So we can always convert aΣ01 sound interpretation

of Q into aΣ01 -sound interpretation ofF.

The definition of trustworthiness is ‘neutral’ w.r.t. arithmetical theories and the like,in that it does not mention the presence of any device allowing coding. It does noteven mention specific signatures. Thus it is remarkable that a theory involving codingis connected via (2) of the theorem to trustworthiness. InAppendix C, we will discussa nice alternative formulation of (2) of the theorem. FromTheorem 5.5combined withTheorem 4.1, we may now immediately conclude to Friedman’s Theorem.

Corollary 5.6 (Friedman’s Theorem).Finitely axiomatized, sequential, consistent theo-ries are trustworthy.

Remark 5.7. We have proved Friedman’s Theorem fromTheorem 4.1. It is easily seenthat, conversely, the existence of aΣ0

1 -sound cut again follows fromCorollary 5.6.Consider a finitely axiomatized, numerized, sequential and consistent theoryT = 〈T,N 〉.By Friedman’s Theorem, there is a faithful interpretationM of F in T . Clearly,〈T,M〉 is

10We might want to insist that predicate logic contains identity. In this case it is only necessary that theinterpretation is faithful w.r.t. the fragment of the formulas containing onlyR.


Σ01 -sound. Ergo, byFact 2.2(3) and the upwards persistence ofΣ0

1 -sentences, we can findaT -cut I such that〈T, I 〉 is Σ0

1 -sound. Example 5.8. PA + incon(PA) is not trustworthy. This can be seen e.g. by noting thatPA + incon(PA) PA. Since any interpretation ofPA in PA + incon(PA) is verifiablyan end-extension of the identity interpretation, it will, by the upwards persistence ofΣ0

1 -sentences, satisfyincon(PA). Hence no faithful interpretation ofPA in PA+ incon(PA) ispossible.

In contrast,ACA0+ incon(ACA0) is trustworthy. We may useTheorem 4.15to get a strengthening of Friedman’s Theorem.

Corollary 5.9. Suppose T is consistent, finitely axiomatized and sequential. Suppose Tand U are mutually interpretable. Then U is trustworthy.

Open Question 5.10. We could say that a theoryT is solid if every U that is mutuallyinterpretable withT is trustworthy. Is there a perspicuous characterization of solidtheories?

Note thatPA andPA + incon(PA) are mutually interpretable. So, byExample 5.8, PAis trustworthy but not solid. We proceed with some further corollaries ofTheorem 5.5. The following corollary is easy.

Corollary 5.11. Any subtheory of a trustworthy theory is trustworthy.

Corollary 5.12. Consider Group Theorygroupc, where we allow an extra constant c inthe language. The theorygroupc is trustworthy.

Proof. Tarski constructs, in [15], a modelG of groupc that has as definable inner modelthe natural numbers with plus and times. In other words, he constructs an interpretationKwith K : Th(G) Th(N). It follows that 〈groupc + (

∧F)K,K〉 is Σ0

1 -sound. Ergo, byTheorem 5.5, groupc is trustworthy. Corollary 5.13. Any trustworthy theory is of degree0′.

Proof. This is immediate byTheorem 3.6. Open Question 5.14. What is the complexity of trustworthiness? Our characterizationshows that this complexity is at mostΣ0

3 . I conjecture that it is completeΣ03 .

6. On the nature of trustworthiness

The notion of trustworthiness may, at first sight, seem to be somewhat artificial. Thus,one may wonder what structure is ‘the natural home’ of the notion. I am not sure thisquestion has a unique answer. However, the answer given below is a good candidate. Theanswer will be that the relevant ‘structure’ is the embedding functor of two preorders.

Consider the preorderPFI of consistent theories ordered by the relationf, whereU f V if U is faithfully interpretable inV . We writeU ≡f V for: U f V andV f U .Consider also the preorderPI of consistent theories ordered by the relation, whereU Vif U is interpretable inV . We writeU ≡ V for: U V andV U .


These preorders can be viewed as categories in the usual way. If we divide outisomorphisms, we get the partial orderings of degrees of faithful interpretability and ofdegrees of interpretability.

Let emb be the identical embedding functor fromPFI to PI. We will show thatembhas a right adjoint,(·), i.e. a mapping from theories to theories satisfying the magicalequation:11

U f V ⇔ emb(U) V.

