Local collection scheme and end-extensions of models of compositional truth
aa r X i v : . [ m a t h . L O ] J un Local collection scheme and end-extensions ofmodels of compositional truth
Mateusz Łełyk, Bartosz WcisłoJune 22, 2020
Abstract
We introduce a principle of local collection for compositional truthpredicates and show that it is conservative over the classically compo-sitional theory of truth in the arithmetical setting. This axiom statesthat upon restriction to formulae of any syntactic complexity, the re-sulting predicate satisfies full collection. In particular, arguments us-ing collection for the truth predicate applied to sentences occurring inany given (code of a) proof do not suffice to show that the conclusionof that proof is true, in stark contrast to the case of induction scheme.We analyse various further results concerning end-extensions ofmodels of compositional truth and collection scheme for the composi-tional truth predicate.
The area of axiomatic truth theories investigates extensions of foundationaltheories, such as Peano Arithmetic (PA) with an additional predicate T whichis intended to denote the set of (codes of) true sentences.One of the canonical examples of these theories is CT − (CompositionalTruth). It is a theory of truth over PA whose axioms state that the predicate T satisfies Tarski’s compositional conditions for arithmetical sentences. Forinstance, a disjunction of two sentences is true if either of the disjuncts is.However, we do not assume that the truth predicate satisfies any inductionwhatsoever. All purely arithmetical formulae satisfy the induction schemebecause CT − by definition contains the whole PA.It is a very simple and classical fact that CT − with the full induction,called CT, is not conservative over PA. By induction on the length of proofs,1e can show that whatever is provable in PA is true and thus show the con-sistency of arithmetic. By a theorem of Kotlarski, Krajewski, and Lachlan,CT − itself is conservative over PA. In fact, not much induction is neededto yield non-conservativeness. It has been shown in [21], Theorem 13, thatalready CT − with induction for ∆ -formulae proves new arithmetical sen-tences.Richard Kaye asked whether the conservativity result remains true ifCT − is enriched with full scheme of collection for the sentences containingthe truth predicate. It is known that in presence of ∆ induction the fullschemes of collection and induction are equivalent. However, without theaccess to this small amount of induction, collection seems to be a very weakprinciple. If we add the full collection scheme to PA − (the theory of thepositive part of a discretely ordered semiring), then this extension is Π -conservative over PA − . One could hope for an analogous result for thecompositional truth predicate. Unfortunately, the methods used by Kayecannot be implemented directly in the setting of the truth predicate wherethe conservativity of collection appears to be a much harder problem. Inparticular, as shown by Smith [22] there are countable models M | = CT − with no proper end-extensions and Kaye’s argument rests on the fact thatevery model of PA − can be properly end-extended.In this paper, we provide a partial answer to the question of Kaye. Weintroduce a principle of local collection . It states that if we restrict our com-positional truth predicate to sentences of any syntactic depth c , the resultingtruncated predicate satisfies full collection. We show that the principle oflocal collection for the compositional truth predicate is conservative overPA.Already this result shows that there is no full analogy between collec-tion and induction in the setting of truth theories. One could introduce asimilar scheme of local induction saying that the truth predicate truncatedto sentences of any fixed syntactic depth c satisfies full induction. We couldreadily check that this weaker form of induction is enough to show thatthere are no proofs of contradiction in PA, since any given proof d involvesonly formulae of some bounded syntactic depth c , so we can check by in-duction that all formulae in d are true. This shows that local induction isnot conservative over PA, in contrast to local collection. The question was posed on a session of Midlands Logic Seminar, see [12]. To our best knowledge, this result first appeared as Exercise 7.7 in [11]. Preliminaries
In this section, we present some basic definitions and background results.
This paper deals with extensions of Peano Arithmetic (PA). This is a the-ory in the language L PA = { , S, + , ×} consisting of finitely many basicaxioms of Robinson’s Arithmetic Q which essentially say how + and × canbe defined inductively in terms of the successor function, and the inductionscheme.Full induction scheme is equivalent to induction for ∆ –formulae to-gether with full collection scheme , Coll. The latter consists of all formulaeof the following form (where we allow φ ( x, y ) to contain more free variablesthan just x, y ): ∀ x < a ∃ y φ ( x, y ) → ∃ b ∀ x < a ∃ y < b φ ( x, y ) . Intuitively, collection scheme expresses that any function with a boundeddomain has bounded range. This is clearly true in the natural numbers,since bounded segments of N are finite and hence the image of any suchset is also finite. Induction for ∆ -formulae is crucial for the equivalencebetween induction and collection as shown by Kaye. Theorem 1 (Kaye) . PA − with full collection scheme (but no induction) is conser-vative for Π -formulae over PA − . Peano arithmetic, and its much weaker fragments are capable of rep-resenting syntactic notions. Below, we list formulae representing syntacticnotions which we will use throughout the paper.
