aa r X i v : . [ m a t h . L O ] A p r A MATHEMATICAL COMMITMENT WITHOUTCOMPUTATIONAL STRENGTH
ANTON FREUND
Abstract.
We present a new manifestation of G¨odel’s second incomplete-ness theorem and discuss its foundational significance, in particular with re-spect to Hilbert’s program. Specifically, we consider a proper extension ofPeano arithmetic ( PA ) by a mathematically meaningful axiom scheme thatconsists of Σ -sentences. These sentences assert that each computably enu-merable (Σ -definable without parameters) property of finite binary trees hasa finite basis. Since this fact entails the existence of polynomial time al-gorithms, it is important for computer science. On a technical level, our ax-iom scheme is a variant of an independence result due to Harvey Friedman. Atthe same time, the meta-mathematical properties of our axiom scheme distin-guish it from most known independence results: Due to its logical complexity,our axiom scheme does not add computational strength. The only knownmethod to establish its independence relies on G¨odel’s second incompletenesstheorem. In contrast, G¨odel’s theorem is not needed for typical examples ofΠ -independence (such as the Paris-Harrington principle), since computationalstrength provides an extensional invariant on the level of Π -sentences. Summary of mathematical results
This paper consists of mathematical results and a foundational discussion. Theformer are summarized in the present section; the latter can be found in Section 2.In the remaining sections we provide detailed proofs of all mathematical claims.First and foremost, our paper is based on a result by Dick de Jongh (unpublished;cf. the introduction to [34]) and Diana Schmidt [35]: The embeddability relation onfinite binary trees yields a well partial order with maximal order type ε (see belowfor an explanation). Harvey Friedman [37] has show that this type of result yieldsstatements of finite combinatorics that are independent of important mathematicalaxiom systems. Against this background, many arguments in the present papermay be considered folklore. Nevertheless we find it worthwhile to give an explicitpresentation, not least because the arguments are rather sensitive with respect toquantifier complexity and the presence of parameters. At some places we providemore details than the expert may find necessary. The aim is to make the paper asaccessible and self-contained as possible.We write B for the set of finite binary trees. More precisely, we assume that eachtree has a distinguished root node, that nodes have either zero or two children, andthat left and right child can be distinguished. Furthermore, we identify isomorphictrees. Formally, we view B as the least fixed point of the following inductive clauses:(i) There is an element ◦ ∈ B (the tree that consists of a single root node). Mathematics Subject Classification.
Key words and phrases.
Independence, computational strength, G¨odel’s second incompletenesstheorem, Hilbert’s program, Kruskal’s theorem, polynomial-time algorithm. (ii) Given s and t in B , we obtain an element ◦ ( s, t ) ∈ B (the tree in which theroot has left subtree s and right subtree t ).For s, t ∈ B we write s ≤ B t if there is a tree embedding of s into t . Such anembedding can either map the root to the root and the immediate subtrees of s into the corresponding subtrees of t ; or it maps all of s into one subtree of t . Hencewe have ◦ ≤ B t for any t ∈ B ; we have s ≤ B ◦ precisely for s = ◦ ; and we have ◦ ( s , s ) ≤ B ◦ ( t , t ) ⇔ ( s ≤ B t and s ≤ B t , or ◦ ( s , s ) ≤ B t i for some i ∈ { , } . These clauses provide a recursive definition of ≤ B .Recall that a partial order consists of a set X and a binary relation ≤ X on X thatis reflexive, antisymmetric and transitive. A finite or infinite sequence x , x , . . . in X is called good if there are indices i < j such that we have x i ≤ X x j ; otherwise,the sequence is called bad. If there is no infinite bad sequence, then ( X, ≤ X ) iscalled a well partial order (wpo). Equivalently, a partial order ( X, ≤ X ) is a wpoif, and only if, every subset Y ⊆ X has a finite “basis” a ⊆ Y with the followingproperty: for any y ∈ Y there is an x ∈ a with x ≤ X y (cf. the argument inRemark 3.1 below).If X is a wpo, then all its linearizations are well orders (since a strictly decreasingsequence in a linearization would be a bad sequence in X ). Hence the order type ofeach linearization is an ordinal number. The supremum of these ordinals is calledthe maximal order type of X . As shown by D. de Jongh and R. Parikh [22], themaximal order type of any wpo is realized by one of its linearizations (i. e. thesupremum is a maximum).Kruskal’s theorem [27] implies that ( B , ≤ B ) is a well partial order. We point outthat the theorem applies to arbitrary (i. e. not necessarily binary) finite trees; the“most general” version of Kruskal’s theorem is investigated in [12]. Concerning thebinary case, de Jongh and Schmidt have proved the finer result that B has maximalorder type ε , which is the least fixed point of ordinal exponentiation with base ω (read [35, Theorem II.2] in combination with the example after [35, Definition I.15]).A classical result of G. Gentzen [16, 17] establishes ε as the proof theoretic ordinalof Peano arithmetic ( PA ). This explains the connection with independence results.In the present paper we consider the binary Kruskal theorem in the context offirst order arithmetic; an introdution to this setting can be found in [18]. We will beparticularly interested in questions of quantifier complexity: Recall that a formulalies in the class ∆ = Σ = Π if it does only contain bounded quantifiers. Sincethe latter range over a finite domain, the truth of closed ∆ -formulas is uniformlydecidable. A Σ n +1 -formula (Π n +1 -formula) has the form ∃ x ϕ (the form ∀ x ϕ ),where ϕ is a Π n -formula (Σ n -formula). Recall that the Σ -formulas correspondto the computably enumerable relations. A relation is ∆ -definable (in PA ) if ithas a Σ -definition and a Π -definition (which PA proves to be equivalent). The∆ -relations coincide with the decidable ones.Working in PA , the elements of B can be represented by numerical codes forfinite sets of sequences with entries from { , } . Note that the relations s ∈ B and s ≤ B t are ∆ -definable in PA . As mentioned above, the fact that B is a wpo canbe expressed in terms of a finite basis property. To state the latter we abbreviate ∃ fin a ψ ( a ) : ≡ ∃ a (“ a ∈ N codes a finite set” ∧ ψ ( a )) . MATHEMATICAL COMMITMENT WITHOUT COMPUTATIONAL STRENGTH 3
In the context of PA it is natural to focus on definable sets. Given a formula ϕ ( s )with a distinguished free variable, the finite basis property for { s ∈ B | ϕ ( s ) } ⊆ B can be formalized as K ϕ : ≡ ∃ fin a ⊆B ( ∀ s ∈ a ϕ ( s ) ∧ ∀ t ∈B ( ϕ ( t ) → ∃ s ∈ a s ≤ B t )) . Note that the quantifiers with subscript s ∈ a are bounded, since a is a code for afinite set (cf. [18, Lemma I.1.32]); in contrast, the quantifiers with subscripts a ⊆ B and t ∈ B are unbounded. The symbol K alludes to Kruskal’s theorem, whichimplies that all instances K ϕ are true (see Remark 3.1 for details). We will be mostinterested in the axiom scheme K Σ − := {K ϕ | “ ϕ a Σ -formula with exactly one free variable” } . The superscript of Σ − emphasizes the fact that no further free variables are allowed.This ensures that each instance of K Σ − is a closed Σ -formula.To motivate the restrictions on the quantifier complexity and the parameters,we recall the notion of computational strength: A computable function f : N → N is provably total in a suitable theory T if the latter proves ∀ x ∃ ! y ϕ ( x, y ) for someΣ -definition ϕ of the graph of f (where ∃ ! abbreviates the existence of a uniquewitness). The computational strength of a theory is commonly identified with thecollection of its provably total computable functions.It is known that the computational strength of a theory does not increase whenwe add a true Π -sentence ψ as an axiom. Essentially, this is due to the fact that theΣ -formula ψ → ϕ ( x, y ) defines the same graph as ϕ ( x, y ) (note that the definitionof provably total function is extensional). A simple but fundamental observationshows that the same is true for closed Σ -axioms: It suffices to note that any trueΣ -sentence ∃ x ψ ( x ) follows from some true Π -instance ψ ( n ) (see Proposition 3.2for details). Note that we may not be able to compute the correct witness n ∈ N ;this issue will resurface at the end of the present section.The general facts from the previous paragraph imply that PA + K Σ − has thesame computational strength as PA . At this point it is crucial that we excludeparameters: If the Σ -formula ϕ contains further free variables, then the universalclosure of K ϕ is a Π -formula, so that our argument does no longer apply. Notethat the version with parameters can be expressed by a single Π -sentence (ratherthan a scheme), due to the existence of a universal computably enumerable set.Next, we explain why PA + K Σ − is a proper extension of PA . Based on a nota-tion system for the ordinal ε (see Section 4 for details), transfinite induction canbe expressed in first order arithmetic: Given a formula ψ ( α ) with a distinguishedfree variable, we set T I ( ε , ψ ) : ≡ ∀ γ ≺ ε ( ∀ β ≺ γ ψ ( β ) → ψ ( γ )) → ∀ α ≺ ε ψ ( α ) . The scheme of parameter-free Π -induction up to ε is the collection T I ( ε , Π − ) := {T I ( ε , ψ ) | “ ψ a Π -formula with exactly one free variable” } . In Section 4 we show that each instance of
T I ( ε , Π − ) can be proved in PA + K Σ − .This is a straightforward consequence of the fact that ε is bounded by (and in factequal to) the maximal order type of B . Nevertheless we find it worthwhile togive a detailed proof, which pays attention to the quantifier complexities and therole of parameters. Gentzen [16] has used Π -induction up to ε to establish theconsistency of PA . This induction does not require parameters, as we will check in ANTON FREUND
Section 5. Hence the consistency of PA can be proved in PA + K Σ − . The lattermust thus be a proper extension, due to G¨odel’s second incompleteness theorem.In Section 6 we review the proof that B has maximal order type ε . Based on thisfact, we can also show that each instance of K Σ − is provable in PA + T I ( ε , Π − ). Tocomplete the picture, we relate transfinite induction and reflection. Let Pr PA ( ϕ ) bea standard formalization of the statement that the formula with code ϕ is provablein PA (see [18, Section I.4(a)]; we will also write ϕ for p ϕ q ). Given a sentence ϕ of first order arithmetic, we putRfn PA ( ϕ ) : ≡ Pr PA ( ϕ ) → ϕ. The local (i. e. parameter-free) Σ -reflection principle over PA is the collectionRfn PA (Σ ) : ≡ { Rfn PA ( ϕ ) | “ ϕ a closed Σ -formula” } . Due to G. Kreisel and A. L´evy [26], uniform reflection over PA is equivalent to ε -induction for formulas with parameters. We will show that the proof can beadapted to the parameter free case. This results in Theorem 7.3, which asserts PA + K Σ − ≡ PA + T I ( ε , Π − ) ≡ PA + Rfn PA (Σ ) . In view of Goryachev’s theorem, we can conclude the following (see Corollary 7.4):Over Peano arithmetic, the Π -consequences of K Σ − are precisely those of the fi-nitely iterated consistency statements for PA . Due to another result of Kreisel andL´evy [26], we can also deduce that PA + K Σ − is not contained in any consistent ex-tension of PA by a computably enumerable set of Π -sentences (see Corollary 5.2). Acknowledgements.