From this equation, the following facts are immediate consequences.

(1) (·) is a functor.(2) The theoryV is trustworthy.(3) V ≡ V . So, every degree of interpretability has a trustworthy element.(4) V is trustworthy iffV ≡f V .

We specify(·). Consider a theoryT . We expand the signature ofT with a unary predicateP and with a binary predicateR. The theoryT is the theory axiomatized by the axiomsof T where we relativize the quantifiers toP. No non-logical principles concerningRare added. (The logical axioms concerning identity belong to predicate logic and are leftunrelativized.) It is easily seen that (a)T ≡ T . By a simple model-theoretical argument, wemay show thatT is conservative over predicate logic with just the binary relation symbolR.Hence, by translatingR asR(x, y), the theoryT faithfully interprets predicate logic withjust the binary relation symbolR. By Theorem 5.5(4), it follows that (b)T is trustworthy.From (a) and (b), it is immediate that(·) is right adjoint toemb.

In caseT has an infinite model, we can skip the relativization toP in the constructionof T . Thus we only need to expand the signature withR. Note that sequential theoriesare not closed under relativization of the domain. However, sequential theories are closedunder adding predicate symbols. By the preceding observation, the mappingadd a binaryrelation symbolwill be right adjoint of the embedding functor, if we restrict both preordersto consistent sequential theories.

In case a numerization〈T,N 〉 satisfies full induction, we can also take forT , the theoryPA + conn(T) | n ∈ ω, whereconn(T) means consistency of the set of axioms ofT with Godel number less than or equal ton. It follows that we can find an appropriateright adjoint, if we restrict both preorders to consistent extensions ofPA in the arithmeticallanguage.

By Theorem 5.9, consistent, finitely axiomatized, sequential theoriesT have the furtherproperty that ifT ≡ U , thenT ≡f U . It is easy to see that this property is equivalent to theproperty of solidity introduced inQuestion 5.10.

Acknowledgements

I thank Lev Beklemishev and Volodya Shavrukov for providing me with pointers to theliterature. I thank Lev also for his comments on the penultimate version of the paper. I am

11See [8], for the basic facts on adjunctions.


grateful to Harvey Friedman who reminded me of his theorem. I thank Warren Goldfarband Volodya Shavrukov for e-mails clarifying the history of the FGH Theorem.

Appendix A. A notational convention

In this appendix we make the convention for the use of two kinds of variables andof boxes precise. LetT = 〈T,N 〉, T ′ = 〈T ′,N 〉, . . ., be numerized theories and letU,U ′, . . ., be arbitrary theories. We assume that each theory comes equipped with a∆b

1-formula defining the axiom set. We treat the case, where we just have ordinary boxes in thelanguage. Addition of e.g.T ,n andU,n is entirely analogous.

We assume the languageLT of T has variablesξ, ξ ′, . . .. We enrichLT to a languageLT with a second kind of variablesx, x′, . . . and with unary operatorsU andT ′ , forvariousU andT ′. The terms of the extended language are the smallest set containing bothsets of variables and closed under the term-forming operations ofLT . The set of formulasof LT is the smallest setF such that:

• P(t0, . . . , tn−1) is in F , if the ti are terms of the extended language andP is ann-arypredicate symbol ofLT ;

• F is closed under the propositional connectives and under the quantifiers∀x, ∃x, ∀ξ ,∃ξ , for all variablesx andξ ;

• If A is a sentence ofLU , thenU A is in F ;• If A is a formula ofLT ′ with only free variables inx, x′, y, . . ., thenT ′ A is in F .

We can give the formulas ofLT their desired translations intoLT via the translation(·)T . We arrange it so that we have infinitely many variablesη, η′, . . . available inLT

distinct from the variablesξ, ξ ′, . . .. We translate the terms by replacingx by η, x′ by η′,etc.

• (P(t0, . . . , tn−1))T := P(tT0 , . . . , tTn−1);

• (·)T commutes with the propositional connectives and with∀ξ , ∃ξ ;• (∀x A)T := ∀η (δN (η)→ AT );• (∃x A)T := ∃η (δN (η) ∧ AT );• (U A)T := provU (#A);• (T ′ A)T := provT ′(#(AT ′)).

(The numerical variables inA are treated in the usual way.)