Definition 2. • Term L PA ( x ) states that x is (a code of) an arithmeticalterm. • TermSeq L PA ( x ) states that x is (a code of) a sequence of arithmeticalterms. • ClTerm L PA ( x ) states that x is (a code of) a closed arithmetical term. • ClTermSeq L PA ( x ) states that x is (a code of) a sequence of closed arith-metical terms. As we already indicated, this appears as Exercise 7.7 in [11]. Var ( x ) states that x is (a code of) a first-order variable. • y = FV ( x ) states that y is the set of free variables of x (which is eithera term or a formula in the language of arithmetic). • Form L PA ( x ) states that x is (a code of) an arithmetical formula. • Form ≤ L PA ( x ) states that x is (a code of) an arithmetical formula with atmost one free variable. • Sent L PA ( x ) states that x is (a code of) an arithmetical sentence. • y = x is a binary formula which states that y is (a code of) a numeraldenoting the number x . • y = x ◦ states that x is a closed arithmetical term and y is its value. Forinstance ( N , S, + , × ) | = ( p S (0) + S ( S (0)) q ) ◦ = 3 . • Asn ( α, x ) states that α is an assignment for x , i.e., a finite functionwhose domain contains all free variables of s , where x is either a for-mula or a term. We will use Asn ( x ) to denote the set of assignments of x and write α ∈ Asn ( x ) instead of Asn ( α, x ) . If α is an assignment fora formula φ , then by φ [ α ] , we mean a sentence in which α ( v ) has beensubstituted for v , for every v free variable of φ . If α is an assignmentfor a term t , then t α denotes the value of t under this assignment. • β ∼ v α means that β and α are assignments, the domain of β is dom ( α ) ∪{ v } (which is possibly the same as dom ( α ) ), and the values of β arethe same as that of α , possibly except for β ( v ) .We will use some conventions to improve readability of the paper. Wewill write provably functional formulae as if they were function symbols(which we already started doing above). For instance, we will use the ex-pression x like a term. In particular, we will typically be suppressing for-mulae describing syntactic operations and simply write the results of theseoperations. For instance, if φ and ψ are codes of sentences, then T ( φ ∧ ψ ) isan abbreviation for "For all z , if z is the conjunction of φ and ψ , then T ( z ) ."We will sometimes confuse formulae with sets defined with these formulae,e.g., writing x ∈ Form L PA instead of Form L PA ( x ) .The notion of syntactic depth plays an important role in this paper. Definition 3.
Let φ be a formula. By syntactic depth of φ , we mean themaximal depth of nesting of connectives and quantifiers in φ . We will de-note this by dp ( φ ) . By dp ( x ) , we will also mean an arithmetical formularepresenting this function. 4 .2 Models of arithmetic In this paper, we will make extensive use of model-theoretic techniques. Allrelevant model-theoretic background may be found in [11]. Let us discusssome results of particular importance.
Definition 4.
Let M be any model over a finite language. A set p of for-mulae is a type if at most one free variable v and finitely many parame-tres a , . . . , a n occur in formulae contained in p , and for every finite subset φ ( v ) , . . . , φ n ( v ) there is an element a from M such that M | = φ i ( a ) forall i ≤ n . The type is realised if there is an element a ∈ M which satisfiesall formulae in p . We say that p is recursive (or computable) if the set of theGödel codes of formulae from p is computable. We say that M is recursivelysaturated if any recursive type over M is realised in M . Recursive saturation is of crucial importance due to the following theo-rem:
Theorem 5 (Barwise–Schlipf–Ressayre) . If M is a countable recursively satu-rated model of Th ⊃ PA and Th ′ is a computable theory consistent with the ele-mentary diagram of M , then M can be expanded to a model of Th ′ . Moreover, one can prove that there is always an expansion of M satisfy-ing Th ′ which is once again recursively saturated and thus also satisfies theassumptions of the above theorem. This property of countable recursivelysaturated models of PA is called chronic resplendence . Another importantproperty of recursively saturated models is that they can be relatively easilyclassified. Definition 6.
Let M | = PA. By the standard system of M , we mean thefamily of X ⊆ N such that X = A ∩ N , where A is definable with parametresin M . (Here and everywhere hereafter in the article, we identify the initialsegment in a model of PA isomorphic with natural numbers with the N itself.) Theorem 7 (Paris–Friedman) . Suppose that
M, N | = PA are countable and re-cursively saturated. Then M ≃ N iff they satisfy the same sentences and haveexactly the same standard systems. In the name "recursively saturated," there is admittedly slight tension with the currentnaming conventions where "computable" is the preferred expression, but "computably sat-urated" sounds extremely awkward.
Theorem 8 (MacDowell–Specker) . Let M be a model over a countable languagecontaining the language of arithmetic and suppose that it satisfies full inductionscheme for that language. Then there exists an elementary extension N ≻ M suchthat M and N have the same cardinality and for every a ∈ N \ M and every b ∈ M , N | = a > b . In fact, if we restrict ourselves to countable models, it is enough to as-sume that the model satisfies collection. This was proved in [13], Theorem28. For a general overview of model theory of collection scheme, see [5](where this result occurs as a part of Theorem 1.2 in a more general contextof models with a linear order).
Theorem 9 (Keisler) . Let M be a countable model over a countable language con-taining the language of arithmetic. Suppose that M satisfies full collection schemefor that language. Then there exists an elementary extension N ≻ M such that M and N have the same cardinality and for every a ∈ N \ M and every b ∈ M , N | = a > b . If N is an extension of M such that every new element in N is greaterthan all elements in M , then N is called an end-extension of M . If N isan end-extension of M , this is denoted by M ⊂ e N or M ≺ e N if it is, inaddition, elementary. In effect, MacDowell–Specker theorem states that anytheory in a countable language which extends PA and proves full inductionscheme for its language has a proper elementary end-extension of the samecardinality.By taking an arbitrary countable model M of PA, taking elementary end-extensions, and taking unions in the limit steps, we can construct a model M ′ ≻ e M which has cardinality ℵ , but whose all initial segments are count-able. Such models are called ω -like models . In a similar manner, we candefine κ -like models for an arbitrary cardinal κ . Note that if M is a κ -likemodel for a regular κ , then it must have cofinality κ which means that everysubset A ⊂ M of cardinality less than κ is bounded.An easy argument shows that if M ⊂ e N | = PA are nonstandard, then M and N have exactly the same standard systems. In particular, if M (cid:22) e N are countable and recursively saturated, then by Paris–Friedman Theorem(Theorem 7), M ≃ N . Moreover, this also holds if M, N are models of somecountable theory extending PA which has full induction.6 .3 Truth
This paper deals with compositional truth theories. Let us now introducesome of them. A systematic treatment can be found in [10]. See also [3],where the reader can find more information on arithmetic strength of clas-sical compositional truth theories. We will not begin with the most canon-ical example called CT − in which the truth predicate T is compositionalfor arithmetical formulae, but rather with its generalisation which plays acrucial technical role in our paper. Definition 10.