I am very grateful to Lev Beklemishev for our inspiring discus-sions and his helpful comments on a first version of this paper.2.
Foundational considerations
In the previous section we have presented an extension of Peano arithmetic byan axiom scheme K Σ − that is related to Kruskal’s theorem. The present section isconcerned with the foundational significance of this extension.Let us first recall some aspects of Hilbert’s program; for a more thorough discus-sion and further references we refer to the introduction by R. Zach [47]. To securethe abstract methods that are central to modern mathematics, Hilbert wanted tojustify them by finitist reasoning about natural numbers, which he views as “extra-logical concrete objects that are intuitively present as immediate experience priorto all throught” [20, p. 171]. (All quotations from [20, 21] are translated as in [46].)The status of the natural numbers entails that certain statements about them arefinitistically meaningful. This includes, first of all, statements which assert that agiven tuple of numbers satisfies some primitive recursive relation. Such a statementcan be verified explicitly, which explains its priviledged role, but also entails—asHilbert [20, p. 165] puts it—that it is “of no essential interest when considered byitself”. In addition, one admits universal statements with verifiable instances. Ac-cording to Hilbert [20, p. 173], such a statement can be accepted as “a hypotheticaljudgement that comes to assert something when a numeral is given”. In contrast,unbounded existential statements are not seen as finitistically meaningful, as “onecannot [. . . ] try out all numbers” [21, p. 73]. At the same time, Hilbert [21, p. 77f]emphasizes the fact that existential statements play an extremely fruitful role inabstract mathematics. One could even be tempted to say that abstract notions MATHEMATICAL COMMITMENT WITHOUT COMPUTATIONAL STRENGTH 5 acquire meaning through their role in the mathematical development, a positionthat seems to resonate with the following statement by Hilbert [21, p. 79]:“To make it a universal requirement that each individual formula[. . . ] be interpretable by itself is by no means reasonable; on thecontrary, a theory by its very nature is such that we do not need tofall back upon intuition or meaning in the midst of some argument.”However, such a conception of meaning is very different from the finitist one.The extent of finitist reasoning is commonly identified with primitive recursivearithmetic (
PRA ). This identification has been justified by W. Tait [42]; in [43]he lists and refutes some objections. A quantifier-free formulation seems to bemost appropriate: In a such a setting, one can only express statements that arefinistically meaningful; universal statements correspond to open formulas. To makeour considerations as accessible as possible, we will, nevertheless, work in the usualframework of first order arithmetic with quantifiers. Following C. Smorynski [40],we agree to identify the finitistically meaningful statements with the Π -sentences.More specifically, then, Hilbert’s program suggested to formalize all of abstractmathematics as an axiom system T . In order to obtain a finitist justification,one was supposed to prove the consistency of T in the theory PRA . At thispoint it is important to note that consistency is not merely a minimal requirement:If the consistency of a theory T is provable in PRA , then the latter proves allΠ -theorems of T , i. e. all results that are finitistically meaningful (see [21, p. 78f]).G¨odel’s incompleteness theorems show that Hilbert’s program cannot be carriedout: It is impossible for T to prove its own consistency; a fortiori, the consistencyof T cannot be established in the weaker theory PRA .Despite G¨odel’s theorems, the aims of Hilbert’s program have been achieved toan astonishing extent: A substantial part of contemporary mathematics can indeedbe formalized in rather weak axiom systems (see e. g. the work of S. Feferman [8],as well as U. Kohlenbach’s proof mining program [23]). In view of these positiveresults, it is all the more intriguing to ask: Are there natural mathematical theoremsthat can be expressed but not proved in
PRA , or in some stronger theory? To countas a natural theorem, the unprovable statement should arise from mathematicalpractice; it should not involve the logical notions of proof or model. In particular,consistency statements (which are unprovable by G¨odel’s theorem) are not seen asexamples of this type.We do have good examples of true Π -statements that are unprovable in relevantaxiom systems: The Paris-Harrington principle cannot be proved in Peano arith-metic [30]; Friedman’s miniaturization of Kruskal’s theorem is independent of aneven stronger system [37], which is associated with predicative mathematics. Thesituation is less satisfactory when it comes to Π -sentences, which are most import-ant from the finitist viewpoint: The independent statement due to S. Shelah [36]involves notions from model theory, so that its status as a natural mathematicaltheorem can be questioned. Friedman has presented work on Π -independence fromZermelo-Fraenkel set theory (see e. g. [13]), but his results are not yet published infinal form. In our opinion, the search for mathematical Π -sentences that are inde-pendent of relevant axiom systems remains one of the most interesting challengesin mathematical logic.The axiom scheme K Σ − from the previous section does not settle the challengeof natural Π -independence. The latter can, nevertheless, serve as a benchmark ANTON FREUND that helps us to assess the foundational significance of K Σ − . In the rest of thissection we carry out such an assessment.First, we will argue that K Σ − is a natural mathematical commitment. In theprevious section we have seen that K Σ − is a restricted version of Kruskal’s theorem.The latter is firmly established as a natural result of mathematical practice. Henceit remains to argue that the restrictions that lead to K Σ − are natural as well.In formulating K Σ − , we have restricted Kruskal’s theorem in two ways: Firstly,we have decided to work with binary rather than arbitrary finite trees. This restric-tion makes it easier to determine the precise strength of K Σ − (i. e. to prove the equi-valence with transfinite induction and local reflection), but it is not essential: If weextend our axiom scheme to arbitrary finite trees, then it will imply the consistencyof stronger axiom systems; at the same time, it will still not increase the computa-tional strength, since it also consists of Σ -statements. The graph minor theoremof N. Robertson and P. Seymour [33] suggests a very intriguing axiom scheme thatis even stronger (cf. [14]) but does not have computational strength either (forthe same general reason). In summary, the restriction to binary trees is purelypragmatic and does not change the general foundational behaviour. Secondly, thescheme K Σ − is a restriction of Kruskal’s theorem insofar as it demands a finite basisfor computably enumerable—rather than arbitrary—sets of trees. In the followingwe give two justifications for the restriction to computably enumerable sets.The first justification is that K Σ − suffices for certain applications in computerscience: Assume that P is an upwards closed property of finite binary trees, whichmeans that P ( s ) and s ≤ B t imply P ( t ). Often (but not always, cf. [9, Theorem 3])one will already know that P is decidable. Then P can be defined by a Σ -formula,and K Σ − yields a finite a ⊆ B such that P ( t ) is equivalent to ∃ s ∈ a s ≤ B t . Thelatter can be decided in polynomial time (in the size of t ). The author knows of noconcrete applications in the context of trees, but the analogous argument for thegraph minor relation has many applications (see e. g. [10]).The second justification for the restriction to computably enumerable sets isbased on the idea that one can have reasons to accept K Σ − but not the full Kruskaltheorem for binary trees. To make this plausible we recall that K Σ − is equivalentto parameter-free Π -induction up to ε . The latter is no stronger than inductionfor decidable (i. e. finitistically meaningful) properties, still up to ε (see e. g. [41,Lemma 4.5]). From a finitist standpoint it makes sense to accept this inductionprinciple but not the second order statement that ε is well-founded, which wouldbe required for the binary Kruskal theorem. Indeed, Tait [43, p. 411] states thatKreisel [24] accepts quantifier-free induction up to each ordinal below ε as finit-ist. Also, G. Takeuti’s justification of transfinite induction is supposed to “involve‘Gedankenexperimente’ [thought experiments] only on clearly defined operationsapplied to some concretely given figures” [44, p. 97].Next, we discuss the fact that K Σ − is a scheme rather than a single statement.In the previous section we have explained that PA + K Σ − proves the consistencyof PA . Of course, this proof involves only finitely many instances K ϕ , . . . , K ϕ n .However, we see no basis for the claim that these particular instances constitute anatural mathematical commitment—in contrast to the axiom scheme as a whole. Inthis sense our reference to an axiom scheme is essential. What does this entail? Wethink that the answer depends on our attitude towards independence phenomena. MATHEMATICAL COMMITMENT WITHOUT COMPUTATIONAL STRENGTH 7