Appendix B. Conservativity

In this appendix, we proveLemma 5.2. Let T = 〈T,N 〉 be a numerized theory. LetΓbe any class ofT-sentences for whichT contains a definable truth predicate, sayTRUE.We only need thatTRUE satisfies Tarski’s convention. We assume that the set of codes ofelements ofΓ has a fixed binumeration inT . We show that there is a unary predicate ofnumbersA(x), such thatT (A(x) ∧ A(y))→ x = y, such that, for anyn, T + A(n) isΓ -conservative overT .

We define, inT , using the G¨odel Fixed Point Lemma, the formulaA(x) as follows.


A(x) ↔ ∃p ∃C∈Γ ( proofT (p, A(x)→ C) ∧ ¬TRUE(C) ∧∀q < p ∀D∈Γ ∀y ( proofT (q, A(y)→ D) → TRUE(D) ) ).

We assume that the formalization ofproof is standard, so that every proof has a singleconclusionC with C < p, etc. We first prove the uniqueness clause. Reason inT . Supposethatx #= y andA(x) andA(y). Let p be a witness forA(x) and letq be a witness ofA(y).By our assumption about the proof predicate, it follows thatp #= q. Since inF, we havethe linearity of<, it follows that p < q or q < p. By the specification ofA, it follows thatthis is impossible.

We move to the metatheory again. We prove our theorem by induction onT -proofs.Suppose, that for allT-proofsq < p, we have, ifq : T A(m) → D, for somem andfor someD ∈ Γ , thenT D. (‘r : T E’ means:r is aT -proof of E.) Suppose furtherthat p : T A(n)→ C, for C ∈ Γ . We showT C. From our assumptions, we have thefollowing propositions.

T (C ∈ Γ ) (B.1)

p : T A(n)→ C. (B.2)

It follows that:

T +¬C ¬A(n) (B.3)

T +¬C C ∈ Γ ∧ proofT (p, A(n)→ C) ∧ ¬TRUE(C). (B.4)

Using (B.3) and (B.4) and the specification ofA, we may conclude that:

T +¬C ∃q < p ∃D∈Γ ∃y ( proofT (q, A(y)→ D) ∧ ¬TRUE(D) ). (B.5)

It follows that:

T +¬C ∨

q<p,D<p,D∈Γ ,m<p

( proofT (q, A(m)→ D) ∧ ¬ D ). (B.6)

Consider anyq < p, D < p with D ∈ Γ , andm < p. In case we have:q : T A(m) →D, it follows, by the minimality ofp, thatT D. In this case the disjunct corresponding toq in (B.6) is T -provably equivalent to absurdity and may be omitted. Suppose thatq doesnot witnessT A(m) → D, then, byΣ -completeness,T ¬ProofT (q, A(m) → D).So again we may omit the disjunct corresponding toq. Thus the whole disjunction of (B.6)reduces to⊥. We may conclude:T C. Quod erat demonstrandum.

Remark B.1. Let’s assume thatΓ is closed under disjunction. LetW be the theoryaxiomatized by the axioms ofT , plus the negations of falseΓ -sentences. We use theobvious formula for the axiom set ofW in T . We write∗ for provability inW . SupposeB is of the form∃y B0. We write∃∃x B, for ∃y ∃x B0. Under these conventions we canrewrite the specification ofA as follows.

T A(x)↔ ∗¬A(x) ≤ ∃∃y ∗¬A(y).

It would be interesting to see a modal treatment of our argument.


Appendix C. Derivable consequence

In this appendix, we provide reformulations of some of our results in terms of derivableconsequence. LetT be a theory and letτ be a signature. We define some consequencerelations for signatureτ . Let Γ and∆ be sets of sentences of the language of signatureτ

and letA be a sentence of the language of signatureτ .

• Γ | ∆ ∗T A :⇔ ∀K (T ΓK ⇒ T +∆K AK).

• ∆ T A :⇔ ∅ | ∆ ∗T A.

• Γ ∼T A :⇔ Γ | ∅ ∗T A.

• ΛτT := A | ∅ T A.

Here it is implicitly assumed that the ‘K’ are interpretations forτ . (If τ is not clear fromthe context, we will exhibit it as superscript.) is the relation ofT -derivable consequenceand ∼ is the relation ofT -admissible consequence. Λτ

T is the predicate logic ofT (forsignatureτ ). For some remarks on admissible consequence, see [21, Appendix A]. Forsome information about derivable consequence, see [20, Subsection 12.3]. For a study ofpredicate logics of classical theories, see [23]. Here are some elementary facts about thesenotions.