By CT − ↾ X , we mean a theory in the language of second-order arithmetic containing arithmetical symbols of L PA , a unary predicate T and the membership relation x ∈ X between first-order elements andsets. To the axioms of PA, we add the following formulae containing a freesecond-order variable X :1. ∀ x T ( x ) → x ∈ Sent L PA ∧ dp ( x ) ∈ X ∀ s, t ∈ ClTerm L PA T ( s = t ) ≡ ( s ◦ = t ◦ ) . ∀ φ ∈ Sent L PA (cid:16) dp ( ¬ φ ) ∈ X → T ¬ φ ≡ ¬ T φ (cid:17) . ∀ φ, ψ ∈ Sent L PA (cid:16) dp ( φ ∨ ψ ) ∈ X → T ( φ ∨ ψ ) ≡ T φ ∨ T ψ (cid:17) . ∀ φ ∈ Form ≤ L PA (cid:16) dp ( ∃ vφ ) ∈ X ∧ FV ( φ ) ⊆ { v } → T ∃ vφ ≡ ∃ xT φ [ x/v ] (cid:17) . ∀ ¯ s, ¯ t ∈ ClTermSeq L PA ∀ φ ∈ Form PA (cid:0) ¯ s ◦ = ¯ t ◦ → T φ (¯ t ) ≡ T φ (¯ s ) (cid:1) .Hence, CT − ↾ X states that T is a compositional truth predicate which be-haves well on formulae whose syntactic depth is in X . Moreover, by condition1. no other formulae are within the range of T . Formally, models for CT − ↾ X are models for the language L PA ∪ { T } with an extra assignment for X . Weoften employ standard model-theoretic conventions and write the interpre-tation of X in place of the variable X , for example ( M, T ) | = CT − ↾ M . Inpractice, we shall also treat X as an additional predicate and write modelsfor CT − ↾ X in the form ( M, T, X ) . We will also be writing ( M, T ) | = CT − ↾ X for X ⊂ M with the obvious meaning. We will essentially use the notationCT − ↾ X in one context: when X is a nonstandard initial segment (possiblywith the largest element).By CT ↾ X , we mean CT − ↾ X along with the full induction scheme for theextended language (i.e. formulae which may use the predicate T and the7ree variable X ). By CT − we mean a theory of an unrestricted compositionaltruth predicate (i.e., we do not restrict axioms of CT − ↾ X to formulae fromthe set X ). By CT we mean CT − with full induction.In our proof, we will need to impose an unpleasantly technical regularitycondition on the truth predicates. Essentially, we want to consider truthpredicates which only see syntactic trees of considered formulae and thevalues of terms which we plug in rather than specific terms and specificvariables over which we quantify.First, we introduce the notion of the structural template. Definition 11. If φ ∈ Form L PA , we say that b φ is its structural template iff • No constant symbol occurs in b φ . • No free variable occurs in b φ twice. • For every term t occurring in b φ , if all variables in t are free, then t is afree variable. • No variable occurs in b φ both as a bounded and as a free variable. • The formula φ can be obtained from b φ by renaming bounded variablesand substituting terms for free variables in such a way that no variableappearing in those terms becomes bounded. • b φ is the smallest formula with those properties.We say that formulae φ, ψ are structurally similar , φ ∼ ψ iff b φ = b ψ. Example 12. If φ = ∀ x ∃ x ∀ z ( x + S × ((( y × S
0) + 0) + ( y + z )) = ( z × ( x + S × Sy , then b φ = ∀ w ∃ w ∀ w ( w + v ) × ( v + ( v + w )) = ( w × ( w + v )) × v , where w i , v i are chosen so as to minimise the formula. Example 13.
1. The formulae φ = ∃ x ∀ x x + y = 0 and φ = ∃ y ∀ y y + x = z × SSS are structurally similar.2. The formulae ψ = ∃ x ∀ y x + y = 0 and ψ = ∃ y ∀ y x + y = 0 arenot structurally similar, because the first quantifies over two distinctvariables and the second does not.8. The formulae η = ∃ x ∀ y x + y = 0 and η = ∃ x ∀ y x + y = y are notstructurally similar because in the first one, the universally quantifiedvariable occurs only once under the scope of the quantifier.By induction on the complexity of formulae, one can easily check thatany formula has its structural template. By minimality, it is unique. Now,we are able to define our desired notion. Definition 14.
Let φ, ψ be two sentences. We say that they are structurallyequivalent iff they are structurally similar and there exist two formulae φ ∗ , ψ ∗ which differ from b φ by renaming bounded variables, and sequences ¯ s, ¯ t of closed terms such that ¯ s ◦ = ¯ t ◦ (i.e., all the terms in the sequence havethe same values) for which φ = φ ∗ (¯ t ) and ψ = ψ ∗ (¯ s ) .We denote this relation with φ ≈ ψ . Example 15.
Suppose that φ = ∃ x x + S S and ψ = ∃ y y + ( S × S S . Then φ ≈ ψ. Definition 16.
By the structural regularity principle (SRP) we mean thefollowing axiom: ∀ φ, ψ ∈ Sent L PA (cid:16) φ ≈ ψ → T φ ≡ T ψ (cid:17) . In what follows, we will essentially work with the theory CT − + SRP.One of the fundamental results in the theory of truth states that CT − , thetheory of compositional truth predicate, does not prove any new arithmeti-cal theorems: Theorem 17 (Kotlarski–Krajewski–Lachlan) . CT − is conservative over PA . On the other hand, as proved in [18], the presence of CT − has nontrivialconsequences on the model-theoretic side. Theorem 18 (Lachlan) . If ( M, T ) | = CT − , then M is recursively saturated.Moreover, the same holds for CT − ↾ [0 , c ] for every nonstandard c ∈ M . By a simple compactness argument, using arithmetical partial truth pred-icates (say, for Σ n classes, as every formula in Σ n has depth ≤ n ), one canshow that the theory of the restricted compositional truth predicate is stillconservative also when we consider the fully inductive variant. Theorem 19.