One possibility is to think of independent statements as “unsolvable conjectures”.More explicitly, one might imagine a mathematician immersed in Peano arithmetic,who is challenged to prove or refute the Paris-Harrington principle. The independ-ence result tells us that this mathematician can never succeed. This conceptionof independence is clearly concerned with single statements rather than schemes.However, one can also think of independence in terms of “potential axioms”. Forexample, one may view the principle of induction for arbitrary first order formulasas a mathematical commitment beyond the finitist standpoint. This example showsthat schemes play a natural role within such a conception of independence.A broad conception of independence may even incorporate rules, in addition toaxiom schemes. In the present context it is interesting to consider the rule ∀ γ ≺ ε ( ∀ β ≺ γ ψ ( β ) → ψ ( γ )) ∀ α ≺ ε ψ ( α )of Π -induction along ε , which allows us to infer ∀ α ≺ ε ψ ( α ) once we have givena proof of ∀ γ ≺ ε ( ∀ β ≺ γ ψ ( β ) → ψ ( γ )), where ψ ( α ) can be any Π -formula withoutfurther free variables. Note that the rule does not commit us to the contrapositive ofthe corresponding axiom, i. e. to the least element principle. Hence the rule avoidscertain existential commitments, which is well motivated in a finitist context. Asshown by L. Beklemishev [3, Theorem 3], the closure of PA under the rule ofΠ -induction along ε proves the same theorems as the extension of PA by finitelyiterated consistency statements. Note that the rule does not refer to logical notionssuch as proof or model. Insofar as induction up to ε is a result of mathematicalpractice, we have a mathematical commitment on the level of Π -statements.Finally, we discuss the fact that K Σ − consists of Σ -statements rather thanΠ -statements. At the end of the previous section we have mentioned that thereis no computably enumerable set Ψ of Π -sentences (or even Π -sentences) suchthat PA + Ψ is consistent and contains PA + K Σ − . This shows that our use ofΣ -sentences is essential in a rather strong sense.As mentioned above, many of the known independence results for PA are con-cerned with Π -sentences. Extending Hilbert’s view on Π -sentences, one couldsee Π -sentences as “hypothetical judgement[s]” [20, p. 173] of complexity Σ .This might suggest that Π -sentences are less abstract—in the finitist sense—thanΣ -statements. From this viewpoint, the independence of K Σ − would be less signi-ficant than the known independence results. An argument that supports the signi-ficance of Σ -independence will be given below. First, we give another explanationfor the fact that Π -independence is more prominent in the existing literature.Gentzen’s ordinal analysis shows that each purported proof of a contradictioncan be reduced to a proof with smaller ordinal label. To establish consistency, onecan use this reduction in two different ways: In the present paper, we invoke induc-tion on α ≺ ε to show that no proof with label α can produce a contradiction. Thisavoids parameters but involves a universal quantification over proofs with given or-dinal label; it leads to local Σ -reflection, which has complexity Σ . Alternatively,Gentzen’s reduction shows that a purported proof p of a contradiction leads toa strictly decreasing sequence of ordinals, which is primitive recursive with para-meter p . One can then invoke the primitive recursive well-foundedness of ε . Thisleads to uniform Σ -reflection (see [15, Theorem 4.5]), which is a Π -statement. Itseems that the second approach is preferred in the finitist literature. For example,Takeuti writes that the consistency proof is based on the following [44, p. 92]: ANTON FREUND “Whenever a concrete method of constructing decreasing sequencesof ordinals is given, any such decreasing sequence must be finite.”This preference may help to explain the pre-eminence of Π -independence. As anexception, we mention that L. Beklemishev and A. Visser [2] have characterizedthe Σ n -consequences of PA (and of its fragments) in terms of iterated reflection.Kreisel [25] has initiated work on finiteness theorems of complexity Σ , but herethe focus is on proof-mining rather than independence.The significance of Σ -independence is related to the notions of provably totalfunction and computational strength, which we have recalled in the previous section.An independent Π -statement will typically add a provably total function: For theParis-Harrington principle this is the case by [30, Theorem 3.2]; the general claimis plausible in view of [15, Theorems 2.24 and 4.5] and [39, Theorem 5]. In contrast,we have seen that K Σ − does not increase the computational strength of PA .The fact that K Σ − does not add provably total functions is interesting in itsown right, but it becomes even more relevant in view of the following: The notionof computational strength is a relatively robust extensional invariant. Bounds onprovably total functions can be established without the use of G¨odel’s theorem,e. g. by induction over cut-free infinite proofs (see [5]). This means that G¨odel’stheorem is not needed to prove that the Paris-Harrington principle is independentof PA (see [6] for an analogous argument with respect to Goodstein’s theorem). Itappears that no similar invariants are available on the level of Σ -statements. Theonly known proof of the fact that PA does not prove all instances of K Σ − appealsto G¨odel’s theorem. In our opinion, this means that K Σ − is a conceptually differentand foundationally significant manifestation of mathematical independence.3. Analyzing the computational strength
In this section we give a detailed proof of the claim that K Σ − does not increasethe computational strength of PA . As preparation, we need to show that all in-stances of K Σ − are true. In the following remark we argue in a strong meta theory;this will later be superseded by a proof in PA + T I ( ε , Π − ) (see Proposition 7.2). Remark 3.1.
As a consequence of Kruskal’s theorem [27], the partial order ( B , ≤ B )does not contain any infinite bad sequence. We will use this fact to justify anarbitrary instance K ϕ ≡ ∃ fin a ⊆B ( ∀ s ∈ a ϕ ( s ) ∧ ∀ t ∈B ( ϕ ( t ) → ∃ s ∈ a s ≤ B t ))of the axiom scheme K Σ − . Aiming at a contradiction, assume that K ϕ is false.By a bad ϕ -sequence we mean a bad sequence t , t , . . . ⊆ B such that ϕ ( t i ) holdsfor each i . Note that the empty sequence is a bad ϕ -sequence. Furthermore, eachbad ϕ -sequence t , . . . , t n − can be extended into a bad ϕ -sequence t , . . . , t n − , t n .To see that this is the case, consider a := { t , . . . , t n − } . As ∀ s ∈ a ϕ ( s ) holds, theassumption that K ϕ is false yields an element t n ∈ B with ϕ ( t n ) and ∀ s ∈ a s B t n .The latter ensures that t , . . . , t n − , t n is still bad. By dependent choice we nowget an infinite bad ϕ -sequence, which contradicts Kruskal’s theorem.As explained in the introduction, the following is due to the general fact that K Σ − consists of true Σ -sentences. The argument is folklore, but we provide detailsin order to make the paper as accessible as possible. Proposition 3.2.
The provably total functions of PA + K Σ − and of PA coincide. MATHEMATICAL COMMITMENT WITHOUT COMPUTATIONAL STRENGTH 9
Proof.