Fact C.1.

(1) Γ T A⇔ ∀U ⊇ T Γ ∼U A.Here ‘U’ ranges over theories with arbitrarily complex axiom sets.

(2) A τT B ⇔ Λτ

T (A→ B).(3) If T τ A and A ∼τ

T B, then AτT B.

(4) We can find T, A, B, such that A∼τ

T B, but not AτT B.

Proof. Ad (3). SupposeK : T τ A and A ∼τ

T B. Consider any interpretationM forτ . We construct a new interpretationP as follows:P is K if ¬AM andP is M if AM.Clearly,P : T τ A. By A ∼τ

T B, it follows thatP : T τ B. Ergo:T (AM → BM).Ad (4). Letσ be the signature of arithmetic. We have

∧F ∧ con(T) ∼σ

T ⊥, for anyT . This is, in fact, Pudl´ak’s strong version of the Second Incompleteness Theorem. On theother hand, if we takeT e.g.PA, clearly

∧F ∧ con(T) #σ

T ⊥. A T-modelN of signatureτ , is a model for signatureτ that is isomorphic to an internalmodel of a modelM of T . Internal models are given by interpretationsK. We could callthe internal model ofM given byK: KM. Thus,N is a T-model iff, for some modelM |= T and for some interpretation for signatureτ , N is isomorphic toKM. We canunderstand in terms ofT-models, as follows.

Fact C.2. ∆ T A :⇔ for all T -modelsN , (N |= ∆⇒ N |= A).

Here is an example illustrating the non-compactness of∼ .

Example C.3. Let σ be the signature of arithmetic. LetT be a finitely axiomatized,consistent, sequential theory. LetU := F + conn(T) | n ∈ ω. (Here, we can useeither the complexity measure ‘depth of connectives’ or the measure ‘depth of quantifierchanges’.) Then, sinceU is locally, but not globally interpretable inT , we find that ∼ isnot compact.


To provide an example to illustrate the non-compactness of, we need a result of JanKrajıcek.

Theorem C.4. Let T = 〈T,N 〉 be I∆0 or let T be finitely axiomatized, consistent andsequential. There is a mapping I→ kI , from T -cuts to natural numbers, such that thetheory

kraj(T ) := T + inconIkI

(T) | I is a T -cutis locally interpretable inT , and, hence, consistent.12 Here the complexity measure usedis depth of connectives. We can, however, also use depth of quantifier changes.

For a proof, see [6, Section 3]. The functionality suggested by our notation ‘kraj(T )’ is parabus de langage, since the theory does not seem to be uniquely determined by the data. Infact, I have the following conjecture.

Open Question C.5. Prove or refute the following conjecture. There are infinitelymany theories satisfying the description ofkraj(T ) that are pairwise not mutuallyinterpretable.

By construction, the theorykraj(T ) is not trustworthy. It follows fromTheorem 5.9thatkraj(T ) is not globally interpretable inT . We can now present the promised example forthe non-compactness of.

Example C.6. Let T = 〈T,N 〉 be I ∆0 or let T be finitely axiomatized, consistent andsequential. LetU := F + conn(T) | n ∈ ω. Now it is easy to see thatU kraj(T ) ⊥,but that, for no finite subtheoryU0 of U , we haveU0 kraj(T ) ⊥. Hence, is notcompact.

Our notions have at most complexityΠ 02 . The following theorem shows that the worst may

happen.

Theorem C.7. There is a theory W such thatΛσW is completeΠ 0

2 . Hereσ is the signatureof arithmetic. It follows that∗, , ∼ andΛ assume their maximal possible complexities.

Proof. Let T = 〈T,N 〉 be I ∆0 or finitely axiomatized, consistent and sequential. LetW = 〈W,N 〉 := kraj(T ) as inTheorem C.4. We show thatΛσ

W is completeΠ 02 .

Consider the sentenceA := ∀x ∃y A0(x, y), whereA0 ∈ ∆0. Let Sx := ∃y A0(x, y).Let Rx be the FGH sentence forW andSx. We define:

Q := ∀x (conx(T) → Rx).