The theory saying " I is a nonstandard initial segment and CT ↾ I "is conservative over PA . ↾ I is simply a theory of truth predicate which satisfiesfull induction and compositional conditions for formulae whose depth is inan unspecified initial segment I . The obvious common strengthening ofthe two theories, i.e. CT, is much stronger than PA. For example, arguingby induction on the length of proofs, we can easily see that CT proves theconsistency of PA. Theorem 20. CT is not conservative over PA . More generally, using essentially the same proof, we can show the fol-lowing
Theorem 21. If ( M, T ) | = CT ↾ [0 , c ] , then every proof of contradiction of PA in M contains a formula of depth > c . Interestingly, we can conservatively add to CT − some specific form ofthe induction scheme. By internal induction we mean the axiom: ∀ φ ∈ Form ≤ L PA (cid:16) T φ (0) ∧ ∀ x ( T φ ( x ) → T φ ( x + 1)) → ∀ xT φ ( x ) (cid:17) . (INT)It essentially states that any set defined with a (possibly nonstandard) for-mula under the truth predicate satisfies the induction scheme. As we havealready mentioned, the following holds: Theorem 22 (Kotlarski-Krajewski-Lachlan) . CT − + INT is conservative over PA . The same result holds if we consider a theory CT − + INT + SRP. Wewill not show it, but we will discuss the proof in the Appendix, since it is aslight modification of the proof of Lemma 29.
Theorem 23. CT − + SRP + INT is conservative over PA . In this section, we will prove our main theorem. As we have written in theintroduction, Richard Kaye asked whether CT − with full collection scheme,but without any induction, is conservative over PA. We provide a partial The result is announced, but not really proved, in [16] and [6]. A complete proof occursin [19]. local collection ,LocColl, which consists of the following formulae: ∀ c (cid:16) ∀ x < a ∃ y φ [ T c /T ]( x, y ) → ∃ b ∀ x < a ∃ y < b φ [ T c /T ]( x, y ) (cid:17) , where φ is an arbitrary formula of L PA extended with a truth predicate and T c ( x ) abbreviates T ( x ) ∧ dp ( x ) ≤ c . Local collection expresses that any suchrestriction of the truth predicate, T c , satisfies full collection scheme. Theorem 24. CT − + LocColl is conservative over PA . The proof of the main result relies on the following simple and well-known observation.
Proposition 25.
Let M | = PA be a κ -like model for some regular cardinal κ . Let T ⊂ M . Then ( M, T ) satisfies full collection scheme.Proof. For any function f : M → M and any a ∈ M , the image of theinitial segment f [[0 , a ]] has less then κ elements and thus it is bounded. Thisimmediately implies that the collection scheme holds in M expanded withan arbitrary predicate.The observation suggests one possible strategy of the proof that collec-tion for the truth predicate is conservative over PA. If up to elementaryequivalence, for any countable M | = PA, we can find an ω -like elemen-tary end-extension M ′ with a truth predicate T , then this predicate T mustautomatically satisfy the collection scheme, which in turn implies that col-lection is conservative over PA. Basing on this approach, we are able toshow conservativity of local collection.We shall rely heavily on the construction of the disintegration of a truthpredicate. To better understand it, observe that in the arithmetical context,the truth predicate canonically determines a satisfaction relation S T ( x, y ) via the definition: S T ( φ, α ) := Form L PA ( φ ) ∧ Asn ( α, φ ) ∧ T ( φ [ α ]) . Then, the disintegration of a truth predicate is simply an infinite family ofprojections of S T along the singleton sets { φ } , for each formula φ in theconsidered model. This strategy of proof was explicitly suggested by Richard Kaye. However, as we alreadymentioned, the most obvious strategy of building ω -like models of CT − simply does notwork, due to the fact that not every model of CT − has an end-extension. efinition 26. Let ( M, T, X ) | = CT − ↾ X . The disintegration of T is a familyof predicates { X T φ } φ ∈ M which are interpreted in ( M, T, X ) by the condition α ∈ X T φ iff Asn ( α, φ ) ∧ T ( φ [ α ]) . The idea of disintegration is that we expand M with all relations (pos-sibly of nonstandard arity) which are arithmetically definable with possi-bly nonstandard formulae using the predicate T . Let us notice that thisnewly obtained structure corresponds to the original one in a very directway. Namely, the following equivalence holds for any η ∈ Form L PA ( M ) : ( M, T, X
T φ ) φ ∈ M | = ∀ α ∈ Asn ( η ) X T η ( α ) ≡ T η [ α ] . Let us proceed to the lemma which is the technical core of our proof.Note that, in view of Theorem 21, already here a dramatic difference fromthe induction scheme becomes apparent. In what follows, we will denotethe elementary diagram of a model M by ElDiag ( M ) . Lemma 27.
For any M | = PA and any c ∈ M , the theory ElDiag ( M )+ CT − ↾ [0 , c ]+ Coll is consistent.
Recall that CT − ↾ [0 , c ] is the theory of a compositional truth predicate forformulae of depth at most c .Since, we will need to use the above lemma iteratively, we will needits strengthening which is proved in almost the same way, so we will onlypresent the proof of the following version: Lemma 28.
Let M | = PA be a countable recursively saturated model and let c < d be any two elements. Suppose that ( M, T ) | = CT − ↾ [0 , c ] + Coll + SRP is recur-sively saturated in the expanded language. Then there exists T ′ ⊃ T such that ( M, T ′ ) | = CT − ↾ [0 , d ] + Coll + SRP . In [6], the conservativity of CT − has been demonstrated with an ele-gant, model-theoretic reasoning. The proof presented there allows numer-ous modifications in order to obtain finer results. We will make use of onesuch strengthening. We will use it in the proof of Lemma 28. Lemma 29.