Consider a provably total function f : N → N of PA + K Σ − . For someΣ -definition θ ( x, y ) of the graph of f , there are Σ -formulas ϕ , . . . , ϕ n − (eachwith a single free variable) such that we have PA + {K ϕ i | i < n } ⊢ ∀ x ∃ ! y θ ( x, y ) . To show that f is a provably total function of PA , we will define the graph of f bya modified Σ -formula θ ′ ( x, y ) such that PA alone proves ∀ x ∃ ! y θ ( x, y ) ′ . For thispurpose we observe that the conjunction K ϕ ∧ · · · ∧ K ϕ n − is equivalent to a trueΣ -sentence ∃ m ψ ( m ). Pick a number n ∈ N such that the Π -sentence ψ ( n ) is true.Then write ∃ z θ ( x, y, z ) ≡ ψ ( n ) → θ ( x, y )for a ∆ -formula θ . Since ψ ( n ) is true and implies each instance K ϕ i , we do have f ( k ) = m ⇔ N (cid:15) ∃ z θ ( k, m, z ) , PA ⊢ ∀ x ∃ y ∃ z θ ( x, y, z ) . However, if PA does not prove ψ ( n ), then it will not prove that the value y is unique.It is well known that one can restore uniqueness by minimizing over the code of thepair h y, z i . Note that minimizing over y alone would lead out of the Σ -formulas:the minimal y that satisfies ∃ z θ ( x, y, z ) is specified by a ∆ -formula. To providedetails we write w = h y, z i for a ∆ -definition of Cantor’s pairing function; recallthat w = h y, z i implies y, z ≤ w . Let θ ′ ( x, y ) be the Σ -formula ∃ w ( ∃ z ≤ w ( w = h y, z i ∧ θ ( x, y, z )) ∧ ∀ w ′ In this section we show that PA + K Σ − proves each instance of T I ( ε , Π − ). Aswe will see, it follows that PA + K Σ − is a proper extension of PA . The result ofthis section is a relatively straightforward consequence of the existing literature. Weprovide details in order to demonstrate that the argument works out with respectto formula complexity and the role of parameters.Let us first recall the usual notation system for ordinals below ε . According toCantor’s normal form theorem, any ordinal α can be uniquely written as α = ω α + · · · + ω α n − with α (cid:23) α (cid:23) · · · (cid:23) α n − , where α = 0 arises from n = 0. For α ≺ ε = min { γ | ω γ = γ } we have α ≺ α .Recursively, this yields finite terms that represent all ordinals below ε . Workingin PA , one can develop basic ordinal arithmetic in terms of the resulting notationsystem (see e. g. [31, 41]). In the following we always refer to term representationsrather than actual ordinals.In the introduction we have defined a set B of binary trees and an embeddabilityrelation ≤ B . To establish a connection with the ordinals below ε , it is convenientto have a binary normal form: If α ≻ α = NF ω β + γ for β = α and γ = ω α + · · · + ω α n − . Note that β and γ can be seen as proper subterms of α . The following constructionis well-known (cf. [44, § Definition 4.1 ( PA ) . We construct a function f : ε → B by setting f ( α ) = ( ◦ if α = 0 , ◦ ( f ( β ) , f ( γ )) if α = NF ω β + γ ,which amounts to a recursion over term representations of ordinals.Concerning the formalization in PA , we note that f is primitive recursive. Hence f is PA -provably total. In particular, the graph of f is ∆ -definable in PA . Thefollowing folklore result shows that f satisfies the definition of a quasi embedding. Lemma 4.2 ( PA ) . For α, β ≺ ε , the inequality f ( α ) ≤ B f ( β ) implies α (cid:22) β .Proof. Define a height function h : ε → N by recursion over terms, setting h ( α ) = ( α = 0 , max { h ( γ ) , h ( δ ) } + 1 if α = NF ω γ + δ .The claim from the lemma can now be verified by induction over h ( β ). For α = 0the implication holds because α (cid:22) β is true. In the remaining case we may write α = NF ω γ + δ . By the definition of ≤ B , the inequality f ( α ) = ◦ ( f ( γ ) , f ( δ )) ≤ B f ( β )fails for f ( β ) = ◦ . Hence we may also assume β ≻ 0, say β = NF ω γ ′ + δ ′ . Again bythe definition of ≤ B , the inequality f ( α ) = ◦ ( f ( γ ) , f ( δ )) ≤ B ◦ ( f ( γ ′ ) , f ( δ ′ )) = f ( β )can hold for two reasons: First assume we have f ( γ ) ≤ B f ( γ ′ ) and f ( δ ) ≤ B f ( δ ′ ).In view of h ( γ ′ ) , h ( δ ′ ) < h ( β ), the induction hypothesis yields γ (cid:22) γ ′ and δ (cid:22) δ ′ .By basic ordinal arithmetic we get α = ω γ + δ (cid:22) ω γ ′ + δ ′ = β. Now assume f ( α ) ≤ B f ( β ) holds because we have f ( α ) ≤ B f ( γ ′ ) or f ( α ) ≤ B f ( δ ′ ).Inductively we get α (cid:22) γ ′ (cid:22) ω γ ′ or α (cid:22) δ ′ . Either way we have α (cid:22) ω γ ′ + δ ′ = β . (cid:3) In addition to the lemma itself, we will need the following standard consequence: Corollary 4.3 ( PA ) . The function f : ε → B is injective.Proof. Consider α, β ≺ ε with f ( α ) = f ( β ). A straightforward induction over B shows that ≤ B is reflexive. Hence we have f ( α ) ≤ B f ( β ) and f ( β ) ≤ B f ( α ). Bythe previous lemma this implies α (cid:22) β and β (cid:22) α . Since the order relation on theordinals is antisymmetric, we obtain α = β . (cid:3) We can now show that the finite basis property implies transfinite induction.The converse implication will be established in Section 7. Proposition 4.4. Each instance of T I ( ε , Π − ) can be proved in PA + K Σ − .Proof. Working in PA + K Σ − , we establish T I ( ε , ψ ) for a given Π -formula ψ with a single free variable. For this purpose we consider the formula ϕ ( t ) : ≡ t ∈ B ∧ ∃ α ≺ ε ( f ( α ) = t ∧ ¬ ψ ( α )) , where f : ε → B is the function from Definition 4.1. Since the graph of f is∆ -definable in PA , we see that ϕ ( t ) is (provably equivalent to) a Σ -formula with MATHEMATICAL COMMITMENT WITHOUT COMPUTATIONAL STRENGTH 11 the single free variable t . Hence we may use K ϕ to get (a code for) a finite set a ⊆ B that satisfies ∀ s ∈ a ϕ ( s ) ∧ ∀ t ∈B ( ϕ ( t ) → ∃ s ∈ a s ≤ B t ) . First assume that a is empty. Then ∃ s ∈ a s ≤ B t fails for all t ∈ B , so that the secondconjunct enforces ∀ t ∈B ¬ ϕ ( t ). Given α ≺ ε , it is straightforward to see that ¬ ϕ ( t )for t := f ( α ) ∈ B implies ψ ( α ). We thus have ∀ α ≺ ε ψ ( α ), which is the conclusionof T I ( ε , ψ ). Now assume that the finite set a ⊆ B is non-empty. Due to ∀ s ∈ a ϕ ( s ),we see that a is contained in the range of f . Also recall that f is injective. Byinduction on the cardinality of a , one can infer that there is an ordinal γ ≺ ε with f ( γ ) ∈ a ∧ ∀ δ ≺ γ f ( δ ) / ∈ a. Given an ordinal γ with this property, we now establish ∀ β ≺ γ ψ ( β ) ∧ ¬ ψ ( γ ) , which implies that T I ( ε , ψ ) holds because its antecedent fails. Aiming at the firstconjunct, we consider an ordinal β ≺ γ . If ψ ( β ) was false, then ϕ ( t ) would holdfor t := f ( β ) ∈ B . Since a ⊆ B witnesses the conclusion of K ϕ , we would getan element s ∈ a with s ≤ B t . Writing s = f ( δ ) with δ ≺ ε , we could invokeLemma 4.2 to conclude δ (cid:22) β ≺ γ . By the above this would imply s = f ( δ ) / ∈ a ,which yields the desired contradiction. To establish the second conjunct we observethat f ( γ ) ∈ a implies ϕ ( f ( γ )). According to the definition of ϕ , this means thatthere is an ordinal α ≺ ε with f ( α ) = f ( γ ) and ¬ ψ ( α ). Since f is injective we get α = γ and thus ¬ ψ ( γ ), as required. (cid:3) According to Gentzen’s ordinal analysis [16], the consistency of Peano arithmeticis provable in PA + T I ( ε , Π − ). A detailed proof of a stronger result can be foundin the next section. Together with Proposition 4.4 and G¨odel’s theorem, it followsthat PA + K Σ − is a proper extension of PA .5. From transfinite induction to reflection Working over PA , we show that T I ( ε , Π − ) implies Rfn PA (Σ ). The conversedirection will be established in Section 7. The result is rather similar to one byKreisel and L´evy [26], who show that induction with parameters corresponds to uni-form reflection. The author has found no reference for the parameter-free case. Aswe will see, the connection with reflection implies that PA + K Σ − is not containedin any consistent extension of PA by a computably enumerable set of Π -sentences.As preparation, we review the ordinal analysis of Peano arithmetic and its form-alization in PA itself. First note that we cannot formalize the usual soundnessargument by induction over formal proofs, since there is no arithmetical truthdefinition that would cover all relevant formulas (due to Tarski [45]). Even whenwe restrict attention to theorems of restricted complexity, their proofs may involvedetours through more complex lemmata. The method of cut elimination aims toremove such detours in order to permit a soundness argument that is based onpartial truth definitions (cf. [18, Section I.1(d)]). However, it is not immediatelypossible to eliminate complex lemmata from proofs in Peano arithmetic, which mayuse complex instances of induction in an essential way. To resolve this problem,ordinal analysis transforms the usual finite proofs into infinite proof trees: In therealm of infinite proofs, induction can be deduced from axioms of low complexity, so that cut elimination becomes possible. Soundness can then be proved by transfiniteinduction over the rank of infinite proof trees.Our ordinal analysis works with proofs in a Tait-style sequence calculus. Inparticular, this means that all formulas are in negation normal form, and thatnegation is a defined operation based on Morgan’s laws. Each node in a proof treededuces a sequent, i. e. a finite set Γ = { ϕ , . . . , ϕ n − } of formulas. The latter is tobe interpreted as the disjunction W Γ = ϕ ∨ · · · ∨ ϕ n − . In the context of sequentswe write Γ , ϕ for Γ ∪ { ϕ } . Detours in proofs are implemented via the cut ruleΓ , ϕ Γ , ¬ ϕ ,Γwhich has the following intuitive significance: In order to show W Γ, it suffices to • prove a lemma ϕ (more precisely, the left premise proves W Γ ∨ ϕ ) and to • prove that ϕ implies W Γ (i. e. to prove W Γ ∨ ¬ ϕ , as in the right premise).The crucial feature of the infinite proof system is the ω -ruleΓ , ϕ (0) Γ , ϕ (1) · · · ,Γ , ∀ n ϕ ( n )which allows to infer ∀ n ϕ ( n ) if there is a proof of ϕ ( n ) for each numeral n . Inductioncan be derived from the ω -rule, since ϕ (0) ∧ ∀ m ( ϕ ( m ) → ϕ ( m + 1)) → ϕ ( n )has a straightforward proof for each number n . It follows that any finite proof inPeano arithmetic can be translated (or “embedded”) into the infinite system.It is not immediately clear how infinite proof trees can be formalized in Peanoarithmetic. In the following we recall a very elegant approach due to Buchholz [4](see his paper for all missing details): The idea is to work with a set Z ∗ of finiteterms. Each term names an infinite proof by specifying its role in the cut eliminationprocess. Specifically, each finite proof d in Peano arithmetic gives rise to a constantsymbol [ d ] ∈ Z ∗ , which denotes the translation of d into the infinite system. Foreach term h ∈ Z ∗ there is a term Eh ∈ Z ∗ that names the proof that results from h by a single application of cut elimination. The intermediate steps of cut eliminationgive rise to auxiliary function symbols. By primitive recursion over terms one candefine an ordinal o ( h ) ≺ ε that bounds the rank of the proof tree representedby h ; for example, the well-known fact that cut elimination leads to an exponentialincrease of the ordinal rank suggests the recursive clause o ( Eh ) = ω o ( h ) . Also byrecursion over terms, one can determine the end sequent e ( h ), the last rule r ( h ),the cut rank d ( h ), and terms s ( h, n ) ∈ Z ∗ that denote the immediate subtrees ofthe proof tree that is represented by h . Working in PA (or even in PRA ), onecan show that the term system Z ∗ is “locally correct” (see [4, Theorem 3.8]); inparticular this means that we have o ( s ( h, n )) ≺ o ( s ), except when r ( s ) signifies anaxiom. To ensure “global correctness”, one needs transfinite induction up to ε ,which is not available in PA . In the sequel we abbreviate h ⊢ α Γ : ⇔ h ∈ Z ∗ ∧ o ( h ) = α ∧ d ( h ) = 0 ∧ e ( h ) ⊆ Γ . Intuitively, this asserts that h is a cut-free infinite proof tree with rank α and endsequent Γ (note that W e ( h ) implies W Γ). Crucially, the relation h ⊢ α Γ is primitiverecursive and hence ∆ -definable in PA . This implies that Z ∗ ⊢ α Γ : ⇔ ∃ h ∈ Z ∗ h ⊢ α Γ MATHEMATICAL COMMITMENT WITHOUT COMPUTATIONAL STRENGTH 13 is a Σ -formula with parameters α and Γ. We can now show the promised result: Proposition 5.1. Each instance of Rfn PA (Σ ) can be proved in PA + T I ( ε , Π − ) .Proof. Consider a closed Σ -formula ϕ . Working in PA + T I ( ε , Π − ), we assumethat we have Pr PA ( ϕ ). In order to establish Rfn PA ( ϕ ), we need to derive ϕ . We useBuchholz’ formalization of ordinal analysis, as discussed above. By embedding andcut elimination (cf. [4, Definitions 3.4 and 3.7]), the assumption Pr PA ( ϕ ) implies ∃ α ≺ ε Z ∗ ⊢ α { ϕ } . Write Γ ⊆ { ϕ } ∪ Π − to express that Γ is a sequent that consists of Π -sentencesand (possibly) the formula ϕ . The statement that Γ contains a true Π -sentencecan be expressed by a Π -formula Tr Π − (Γ) (cf. [18, Theorem I.1.75]). Aiming at acontradiction, we assume that ϕ is false. Under this assumption we will derive ∀ α ≺ ε ∀ Γ (Γ ⊆ { ϕ } ∪ Π − ∧ Z ∗ ⊢ α Γ → Tr Π − (Γ)) , arguing by transfinite induction on α ≺ ε . Note that the sentence ϕ is representedby a fixed numeral. Hence α is the only free variable of the induction formula, andthe induction is covered by the scheme T I ( ε , Π − ). Once the induction is carriedout, it is starightforward to derive the desired contradiction: By the above we have Z ∗ ⊢ α { ϕ } for some α ≺ ε . However, we cannot have Tr Π − ( { ϕ } ), since ϕ wasassumed to be false (note that this covers both ϕ ∈ Π ⊆ Σ and ϕ ∈ Σ \ Π ). Itremains to carry out the induction. In the step we consider a sequent Γ ⊆ { ϕ } ∪ Π − and assume h ⊢ α Γ for some h ∈ Z ∗ . We distinguish cases according to the lastrule r ( h ). Note that this cannot be a cut, since h ⊢ α Γ entails d ( h ) = 0. If r ( h )is an axiom, then e ( h ) ⊆ Γ contains a true literal (cf. [4, Definition 2.2]). Tocomplete the proof, we consider the introduction of a quantifier; the introductionof a propositional connective is similar and simpler. First assume that h ends withan ω -rule, which introduces a formula ∀ n θ ( n ) ∈ Γ. Due to Γ ⊆ { ϕ } ∪ Π − we seethat ∀ n θ ( n ) must be a Π -sentence. Local correctness (see [4, Theorem 3.8]) yields Z ∗ ⊢ o ( s ( h,n ))0 Γ , θ ( n ) with o ( s ( h, n )) ≺ o ( h ) = α for all n ∈ N . The induction hypothesis implies that each sequent Γ , θ ( n ) contains atrue Π -sentence. Hence we get such a sentence in Γ, or all instances θ ( n ) are true.In the latter case, it follows that Γ contains the true Π -sentence ∀ n θ ( n ). Finally,assume that r ( h ) introduces an existential formula ∃ n ψ ( n ). In view of Γ ⊆ { ϕ }∪ Π − we must have ∃ n ψ ( n ) ≡ ϕ (note that [4] does not work with bounded quantifiersbut treats primitive recursive relations as atomic). By local correctness there issome existential witness k ∈ N such that we have Z ∗ ⊢ o ( s ( h, Γ , ψ ( k ) with o ( s ( h, ≺ o ( h ) = α. The induction hypothesis yields a true Π -sentence in Γ , ψ ( k ). To establish Tr Π − (Γ)it suffices to show that ψ ( k ) cannot be true: if it was, then ϕ ≡ ∃ n ψ ( n ) would betrue as well, which contradicts our assumption. (cid:3) The following proof is similar to one by Kreisel and L´evy [26, § 8] (see [1,Lemma 2] for an argument that takes the formula complexity into account). Corollary 5.2. There is no computably enumerable set Ψ of Π -sentences suchthat PA + Ψ is consistent and contains PA + K Σ − . In particular, the latter is aproper extension of PA . Proof. Consider a computably enumerable set Ψ of Π -sentences such that PA + Ψproves each instance of K Σ − . We need to show that PA + Ψ is inconsistent.According to [28, Theorem 4], there is a single Π -sentence ψ such that PA + ψ is a Σ -conservative extension of PA + Ψ. In view of conservativity, it suffices toshow that PA + ψ is inconsistent. By Propositions 4.4 and 5.1 we have PA + Rfn PA (Σ ) ⊆ PA + T I ( ε , Π − ) ⊆ PA + K Σ − ⊆ PA + Ψ ⊆ PA + ψ. Hence we can invoke local Σ -reflection to get PA + ψ ⊢ Pr PA ( ¬ ψ ) → ¬ ψ. The contrapositive yields PA + ψ ⊢ ¬ Pr PA ( ¬ ψ ). This means that PA + ψ provesits own consistency, so that it is inconsistent by G¨odel’s theorem. (cid:3) Since any true Σ -sentence follows from a true Π -sentence, there is a set Ξ ofΠ -sentences such that PA +Ξ is consistent and contains PA + K Σ − . The corollarytells us that Ξ cannot be computably enumerable.6. A primitive recursive reification In the rest of this paper we complete the proof that K Σ − , T I ( ε , Π − ) andRfn PA (Σ ) are equivalent over PA . The present section is concerned with a tech-nical result that will be crucial for this purpose.Write Bad( B ) for the set of non-empty finite bad sequences in B . We want toconstruct a primitive recursive function r : Bad( B ) → ε such that we have r ( h t , . . . , t n , t n +1 i ) ≺ r ( h t , . . . , t n i )whenever h t , . . . , t n +1 i is an element of