We show thatA iff ΛσW ∧

F → Q.Suppose thatA. Let K be any interpretation for the signatureσ . Consider the

interpretationM such that, inT ,M isK if (∧

F)K andN , otherwise. Clearly,M : T F.

12By inspecting the argument, it becomes clear that Krajıcek’s theory is recursively enumerable. I did notcheck that the axioms are indeed p-time decidable. However, we can always apply Craig’s trick to obtain ap-time decidable axiomatization. Note that the verification of Craig’s trick demands a metatheory containingΣ0

1 -collection.


By Fact 2.2(3), there is aT-cut I of N , such thatI ≤T M. By the construction ofW, wehaveWinconI

n(T), for somen. It follows thatWinconMn (T). We may conclude thatW((

∧F)K → inconKn (T)). It follows that (a):

W

((∧F)K → (∀x (conx(T)→ x < n))K

).

From A we can infer, for anyk, that Sk. HenceRk ∨ R⊥k . From Rk, we have, byΣ01 -

completeness,W((∧

F)K → RKk ). In case,R⊥k , we haveW Rk, by the definition ofRk,

andW R⊥k , byΣ01 -completeness. HenceW⊥, quod non. Ergo we find:W((

∧F)K →

(∧

k<n Rk)K). Thus, we get (b):

W

((∧F)K → (∀x < n Rx)

K)

.

We may conclude, combining (a) and (b), thatW((∧

F)K → QK). Thus,ΛσW ∧

F →Q.

For the converse, suppose thatΛσW ∧

F → Q. Consider anyn. Pick aT -cut I suchthatWconI

n(T). By our assumption, we haveW QI . Hence,W RIn and, so,W Rn. By

the FGH Theorem, we may conclude thatSn. We show how various notions of this paper can be formulated in a natural way in terms ofderivable and admissible consequence. Letσ be the signature of arithmetic. We need thefollowing definitions.

• Let Tn be0= 0 if n = 0 and the set of trueΠ 0n -sentences otherwise.

• SupposeF ⊆ Γ . We define:Γ nT A :⇔ Γ , Tn T A.

• SupposeF ⊆ Γ . We define:Γ ∼nT A :⇔ Γ | Tn ∗

T A.

• T, n-conadm(U) iff not U ∼nT ⊥.

• T, n-con(U) iff not U nT ⊥.

We now have:

Fact C.8.

(1) T, n-conadm(U) implies T, n-con(U).(2) T, n-con(U) iff there is aΣ0

n -sound T -model of U.(3) A theory T is consistent and numerizable iff T, 0-conadm(F).(4) If T, 0-con(Q), then T is undecidable. (This follows from Tarski’s Theorem that if an

essentially undecidable theory is interpretable in a consistent extension of a given the-ory T , then T is undecidable. In fact T, 0-con(U) iff U is weakly interpretable in T .)

(5) Let T be finitely axiomatized, consistent and sequential. Then, we have T,

1-conadm(Q). (This follows fromTheorem4.1.)(6) T is trustworthy iff T, 1-con(Q). (This follows fromTheorem5.5.) We may conclude

that T, 1-con(Q) implies T, n-con(Q), for all n.

Note thatQ in the above statements can be replaced byF or S12 or I ∆0, by the fact that

these stronger theories are interpretable on a cut inQ.


Remark C.9. Consider a ∆b1-axiomatizable theoryT satisfying T, 0-con(Q). By

Theorem 3.6, T is in Turing degree0′. William Hanf showed that there are finitelyaxiomatizedT in any recursively enumerable Turing degree. (Even that there areessentially undecidable, finitely axiomatized theories of any recursively enumerable degreeof unsolvability.) See [5]. Ergo, there are finitely axiomatized, undecidable theoriesT suchthatT, 0-incon(Q).

Appendix D. On the existential axioms of Q

In this appendix, we discuss a detail of the proof of Wilkie’s Theorem thatI ∆0 isinterpretable on an initial segment inQ. An initial segment, is a definable set of numbersQ-provably closed underS and downwards closed under≤.13

Our presentation is directly dependent on the presentation of Petr Hájek and PavelPudlak in their book [4, pp. 369, 370]. The reader is advised to first look at Hájek andPudlak’s proof. The axioms ofQ are the following.