Suppose that ( M, T, I ) | = CT − ↾ I + Coll + SRP is a countable modelrecursively saturated in the extended language with I an initial segment, possiblyempty. Then there exists T ′ ⊃ T such that ( M, T ′ ) | = CT − + SRP and, more-over, the model ( M, T, X T ′ φ ) φ ∈ M satisfies full collection scheme, where the family { X T ′ φ } φ ∈ M is the disintegration of T ′ . X φ satisfy collection jointly with the original predicate T .The lemma is proved by combining a resplendence argument and theEnayat–Visser construction. The details are standard and are given in theAppendix. We now turn to the proof of Lemma 28. Proof of Lemma 28.
Let ( M, T ) , c, d be as in the assumption. Using Lemma29, we construct a model ( M, T ∗ ) | = CT − + SRP such that T ∗ ⊃ T and thepredicates { X T ∗ φ } φ ∈ M satisfy full collection jointly with T .We will show that M has an elementary end extension M ′ such that forsome T ′ ⊆ M ′ extending T ∗ , ( M ′ , T ′ ) | = CT − ↾ M + Coll + SRP. In particular,it follows that for any c ∈ M , ( M ′ , T ′ ) | = ElDiag ( M ) + CT − ↾ [0 , c ] + Coll + SRP . By resplendence, this will conclude the proof. (Note that T ′ which we con-struct in the proof is not literally the same as T ′ satisfying the conclusion ofthe lemma, but we would like to avoid employing excessively heavy nota-tion.)Since ( M, T, X T ∗ φ ) φ ∈ M satisfies full collection scheme, by Keisler’s The-orem 9, it has an elementary end extension. By taking an ω -chain of suchelementary end-extensions, we obtain a model ( M ′ , X ′ T ∗ φ ) φ ∈ M elementarilyextending ( M, X T ∗ φ ) φ ∈ M , where M ′ is an ω -like model.Now, let T ′ = n φ ( t , . . . , t e ) ∈ Sent L PA ( M ′ ) | φ ∈ Form L PA ( M ) , h t , . . . , t e i ∈ ClTermSeq L PA ( M ′ ) , ( M ′ , X ′ T ∗ φ ) φ ∈ M | = X T ∗ φ ( h t ◦ , . . . , t e ◦ i ) o , where we conflate a sequence of values and a corresponding assignment.Notice that we do not assume that the sequence h t ◦ , . . . , t e ◦ i has standardlength or standard values. Let finally: T ′ = n φ ∈ Sent L PA ( M ′ ) | ∃ ψ ∈ Form L PA ( M ) ∃h t , . . . , t e i ∈ ClTermSeq L PA ( M ′ ) ψ ≈ φ ( t , . . . , t e ) o . We claim that ( M ′ , T ′ ) | = CT − ↾ M + Coll + INT + SRP. This model satis-fies collection scheme by Proposition 25, since M ′ is ω -like. Notice that T ′ was defined only for sentences obtained by substituting terms into formu-lae from M , whereas we want to make sure that it is defined on formulae13hose depth is in M . However, since M ′ is an end-extension of M for anyformula whose depth is in M , its syntactic template is in M as well.We first check that the compositional conditions are satisfied for T ′ andthe sentences φ ( t , . . . , t e ) for φ ∈ Form L PA ( M ) by cases which depend onthe syntactic shape of a formula φ . For example, let φ ( t , . . . , t e ) = ∃ vψ ( v, t , . . . , t e ) where ψ ∈ Form L PA ( M ) and h t , . . . , t e i ∈ M ′ . The equivalence ∀ α ∈ Asn ( φ ) (cid:16) X T ∗ φ ( α ) ≡ ∃ β ∼ v α X T ∗ ψ ( β ) (cid:17) holds in ( M, X T ∗ φ ) φ ∈ M , since T ∗ satisfies compositional conditions. There-fore it must hold in ( M ′ , X ′ T ∗ φ ) φ ∈ M by elementarity. So by definition T ′ sat-isfies the compositional condition for the quantifier for the formula φ . Theother cases are analogous.The compositional conditions are satisfied for other sentences with depthin M as well. Take any formula φ ∈ M ′ such that dp ( φ ) ∈ M . First observethat if φ ∼ ψ and ψ ∈ M , then b φ ∈ M , since by elementarity b ψ ∈ M andthese two are equal. Then we check that T ′ is compositional by case distinc-tion depending on the main connective or quantifier in φ .For instance, suppose that φ = ∃ vη , T ′ φ holds, and φ ≈ ψ = ( ∃ wξ ) ∈ M such that T ′ ψ holds. Then by compositionality of T ′ , there exists x ∈ M ′ such that T ′ ξ ( x ) holds. Now, since η ( x ) ≈ ξ ( x ) , by definition T ′ η ( x ) holdsas well. An analogous reasoning shows that if T ′ η ( x ) holds for some x ∈ M ′ ,then T ′ φ holds. The argument for disjunction is similar.The argument for negation is the only place where we use SRP. Namely,suppose that T ′ ¬ φ holds for some φ ∈ M ′ . We want to show that T ′ φ doesnot hold. Suppose otherwise. By definition of T ′ , there exists ψ ≈ φ suchthat ψ = ψ ∗ ( t , . . . , t n ) for some ψ ∗ ∈ Form L PA ( M ) and T ′ ¬ ψ holds. Bycompositionality, T ′ ψ does not hold. Now, by SRP T ′ η cannot hold for any η ≈ ψ . In particular, it cannot hold for any η ≈ φ . The other implicationfor the compositionality of negation can be proved with a simple argumentsimilar to the argument for the existential quantifier.It follows immediately by the construction that T ′ satisfies the structuralregularity property SRP.Now, we are ready to prove our theorem. Proof of Theorem 24.