Bad( B ), provably in PA . Such a functionis called a reification. It ensures that B is a well partial order with maximal ordertype at most (and in fact equal to) ε .As mentioned in the introduction, the result that B has maximal order type ε isdue to de Jongh and Schmidt. Experience shows that maximal order types can bewitnessed by effective reifications. For the case of finite (and in particular binary)trees this has been established by M. Rathjen and A. Weiermann [32, Section 2].Unfortunately, we cannot simply cite their result: In [32] it is shown that ACA proves the existence of a reification; however, it is not entirely trivial to see thatthe constructed reification is (primitive) recursive. In the rest of this section weverify this fact in detail. Some readers may prefer to skip this verification and tocontinue with the applications in the next section. We point out that the followingpresentation is influenced by the more general construction in [19].The reification of B will depend on reifications of various other orders. In thecontext of first order arithmetic it helps to think of these orders as types, which arerepresented by finite expressions. Definition 6.1 ( PA ) . The following recursive clauses generate a collection of typesand a subcollection of indecomposable types:(i) The symbols B and E are indecomposable types.(ii) If A, B are types, then A + B is a type.(iii) If A, B are indecomposable types, then A × B is an indecomposable type.(iv) If A is any type, then A ∗ is an indecomposable type. MATHEMATICAL COMMITMENT WITHOUT COMPUTATIONAL STRENGTH 15 Note that it is not allowed to form types such as ( A + B ) × C , since A + B isnot indecomposable. This will become important in the proof of Proposition 6.12.The elements of our orders are represented by terms of the corresponding types.To obtain primitive recursive constructions, it is crucial to work with terms of alltypes simultaneously. For example, it is neither possible nor necessary to constructall terms of type A before one constructs a term of type A ∗ . We do not specifyterms of type E , because the latter is supposed to represent the empty order. Definition 6.2 ( PA ) . The following recursive clauses generate a collection ofterms. We simultaneously specify the types of these terms:(i) Each binary tree t ∈ B is a term of type B .(ii) If a is a term of type A and B is a type, then ι B a is a term of type A + B .If b is a term of type B and A is a type, then ι A b is a term of type A + B .(iii) If a and b are terms of types A and B , then h a, b i is a term of type A × B .(iv) If a , . . . , a n − have type A , then h a , . . . , a n − i A is a term of type A ∗ .Note that (iii) does only apply when A and B are indecomposable.One readily constructs a G¨odel numbering s, t < ◦ ( s, t ) for s, t ∈ B , a < ι B a, b < ι A b, a, b < h a, b i , a , h a , . . . , a n i A < h a , . . . , a n i A . We will use this G¨odel numbering to construct primitive recursive functions bycourse-of-values recursion. Binary functions can be constructed with the help ofthe Cantor pairing function, which is monotone in both components. For example,the following definition decides a ≤ A a ′ by recursion over the code of h a, a ′ i . Definition 6.3 ( PA ) . The relation a ≤ A a ′ between terms a and a ′ of the sametype A is generated by the following recursive clauses (i. e. it is the smallest relationthat satisfies them):(i) If s ≤ B t , then s ≤ B t .(ii) If a ≤ A a ′ , then ι B a ≤ A + B ι B a ′ . If b ≤ B b ′ , then ι A b ≤ A + B ι A b ′ .(iii) If a ≤ A a ′ and b ≤ B b ′ , then h a, b i ≤ A × B h a ′ , b ′ i .(iv) If there is a strictly increasing f : { , . . . , m − } → { , . . . , n − } such that a i ≤ A a ′ f ( i ) holds for all i < m , then h a , . . . , a m − i A ≤ A ∗ h a ′ , . . . , a ′ n − i A .Let us record the expected property: Lemma 6.4 ( PA ) . Each relation ≤ A is a partial order on the terms of type A .Proof. First check a ≤ A a by induction over a , simultaneously for all types A .Then use induction over a + a ′ to verify that a ≤ A a ′ and a ′ ≤ A a imply a = a ′ .Finally, show a ≤ A a ′ & a ′ ≤ A a ′′ ⇒ a ≤ A a ′′ by induction over a + a ′ + a ′′ . (cid:3) From now on we write a ∈ A to express that a is a term of type A . Despitethis notation, one should keep in mind that A is a finite expression rather than aninfinite set. The following provides a substitute for the “missing” types A × B . Definition 6.5 ( PA ) . For arbitrary types A and B we recursively define a type A ⊗ B and terms [ a, b ] ∈ A ⊗ B for all a ∈ A and b ∈ B : First put A ⊗ B = A × B and [ a, b ] = h a, b i when A, B are indecomposable . Now consider A = C + D and an arbitrary B . To save parentheses, we assume that ⊗ binds stronger than +. We then define( C + D ) ⊗ B = C ⊗ B + D ⊗ B and [ ι D c, b ] = ι D ⊗ B [ c, b ] , [ ι C d, b ] = ι C ⊗ B [ d, b ] . For indecomposable A and B = C + D we set A ⊗ ( C + D ) = A ⊗ C + A ⊗ D and [ a, ι D c ] = ι A ⊗ D [ a, c ] , [ a, ι C d ] = ι A ⊗ C [ a, d ] . The following is readily checked by induction on a + a ′ + b + b ′ . Lemma 6.6 ( PA ) . We have [ a, b ] ≤ A ⊗ B [ a ′ , b ′ ] ⇔ a ≤ A a ′ and b ≤ B b ′ for arbitrary terms a, a ′ ∈ A and b, b ′ ∈ B . For a ∈ A we will abbreviate a ′ ∈ A a : ⇔ a ′ ∈ A and a A a ′ . The sets A a are important for the analysis of maximal order types, because theycontain all elements that can follow a in a bad sequence. In our setting it will beimportant to have a quasi embedding of A a into a suitable type A ( a ). To saveparentheses we agree on A ⊗ B ⊗ C = ( A ⊗ B ) ⊗ C and [ a, b, c ] = [[ a, b ] , c ]. Thefollowing construction is similar to the one in [19, Definition 5.3 and Example 5.4]. Definition 6.7 ( PA ) . By recursion over a we define a type A ( a ) for each a ∈ A :(i) We have B ( ◦ ) = E and B ( ◦ ( s, t )) = ( B ( s ) + B ( t )) ∗ .(ii) We have ( A + B )( ι B a ) = A ( a ) + B and ( A + B )( ι A b ) = A + B ( b ).(iii) We have ( A × B )( h a, b i ) = A ( a ) ⊗ B + A ⊗ B ( b ).(iv) We have A ∗ ( hi A ) = E and A ∗ ( h a , . . . , a n i A ) = A ( a ) ∗ + A ( a ) ∗ ⊗ A ⊗ A ∗ ( h a , . . . , a n i A ) . As promised, we get the following quasi embeddings: Proposition 6.8. There is a primitive recursive function e such that PA proves thefollowing: For any type A and terms a ∈ A, b ∈ A a we have e A ( a, b ) = e ( a, b ) ∈ A ( a ) (note that A can be inferred from a ). Furthermore we have e A ( a, b ) ≤ A ( a ) e A ( a, b ′ ) ⇒ b ≤ A b ′ for any terms b, b ′ ∈ A a .Proof. The value e A ( a, b ) is defined by recursion over the code of the pair h a, b i ,simultaneously for all types A . Once the construction of e is complete, the secondpart of the proposition can be verified by induction on a + b + b ′ . In thefollowing we distinguish cases according to the form of a .First consider a = ◦ ∈ B = A . Since ◦ ≤ B t is true for any t ∈ B , the set A a isempty and there are no values to define. Now assume a = ◦ ( s , s ) ∈ B = A . Forthe term b = ◦ ∈ B we put e B ( ◦ ( s , s ) , ◦ ) = hi B ( s )+ B ( s ) ∈ ( B ( s ) + B ( s )) ∗ = B ( ◦ ( s , s )) . Now assume that we have b = ◦ ( t , t ) ∈ B . The condition b ∈ A a amounts to ◦ ( s , s ) B ◦ ( t , t ), which yields s B t or s B t . Let us assume that wehave s B t , which amounts to t ∈ B s . We may then refer to the recursivelydefined value e B ( s , t ) ∈ B ( s ) . MATHEMATICAL COMMITMENT WITHOUT COMPUTATIONAL STRENGTH 17 More formally, the recursive definition of e A ( a, b ) and the inductive verification of e A ( a, b ) ∈ A ( a ) should be separated. In order to do so, we can agree on a defaultvalue for the hypothetical case that the decidable property e B ( s , t ) ∈ B ( s ) fails;the induction shows that the default value is never called. By ◦ ( s , s ) B ◦ ( t , t )we also have ◦ ( s , s ) B t , which amounts to t ∈ B ◦ ( s ,s ) and provides e B ( ◦ ( s , s ) , t ) ∈ B ( ◦ ( s , s )) = ( B ( s ) + B ( s )) ∗ . Let us agree to write c ⋆ h c , . . . , c n i C := h c , c , . . . , c n i C ∈ C ∗ for terms c , . . . , c n of a type C . We can now state our recursive clause as e B ( ◦ ( s , s ) , ◦ ( t , t )) = ι B ( s )0 e B ( s , t ) ⋆ e B ( ◦ ( s , s ) , t ) if s B t ,ι B ( s )1 e B ( s , t ) ⋆ e B ( ◦ ( s , s ) , t ) otherwise . To explain the second case we recall that s B t must hold if s B t fails.Before we state the other recursive clauses, let us verify that the second part ofthe proposition holds for A = B . As above