Q1 Sx #= 0,Q2 Sx = Sy → x = y,Q3 x #= 0→ ∃y x = Sy,Q4 x + 0= x,Q5 x + Sy = S(x + y),Q6 x × 0= 0,Q7 x × Sy = (x × y)+ x,Q8 x ≤ y ↔ ∃z z+ x = y.

To prove Wilkie’s Theorem, it is convenient to take≤ as a primitive symbol. If we wouldtake it as defined by∃z z+ x = y, then we would have to state explicitly that on an initialsegmentI , the meaning of≤ is preserved, i.e. that≤I is equal to≤ I . This sameness ofmeaning is important, since we want downwards preservation ofΠ 0

1 -sentences to the initialsegment and upwards preservation ofΣ0

1 -sentences from the segment. These preservationresults are e.g. used to get initial segments with more and more∆0-induction.

Hajek and Pudlák wisely choose to treat≤ as a primitive symbol. However, on p. 369,in their proof of Wilkie’s Theorem, they stumble in the last step. They write: “. . . we cantrivially interpretQ by eliminating≤ from the language and deletingQ8”. In other words,they redefine≤. This argument won’t wash, since they need the new≤ on the initialsegment to be the restriction of the old≤ to the initial segment. Otherwise, the centralargument does not go through.

Fortunately the gap in the argument of Hájek and Pudlák is easily closed by proceedinganalogously to their verification ofQ3 on the initial segmentI : prove, by induction onx,that∀y≤ x ∃z≤x z+ y = x. (We need some auxiliary inductions to show e.g. thatx ≤ xandx ≤ Sx.)

However, the problem to verify the existential axiomsQ3 andQ8 also occurs in the caseof the interpretation of Hájek and Pudlák’s theoryQ+ in Q. For this reason, I prefer anotherstrategy to settle the problem of these axioms for once and for all, right from the start.

13Our initial segmentis Hajek and Pudlák’s cut.


We work in Q. We use the easily verifiable theorem thatx + y = 0 → x = y = 0.A numberx is L-successiveiff ∀y ∀z (y + z = x → Sy + z = Sx). We show that 0 isL-successive. Suppose thaty+z= 0. Theny = z= 0. Moreover,Sy+z= S0+0= S0.Next we show that the L-successive numbers are closed under successor. Supposex is L-successive and supposey+ z= Sx. We want to show thatSy+ z= SSx. In casez= 0,we havey = Sx and, hence,Sy+0= Sy = SSx. In casey = Su, we havey+Su = Sx.So,S(y+ u) = Sx. Ergoy+ u = x. Sincex is L-successive, we haveSy+ u = Sx and,so,S(Sy + u) = SSx. We may conclude thatSy + Su = SSx.

A numberx is a commutatoriff ∀y ∀z (y + z = x → z+ y = x). We say thatx isa strong commutatoriff x is L-successive andx is a commutator. We already know that 0is L-successive. Moreover, ify + z = 0, theny = z = 0, and, hence,z+ y = 0. So 0is a strong commutator. We show that the strong commutators are closed under successor.Supposex is a strong commutator. By the above argument,Sx is L-successive. Supposey + z = Sx. To show:z+ y = Sx. First supposez = 0. We havey = y + 0 = Sx. Sowe need to show that 0+ Sx = Sx. We havex + 0 = x. So, sincex is a commutator, wefind 0+ x = x and, hence 0+ Sx = Sx. Next, supposez = Su. We havey+ Su = Sx.Then,y + u = x. Hence, sincex is a commutator, we haveu + y = x. Ergo, sincex isL-successive,Su+ y = Sx.

Theorem D.1 (in Q). Suppose, that the elements of an initial segment I are allcommutators. Then, the segment I verifiesQ3 andQ8.

Proof. Reason inQ. SupposeSx is in I . We havex + S0 = Sx. Since,Sx is acommutator, we findS0 + x = Sx. Ergo x ≤ Sx. Hencex ∈ I . Thus any non-zeronumber inI has a predecessor inI .

Supposex ≤ y, for y ∈ I . Then, for somez, z+ x = y. Since,y is a commutator, wefind x + z= y, and soz≤ y. Hencez ∈ I .

Now we execute the remaining part of the proof of Wilkie’s Theorem inside thestrong commutators, without worries aboutQ3 andQ8. We need closure of the strongcommutators under successor to construct the appropriate initial segments.

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