Let M | = PA be any countable recursively saturatedmodel. Fix any sequence ( a n ) n ∈ ω cofinal in M . Using Lemma 27 in theinitial step and Lemma 28 and chronic resplendence in the induction step,we construct a sequence of predicates T n such that ( M, T n ) | = CT − ↾ [0 , a n ] + Coll , T := S n ∈ ω T n . Then we readily check that ( M, T ) | = CT − + LocColl. Since M was arbitrary, this concludes the proof. As we have already noted, the behaviour of local collection is in stark con-trast to the behaviour of local induction which is its natural analogue forthe induction scheme. More precisely, let us define the instances of localinduction, LocInd, as follows: ∀ c (cid:16) φ [ T c /T ](0) ∧ ∀ x (cid:0) φ [ T c /T ]( x ) → φ [ T c /T ]( x + 1) (cid:1) → ∀ xφ [ T c /T ]( x ) (cid:17) , where φ is an arbitrary formula in the language L PA extended with a truthpredicate and T c ( x ) is an abbreviation for T ( x ) ∧ dp ( x ) ≤ c . In other words, ( M, T ) | = CT − + LocInd iff ∀ c ∈ M, ( M, T c ) | = CT ↾ [0 , c ] , (LocInd)so local induction scheme expresses that any restricted truth predicate T c satisfies full induction.One can easily observe that local induction is not conservative over PA,since it proves the consistency of PA. Indeed, by composing Theorem 21 andthe above condition (LocInd) one gets that for every c ∈ M , every proof of in PA contains a formula of complexity > c . Let us briefly recall thewhole argument: take any proof d in PA, say, in Hilbert calculus. There ex-ists c such that all sentences occurring in that proof have complexity smallerthan c . Take the restricted predicate T c and show, using local induction, thatevery sentence in that proof is true. Consequently, the conclusion of theproof has to be true, and thus it cannot be of the form " = 0 ."The above proof essentially shows that in CT − + LocInd, we can showthe following principle of global reflection : ∀ φ ∈ Sent L PA Pr PA ( φ ) → T φ, (GR)where Pr PA ( x ) is the canonical provability predicate for PA. In order toprove global reflection, we fix any φ which is provable in PA, we fix anyproof of φ and take any b such that all formulae in the proof have depthsmaller than b . Then we take the restriction T b and show by induction onthe length of derivation that all formulae in the proof are true under allassignments. 15s shown by Kotlarski in [15], CT − with global reflection proves ∆ -induction for the truth predicate. His argument was later refined in twoways by Cezary Cieśliński: firstly, in [2] it was shown that reflection overfirst order logic (i.e. (GR) with Pr PA changed to Pr ∅ ) is sufficient to prove ∆ induction. Secondly, in [1] it was shown that the closure under proposi-tional logic principle, i.e. the sentence ∀ φ Pr T Prop ( φ ) → T ( φ ) , where Pr T Prop ( φ ) expresses that φ is provable from true premises in purepropositional calculus, is enough to yield bounded induction. Kotlarski in [15] characterised the arithmetical strength of global reflec-tion in terms of the following family of theories:Th = PATh n +1 = {∀ xφ ( x ) | φ ( x ) ∈ L PA , ∀ k ∈ ω Th n ⊢ φ ( k ) } Theorem 30 (Kotlarski) . CT − + (GR) is arithmetically conservative over PA + { Con ( Th n ) | n ∈ ω } . An easy argument shows that the above arithmetical theory is equiva-lent to ω -many iterations of the uniform reflection principle over PA. Detailsconcerning the inclusion of Kotlarski’s theory in the iterations of reflectioncan be found in the paper [23] and in the second author’s PhD Thesis, [7].It turns out that the content of LocInd can be characterised in a veryprecise manner. We have just shown that it implies global reflection GR. Itturns out that LocInd is exactly equivalent to GR. Fact 31. CT − + LocInd is equivalent to CT − + GR . Moreover, it was shown in the second author’s PhD thesis [7] that CT − + GR is equivalent to CT , the compositional truth theory CT − extended withbounded induction for the full language which immediately allows us toobtain an equivalent characterisation. Fact 32. CT − + LocInd is equivalent to CT . We note, however, that the last principle expresses closure of the set of true sentencesunder a logical reasoning. Thus we potentially require something more than in the previoustwo reflection principles.
16t is relatively straightforward to show that CT − + GR proves internalinduction, INT. Let ind ( φ ) abbreviate the axiom of induction for a formula φ ∈ Form ≤ L PA , i.e. the sentence φ (0) ∧ ∀ x (cid:0) φ ( x ) → φ ( x + 1) (cid:1) → ∀ xφ ( x ) . We work in CT − + GR. Since for every φ ∈ Form ≤ L PA we have Pr PA ( ind ( φ )) ,by GR it follows that T ( ind ( φ )) holds. By compositional axioms and exten-sionality we obtain the sentence T ( φ (0)) ∧ ∀ x (cid:0) T ( φ ( x )) → T ( φ ( x + 1)) (cid:1) → ∀ xT ( φ ( x )) . It is a classical fact of first-order arithmetic that there exist partial Σ n -truth predicates. More precisely, the following holds provably in PA: Theorem 33.