we write a = ◦ ( s , s ). In the case ofthe term b ′ = ◦ we observe e B ( a, b ) ≤ B ( s ) e B ( a, b ) = hi B ( s )+ B ( s ) ⇒ e B ( a, b ) = hi B ( s )+ B ( s ) . The consequent of this implication can only hold for b = ◦ . In this case b ≤ B b ′ issatisfied for any b ′ ∈ B . Hence it remains to consider terms of the form b = ◦ ( t , t )and b ′ = ◦ ( t ′ , t ′ ). In general we have c ⋆ σ ≤ C ∗ c ′ ⋆ σ ′ ⇔ c ⋆ σ ≤ C ∗ σ ′ or ( c ≤ C c ′ and σ ≤ C ∗ σ ′ ) . First assume that e B ( s, b ) ≤ B ( s ) e B ( s, b ′ ) holds because of e B ( s, b ) ≤ B ( s ) e B ( s, t ′ i ).Then the induction hypothesis yields b ≤ B t ′ i , which implies b ≤ B ◦ ( t ′ , t ′ ) = b ′ .Now assume we have e B ( s, b ) ≤ B ( s ) e B ( s, b ′ ) because there are i, j ∈ { , } with ι B ( s − i ) i e B ( s i , t i ) ≤ B ( s )+ B ( s ) ι B ( s − j ) j e B ( s j , t ′ j ) ,e B ( s, t − i ) ≤ B ( s ) e B ( s, t ′ − j ) . The first inequality can only hold for i = j . It yields e B ( s i , t i ) ≤ B ( s i ) e B ( s i , t ′ i ),which implies t i ≤ B t ′ i by induction hypothesis. From the second inequality we caninfer t − i ≤ B t ′ − i . Together we get b = ◦ ( t , t ) ≤ B ◦ ( t ′ , t ′ ) = b ′ , as desired.Sum and product types are considerably easier to handle. We only state therecursive clauses and leave all verifications to the reader: e A + B ( ι B a, ι B a ′ ) = ι B e A ( a, a ′ ) , e A + B ( ι B a, ι A b ′ ) = ι A ( a )1 b ′ ,e A + B ( ι A b, ι B a ′ ) = ι B ( b )0 a ′ , e A + B ( ι A b, ι A b ′ ) = ι A e B ( b, b ′ ) .e A × B ( h a, b i , h a ′ , b ′ i ) = ι A ⊗ B ( b )0 [ e A ( a, a ′ ) , b ′ ] if a A a ′ ,ι A ( a ) ⊗ B [ a ′ , e B ( b, b ′ )] otherwise . Finally, we consider the case of a type A ∗ . For a = hi A ∈ A ∗ it suffices to observethat ( A ∗ ) a is empty, since hi A ≤ A ∗ τ holds for any τ ∈ A ∗ . Now consider a termof the form a = a ⋆ σ ∈ A ∗ . We write b = h b , . . . , b n − i A ∈ ( A ∗ ) a and distinguishtwo cases. If we have a A b i for all i < n , then we set e A ∗ ( a, b ) = ι A ( a ) ∗ ⊗ A ⊗ A ∗ ( σ )0 h e A ( a , b ) , . . . , e A ( a , b n − ) i A ( a ) . Note that this is an element of A ( a ) ∗ + A ( a ) ∗ ⊗ A ⊗ A ∗ ( σ ) = A ∗ ( a ), as required.Otherwise we fix the smallest number i < n with a ≤ A b i . In view of b ∈ ( A ∗ ) a we must have σ A ∗ h b i +1 , . . . , b n − i A . We can thus define e A ∗ ( a, b ) as ι A ( a ) ∗ [ h e A ( a , b ) , . . . , e A ( a , b i − ) i A ( a ) , b i , e A ∗ ( σ, h b i +1 , . . . , b n − i A )] . Using the induction hypothesis, one readily checks that e A ∗ ( a, b ) ≤ A ∗ ( a ) e A ∗ ( a, b ′ )implies b ≤ A ∗ b ′ . (cid:3) Our next aim is to iterate the previous construction along bad sequences. Givena type A , we write σ ∈ Bad + ( A ) to express that σ is a finite bad sequence in A .This means that we have σ = h a , . . . , a n − i for terms a , . . . , a n − ∈ A that satisfy a i A a j for all i < j < n . If we have σ ∈ Bad + ( A ) and σ is different from theempty sequence hi , then we write σ ∈ Bad( A ). For σ = h a , . . . , a n − i ∈ Bad + ( A )we abbreviate σ ⌢ a = h a , . . . , a n − , a i and put a ∈ A σ : ⇔ a ∈ A and σ ⌢ a ∈ Bad( A ) . The expressions A ( a ) and e A ( a, b ) have only been explained for a ∈ A and b ∈ A a .We will see that the following definition does conform with these restrictions. Inorder to state the definition it is, nevertheless, helpful to realize that the primitiverecursive functions ( A, a ) A ( a ) and ( A, a, b ) e A ( a, b ) can be extended toarbitrary arguments. Definition 6.9 ( PA ) . Consider a type A . For a sequence σ ∈ Bad + ( A ) and aterm b ∈ A σ we define A [ σ ] and ˆ e A ( σ, b ) by the recursive clauses A [ hi ] = A, A [ σ ⌢ a ] = A [ σ ](ˆ e A ( σ, a )) , ˆ e A ( hi , b ) = b, ˆ e A ( σ ⌢ a, b ) = e A [ σ ] (ˆ e A ( σ, a ) , ˆ e A ( σ, b )) . In order to justify the recursion in detail, we consider σ = h a , . . . , a n − i andwrite σ ↾ i = h a , . . . , a i − i . Then A [ σ ↾ i ] and the values ˆ e A ( σ ↾ i, a j ) for i ≤ j < n are constructed simultaneously by recursion on i < n . For σ ′ := σ ⌢ a n with a n := b this also explains the value ˆ e A ( σ, b ) = ˆ e A ( σ ′ ↾ n, a n ). Corollary 6.10 ( PA ) . If σ is a finite bad sequence in the type A , then A [ σ ] is atype. For any b ∈ A σ the value ˆ e A ( σ, b ) is a term of this type. Furthermore we have ˆ e A ( σ, b ) ≤ A [ σ ] ˆ e A ( σ, b ′ ) ⇒ b ≤ A b ′ for any terms b, b ′ ∈ A σ .Proof. We use induction on σ to verify all claims simultaneously. The case of σ = hi is immediate. Now assume that we have σ = σ ⌢ a . The induction hypothesis tellsus that ˆ e A ( σ , a ) is a term of type A [ σ ]. In view of Definition 6.7 it followsthat A [ σ ] = A [ σ ](ˆ e A ( σ , a )) is a type. For b ∈ A σ we have a A b , so that theinduction hypothesis yields ˆ e A ( σ , a ) A [ σ ] ˆ e A ( σ , b ). By Proposition 6.8 we getˆ e A ( σ, b ) = e A [ σ ] (ˆ e A ( σ , a ) , ˆ e A ( σ , b )) ∈ A [ σ ](ˆ e A ( σ , a )) = A [ σ ] . From ˆ e A ( σ, b ) ≤ A [ σ ] ˆ e A ( σ, b ′ ) we can infer ˆ e A ( σ , b ) ≤ A [ σ ] ˆ e A ( σ , b ′ ), also by Pro-position 6.8. Then b ≤ A b ′ follows by induction hypothesis. (cid:3) In order to obtain a reification, it remains to assign a suitable ordinal to eachtype. Let us write α ⊕ β and α ⊗ β for the natural (“Hessenberg”) sum and productof ordinals α, β ≺ ε (see e. g. [38, § MATHEMATICAL COMMITMENT WITHOUT COMPUTATIONAL STRENGTH 19 in both arguments. Ordinals of the form ω γ are additively indecomposable, in thesense that α, β ≺ ω γ implies α ⊕ β ≺ ω γ ; conversely, any additively indecomposableordinal δ = 0 has the form δ = ω γ . For α, β ≺ ω γ := ω ( ω γ ) we have α ⊗ β ≺ ω γ . Definition 6.11 ( PA ) . Let us say that a type is low if it does not involve theconstant symbol B . We recursively assign an ordinal o ( A ) to each low type A : o ( E ) = 0 , o ( A + B ) = o ( A ) ⊕ o ( B ) ,o ( A × B ) = o ( A ) ⊗ o ( B ) , o ( A ∗ ) = ω o ( A )2 . The following is crucial for the construction of a reification. Proposition 6.12 ( PA ) . If A is a low type and a ∈ A is a term, then A ( a ) is alow type and we have o ( A ( a )) ≺ o ( A ) .Proof. As preparation we note that A ⊗ B is low when the same holds for A and B .A straightforward induction shows o ( A ⊗ B ) = o ( A ) ⊗ o ( B ); for example, the dis-tributivity property from [38, Lemma 4.5(8)] accounts for the inductive verification o (( C + D ) ⊗ B ) = o ( C ⊗ B + D ⊗ B ) = o ( C ⊗ B ) ⊕ o ( D ⊗ B ) == ( o ( C ) ⊗ o ( B )) ⊕ ( o ( D ) ⊗ o ( B )) = ( o ( C ) ⊕ o ( D )) ⊗ o ( B ) = o ( C + D ) ⊗ o ( B ) . By induction on A one can show that o ( A ) is additively indecomposable when A is an indecomposable type. The most interesting step concerns a type A = B × C ,where B and C are indecomposable according to Definition 6.1. Inductively wemay write o ( B ) = ω β and o ( C ) = ω γ (unless we have o ( A ) = 0). Then o ( B × C ) = o ( B ) ⊗ o ( C ) = ω β ⊗ ω γ = ω β ⊕ γ is an additively indecomposable ordinal as well. The claim of the proposition cannow be verified by induction over a , for all types A simultaneously. First considerthe case of a term ι B a ∈ A + B . The induction hypothesis tells us that A ( a ) is lowwith o ( A ( a )) ≺ o ( A ). Hence ( A + B )( ι B a ) = A ( a ) + B is low and we have o (( A + B )( ι B )) = o ( A ( a ) + B ) = o ( A ( a )) ⊕ o ( B ) ≺ o ( A ) ⊕ o ( B ) = o ( A + B ) . The case of ι A b ∈ A + B is analogous. Now consider a term h a, b i ∈ A × B . In viewof the above, the induction hypothesis implies that A ( a ) ⊗ B is low with ordinal o ( A ( a ) ⊗ B ) = o ( A ( a )) ⊗ o ( B ) ≺ o ( A ) ⊗ o ( B ) = o ( A × B ) . In the same way we get o ( A ⊗ B ( b )) ≺ o ( A × B ). In view of Definition 6.1, a type ofthe form A × B is always indecomposable. By the above this entails that o ( A × B )is an additively indecomposable ordinal. Hence we obtain o (( A × B )( h a, b i )) = o ( A ( a ) ⊗ B + A ⊗ B ( b )) = o ( A ( a ) ⊗ B ) ⊕ o ( A ⊗ B ( b )) ≺ o ( A × B ) . Finally, we consider the case of a type A ∗ . Concerning the term hi A ∈ A ∗ , we note o ( A ∗ ( hi A )) = o ( E ) = 0 ≺ ω o ( A )2 = o ( A ∗ ) . Now consider a term a⋆σ ∈ A ∗ (see the proof of Proposition 6.8 for the notation). Inview of a, σ < a ⋆ σ the induction hypothesis yields o ( A ∗ ( σ )) ≺ o ( A ∗ ) = ω o ( A )2 and o ( A ( a )) ≺ o ( A ). The latter implies o ( A ( a ) ∗ ) = ω o ( A ( a ))2 ≺ ω o ( A )2 . Since we are concerned with ordinals below ε , we also have o ( A ) ≺ ω o ( A )2 . Using the