For every n , there exists a formula Tr n such that for every sentence φ with dp ( φ ) ≤ n (in fact, for φ ∈ Σ n ), the following equivalence holds: Tr n ( φ ) ≡ φ. This theorem formalises in PA, hence we have: ∀ c ∀ φ ∈ Sent L PA (cid:16) dp ( φ ) ≤ c → Pr PA ( Tr c ( φ ) ≡ φ ) (cid:17) . By GR and the compositional axioms we obtain: ∀ c ∀ φ ∈ Sent L PA (cid:16) dp ( φ ) ≤ c → ( T Tr c ( φ ) ≡ T φ ) (cid:17) . Let Θ c ( x ) be defined as T Tr c ( x ) ∧ x ∈ Sent L PA ∧ dp ( x ) ≤ c . Fix any instanceof the induction scheme containing the truth predicate T c ( x ) : φ [ T c ](0) ∧ ∀ x (cid:0) φ [ T c ]( x ) → φ [ T c ]( x + 1) (cid:1) → ∀ xφ [ T c ]( x ) . Since T c and Θ c are equivalent by the above considerations, the displayedsentence is equivalent to: φ [Θ c ](0) ∧ ∀ x (cid:0) φ [Θ c ]( x ) → φ [Θ c ]( x + 1) (cid:1) → ∀ xφ [Θ c ]( x ) . But, by applying compositional axioms for the full truth predicate T , we can"pull it up" from Θ c to the top of the formula φ , thus obtaining: T φ [ Tr ′ c ](0) ∧ ∀ x (cid:0) T φ [ Tr ′ c ]( x ) → T φ [ Tr ′ c ]( x + 1) (cid:1) → ∀ xT φ [ Tr ′ c ]( x ) , where Tr ′ c ( x ) is the formula Tr c ( x ) ∧ x ∈ Sent L PA ∧ dp ( x ) ≤ c . The lastformula is however an instance of the internal induction axiom and thus isprovable in CT − + GR. This shows that LocInd holds in CT − + GR. For a detailed discussion of arithmetical truth predicates, see [9], Chapter I, Section 1(d),pp.50–61. The strength of B Σ n ( T ) Let us now consider a question whether adding a little bit of collection toCT increases the arithmetical strength of the latter theory. Let us denoteB Σ n ( T ) := CT + Σ n -Coll , where Σ n -Coll is the restriction of full collection scheme to Σ n formulae inthe expanded language. It is easy to observe that, as in the purely arithmeti-cal setting, we have B Σ n +1 ( T ) ⊢ CT n , and CT n +1 ⊢ Con ( CT n ) , hence already B Σ ( T ) is arithmetically non-conservativeover CT . What is left is the case of Σ collection: we shall show that it is Π -conservative over CT (over the full language with the truth predicate),which implies that B Σ ( T ) is arithmetically conservative over CT as forevery arithmetical sentence φ , T ( φ ) is an atomic sentence of the expandedlanguage equivalent to φ (provably in CT − ). More generally, the situationfor fragments of CT parallels the one well known from fragments of PA: Theorem 34.
For every n ≥ , B Σ n +1 ( T ) is Π n +2 conservative over CT n in theextended language. In particular for all n , B Σ n +1 ( T ) is arithmetically conservativeover CT n . Although the proof follows essentially by the same pattern of reasoningas in the classical Paris–Friedmann result (see [9], Theorem 1.61, Chapter IVor [11], Corollary 10.9), one detail has to be taken care of. It is the contentof the following lemma. Let us recall that if we have a model M and a set I ⊆ M , then sup M ( I ) := { x ∈ M | ∃ b ∈ I M | = x < b } . If M is a model of PA − and I is closed under multiplication, then sup M ( I ) isa substructure of M . If additionally M | = CT − , then we can naturally view sup M ( I ) as a substructure of M . Lemma 35.
Suppose M (cid:22) N are models of CT . Then sup N ( M ) | = CT − . Con-sequently, sup N ( M ) | = CT .Proof. The only problematic issue is whether sup N ( M ) satisfies the compo-sitional axiom for the existential quantification, i.e. ∀ φ ∈ Form ≤ PA (cid:16) FV ( φ ) ⊆ { v } → T ∃ vφ ≡ ∃ x T φ ( x ) (cid:17) , φ and v as above and put I = sup N ( M ) . Given that I ⊆ N , the non-obvious part is whether I validates the implication T ∃ vφ → ∃ xT φ ( x ) . Work-ing in I , assume T ( ∃ vφ ) . Let d ∈ M be greater than ∃ vφ (as an element of N ). Consider the following sentence ∃ c ∀ v < d ∀ ψ < d (cid:18)(cid:0) Var ( v ) ∧ Form ≤ L PA ( ψ ) ∧ T ∃ vψ (cid:1) → ∃ x < c T ψ ( x ) (cid:19) . The above is true in M , since it is equivalent to ∃ c ∀ v < d ∀ ψ < d (cid:18)(cid:0) Var ( v ) ∧ Form ≤ L PA ( ψ ) ∧ T d +1 ∃ vψ (cid:1) → ∃ x < c T d +1 ψ ( x ) (cid:19) which is an instance of the strong collection scheme for T d +1 and each re-striction of T is fully inductive by LocInd which is equivalent to CT by Fact32. Fix c ∈ M witnessing the existential quantifier. By elementarity ∀ v < d ∀ ψ < d (cid:18)(cid:0) Var ( v ) ∧ Form ≤ L PA ( ψ ) ∧ T ∃ vψ (cid:1) → ∃ x < c T ψ ( x ) (cid:19) is true in N , but as it is a ∆ ( T ) sentence with parameters from I , it holdsin the latter model as well (by definition I ⊆ e N ). This ends the proof since d majorizes both v and φ in sup N ( M ) .The rest of the proof of Theorem 34 can be carried out as in the case ofarithmetical theories. We will use the following lemma as the key ingredi-ent. Lemma 36.
Let I (cid:22) n N be two models in a language extending L PA , possibly withadditional predicates, where n ∈ N (we allow n = 0 ). If N satisfies Σ n -induction,then I satisfies Σ n +1 -collection. The proof of this fact for the language of arithmetic is given e.g. in[11], Proposition 10.5. It transfers to languages extending L PA after obvi-ous modifications. Proof of Theorem 34.