fact that ω o ( A )2 is additively and multiplicatively indecomposable, we can deduce o ( A ∗ ( a ⋆ σ )) = o ( A ( a ) ∗ + A ( a ) ∗ ⊗ A ⊗ A ∗ ( σ )) == o ( A ( a ) ∗ ) ⊕ o ( A ( a ) ∗ ) ⊗ o ( A ) ⊗ o ( A ∗ ( σ )) ≺ ω o ( A )2 = o ( A ∗ ) , as required. (cid:3) Recall that the terms of type B coincide with the finite binary trees, i. e. withthe element of B . Below we will show that the type B [ σ ] is low for any non-emptybad sequence σ ∈ Bad( B ) = Bad( B ). To state the following definition, we simplyassume that the primitive recursive function o ( · ) is extended to arbitrary arguments. Definition 6.13 ( PA ) . For σ ∈ Bad( B ) we put r ( σ ) := o ( B [ σ ]).Finally, we can deduce the promised result: Corollary 6.14 ( PA ) . The primitive recursive function r : Bad( B ) → ε is areification, i. e. we have r ( h t , . . . , t n , t n +1 i ) ≺ r ( h t , . . . , t n i ) for any bad sequence h t , . . . , t n , t n +1 i in B .Proof. We use induction on σ ∈ Bad( B ) to show that B [ σ ] is a low type. For thispurpose it is crucial to recall that the empty sequence was included in Bad + ( B ) butexcluded from Bad( B ). Hence the base case concerns a sequence of the form σ = h t i .In view of Definition 6.9 we have B [ h t i ] = B [ hi ](ˆ e B ( hi , t )) = B ( t ) . Even though the type B is not low, a straightforward induction on t ∈ B showsthat B ( t ) is a low type. Now consider a sequence σ ⌢ t ∈ Bad( B ) with σ = hi . Theinduction hypothesis ensures that B [ σ ] is a low type. According to Corollary 6.10we have ˆ e B ( σ, t ) ∈ B [ σ ]. By (the easy part of) Proposition 6.12 we conclude that B [ σ ⌢ t ] = B [ σ ](ˆ e B ( σ, t ))is a low type as well. The more substantial part of Proposition 6.12 yields r ( σ ⌢ t ) = o ( B [ σ ⌢ t ]) ≺ o ( B [ σ ]) = r ( σ ) . For σ = h t , . . . , t n i and t = t n +1 this is the claim of the corollary. (cid:3) From reflection to the finite basis property Working over PA , we show that Rfn PA (Σ ) entails T I ( ε , Π − ), which does inturn entail K Σ − . This completes our proof that all three principles are equivalent.Using Goryachev’s theorem, we can deduce a characterization of the Π -sentencesthat are provable in PA + K Σ − .For the case of uniform reflection and induction with parameters, the followinghas been shown by Kreisel and L´evy [26]. Proposition 7.1. Each instance of T I ( ε , Π − ) can be proved in PA + Rfn PA (Σ ) . MATHEMATICAL COMMITMENT WITHOUT COMPUTATIONAL STRENGTH 21 Proof. Consider a Π -formula ψ ( x ) with a single free variable. Arguing in thetheory PA + Rfn PA (Σ ), we establish T I ( ε , ψ ) by contraposition: Assume thatthe conclusion of transfinite induction fails, so that we have ∃ α ≺ ε ¬ ψ ( α ). The latteris a Σ -formula, so that its truth can be established by an explicit verification. Moreformally, we invoke formalized Σ -completeness (cf. [18, Theorem I.1.8]) to obtain ∃ α ≺ ε Pr PA ( ¬ ψ ( ˙ α )) . This uses Feferman’s dot notation: By ψ ( ˙ α ) one denotes the closed object for-mula that result from ψ ( x ) when we substitute x by the α -th numeral, where thecode α is considered as a natural number (cf. the notation in [18, Corollary I.1.76]).Gentzen [17] has shown that PA proves induction up to each fixed ordinal below ε .This result can itself be formalized in Peano arithmetic (and in much weaker the-ories, cf. [11, Section 3]), so that we get ∀ α ≺ ε Pr PA ( ∀ γ ≺ ε ( ∀ β ≺ γ ψ ( β ) → ψ ( γ )) → ψ ( ˙ α )) . Together with the above this yieldsPr PA ( ¬∀ γ ≺ ε ( ∀ β ≺ γ ψ ( β ) → ψ ( γ ))) . By an instance of Rfn PA (Σ ) we get ¬∀ γ ≺ ε ( ∀ β ≺ γ ψ ( β ) → ψ ( γ )), which is (provablyequivalent to) a closed Σ -formula. Hence the premise of T I ( ε , ψ ) fails, so thatour proof by contraposition is complete. (cid:3) The following is a consequence of the result that ( B , ≤ B ) is a well partial orderwith maximal order type ε , which is due to de Jongh (unpublished; cf. the intro-duction to [34]) and Diana Schmidt (see [35, Theorem II.2] in combination with theexample after [35, Definition I.15]). A detailed proof in our setting has been givenin the previous section. Proposition 7.2. Each instance of K Σ − can be proved in PA + TI ( ε , Π − ) .Proof. We fix an instance K ϕ (where ϕ is a Σ -formula with a single free vari-able) and work in PA + TI ( ε , Π − ). It is instructive to recall the argument fromRemark 3.1, which relies on a notion of ϕ -sequence. If { n ∈ N | ϕ ( n ) } is comput-ably enumerable but not decidable, then it is not decidable whether a given finitesequence is a ϕ -sequence. For this reason we now introduce a finer notion: Write ϕ ( x ) ≡ ∃ y θ ( x, y ) with a ∆ -formula θ . As in the previous section we write Bad( B )for the set of non-empty finite bad sequences in B . By a certified ϕ -sequence wemean a finite sequence ( t , c ) , . . . , ( t n , c n ) ⊆ B × N such that we have h t , . . . , t n i ∈ Bad( B ) and θ ( t i , c i ) for all i ≤ n . Note thatthe latter implies ϕ ( t i ). Since θ contains no further free variables, the notion ofcertified ϕ -sequence is defined by a ∆ -formula without parameters. By pickingthe value f ( n ) with minimal code, one can thus define a (possibly partial) function f : N → B × N with the following property: • If the sequence h f (0) , . . . , f ( n − i is defined and can be extended into acertified ϕ -sequence of length n +1, then h f (0) , . . . , f ( n ) i is such a sequence.Note that the relation f ( x ) = y is Σ -definable without parameters. Aiming at acontradiction, we now assume that the instance K ϕ is false. Then all values f ( n ) aredefined: Inductively, we may assume that f ( m ) = ( t m , c m ) is defined for all m < n ;in the case of n > 0, the construction of f ensures that h ( t , c ) , . . . , ( t n − , c n − ) i is a certified ϕ -sequence. To deduce that f ( n ) is defined as well, we consider theset a := { t , . . . , t n − } . As K ϕ is false, we must have ¬∀ s ∈ a ϕ ( s ) ∨ ∃ t ∈B ( ϕ ( t ) ∧ ∀ s ∈ a s B t ) . For s = t m ∈ a , the construction of f ensures θ ( t m , c m ) and thus ϕ ( s ). Hence thesecond disjunct yields an element t n ∈ B with ϕ ( t n ) and t m B t n for all m < n .The latter implies h t , . . . , t n i ∈ Bad( B ). Due to ϕ ( t n ) we can pick a number c n with θ ( t n , c n ). Then h f (0) , . . . , f ( n − , ( t n , c n ) i is a certified ϕ -sequence, and f ( n )is defined as the smallest pair h t n , c n i for which this holds. We can now define atotal computable function g : N → Bad( B ) by setting g ( n ) := h t , . . . , t n i with f ( m ) = ( t m , c m ) . According to Corollary 6.14, there is a primitive recursive reification r : Bad( B ) → ε . It follows that the total computable function r ◦ g : N → ε is strictly decreasing.This is impossible in the presence of TI ( ε , Π − ). To be more precise, we note that r ◦ g ( n ) = α is Σ -definable without parameters. Using TI ( ε , Π − ) one can prove ∀ α ≺ ε ∀ n ∀ δ ≺ ε ( r ◦ g ( n ) = δ → α (cid:22) δ ) . To establish the induction step, it suffices to derive a contradiction from the as-sumption that we have r ◦ g ( n ) ≺ α for some n ∈ N . Since r is a reification, thelatter would lead to r ◦ g ( n + 1) ≺ r ◦ g ( n ) =: γ , which contradicts the inductionhypothesis for γ ≺ α . If we apply the result of the induction to α = r ◦ g (0)+1 ≺ ε , n = 0 and δ = r ◦ g (0), then we get r ◦ g (0) + 1 (cid:22) r ◦ g (0), which is impossible. (cid:3) Together with Propositions 4.4, 5.1 and 7.1 we obtain the following: Theorem 7.3. We have PA + K Σ − ≡ PA + T I ( ε , Π − ) ≡ PA + Rfn PA (Σ ) , i. e. all three theories prove the same theorems. Let Con( PA + ϕ ) be a reasonable formalization of the statement that PA + ϕ is consistent. We consider the recursively generated Π -sentencesCon ( PA ) : ≡ , Con n +1 ( PA ) : ≡ Con( PA + Con n ( PA )) . Note that Con ( PA ) is equivalent to the usual consistency statement. As mentionedin the introduction, we obtain the following: Corollary 7.4. We have PA + K Σ − ≡ Π PA + { Con n ( PA ) | n ∈ N } , i. e. the two theories prove the same Π -sentences.Proof. Let us write Rfn PA for the full local reflection principle, i. e. the collection ofall formulas Pr PA ( ϕ ) → ϕ , where ϕ can be any sentence in the language of first orderarithmetic. According to Goryachev’s theorem (see e. g. [29, Theorem IV.5]), anyΠ -theorem of PA + Rfn PA can be proved in PA + { Con n ( PA ) | n ∈ N } . 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