In order to prove the theorem, it is enough to show thatif φ ∈ Σ n +2 is satisfied in some model of CT n , then it is satisfied in somemodel of B Σ n +1 ( T ) .So fix a model M | = CT n + φ , where φ = ∃ x ∀ yψ ( x, y ) , ψ ∈ Σ n . Let N bean elementary extension of M such that sup N ( M ) = N . Put I = sup N ( M ) .By elementarity, there exists c ∈ M such that N | = ∀ yψ ( c, y ) , ψ ( x, y ) is ∆ , then it automatically follows that the same is true in I , which,by Lemma 35 is a model of CT and thus by Lemma 36, a model of B Σ ( T ) .This concludes the proof for n = 0 .For greater n ’s we have to show that I ≺ n N . This will conclude ourargument since then again we obtain that I | = ∀ yψ ( c, y ) by elementarityand that I | = B Σ n +1 ( T ) by Lemmata 35 and 36.We show Σ n -elementarity: employing the Tarski–Vaught test, it is suffi-cient to show that for all θ ( x, y ) ∈ Σ n and all b ∈ IN | = ∃ xθ ( x, b ) ⇒ ∃ d ∈ I N | = θ ( d, b ) . So fix θ ( x, y ) ∈ Σ n , b ∈ I and assume N | = ∃ xθ ( x, b ) . Fix e ∈ M such that N | = b < e . Since M | = CT n we have an f such that M | = ∀ y < e (cid:0) ∃ xθ ( x, y ) → ∃ x < f θ ( x, y ) (cid:1) , hence the same is true in N by elementarity. It follows that for some d , N | = θ ( d, b ) ∧ d < f . Any such d must belong to sup N ( M ) which concludesthe proof. It follows that sup N ( M ) is a Σ n elementary initial segment of N | = CT n . This concludes the proof.Kotlarski and Ratajczyk, in [17], gave a characterisation of arithmeticalconsequences of CT n in terms of the transfinite induction. For each k ∈ N define (below, α β denotes ordinal exponentiation): ω ( k ) = kω m +1 ( k ) = ω ω m ( k ) . Assume a standard coding of ordinals below φ (0) (see e.g. [8]) and denoteby TI ( α, φ ) the transfinite induction for φ up to α , the formula ∀ β (cid:0) ∀ γ ≺ βφ ( γ ) → φ ( β ) (cid:1) −→ ∀ γ ≺ αφ ( γ ) . Theorem 37 (Kotlarski, Ratajczyk, [17]) . For every n , the sets of arithmeticalconsequences of CT n and PA + (cid:8) TI ( ε ω n ( k ) , φ ) | φ ∈ L PA , k ∈ N (cid:9) coincide. Corollary 38.
For every n , the sets of arithmetical consequences of B Σ n +1 ( T ) and PA + (cid:8) TI ( ε ω n ( k ) , φ ) | φ ∈ L PA , k ∈ N (cid:9) coincide. End-extensions of models of CT − One obvious strategy to show that CT − + Coll is conservative over PA wouldbe to show that any countable model of CT − has a countable end-extensionand thus build an ω -chain of models of CT − . However, in general thisstrategy is doomed to fail, as witnessed by the following result of Smith([22], Theorem 4.3). Theorem 39 (Smith) . There exists a countable model of CT − which has no end-extension.Sketch of a proof. Take a model ( M, T ) | = CT − in which there is a formula φ such that (cid:8) h x, y i ∈ M | ( M, T ) | = T φ ( x, y ) (cid:9) is a bijection from M to itsproper initial segment [0 , a ] . (It can be shown that such a predicate T existsby a modification of Enayat–Visser argument. A more complete argumentcan be found in the paper [22] of Smith.)Now, all the following sentences are in T :1. ∀ x, y (cid:16) φ ( x, y ) → y < a (cid:17) . ∀ x , x , y (cid:16) φ ( x , y ) ∧ φ ( x , y ) → x = x (cid:17) .3. ∀ x, y , y (cid:16) φ ( x, y ) ∧ φ ( x, y ) → y = y (cid:17) .4. ∀ x ∃ yφ ( x, y ) . Therefore, if ( M, T ) has an end-extension ( N, T ′ ) , then all the above sen-tences will be in T ′ . This means that in ( N, T ′ ) , the formula φ defines abijection from N to [0 , a ] . However, this is impossible, since all elements in [0 , a ] are already values of elements from M under this bijection.In the light of the above result, once could hope to find some extension Θ of CT − such that:1. Θ is conservative over PA.2. Any countable model from Θ has an end-extension to a model of Θ .3. Θ is closed under taking unions of end-extensions.Our initial hope was that CT − + INT may fit the bill. Note that the lack ofinternal induction (or, in fact, of internal collection) is exactly the obstruc-tion which makes it impossible for a model introduced by Smith to have21n end-extension. Unfortunately, we did not manage to settle the questionwhether any countable model ( M, T ) | = CT − + INT has an end-extensionto a model of CT − + INT . However, we managed to obtain the followingpartial result:
Theorem 40. If ( M, T ) | = CT − + INT + SRP is a countable model recursivelysaturated in the extended language, then there exists an end extension ( N, T ′ ) | = CT − + INT + SRP which is also recursively saturated in the extended language.
Before we proceed to the proof, we need one more lemma:
Lemma 41.
Let ( M, T, I ) | = CT − ↾ I + SRP + INT be any countable model re-cursively saturated in the expanded language. Then there exists T ′ ⊃ T such that ( M, T ) | = CT − + SRP + INT . The lemma is proved by using resplendence and a slight modificationof the Enayat–Visser argument. We will briefly discuss its proof in the Ap-pendix.
Proof of Theorem 40.
Let ( M, T ) | = CT − + INT + SRP be countable and re-cursively saturated. As in the proof of Lemma 27, let X T φ , φ ∈ Form L PA ( M ) be the disintegration of T . The model ( M, X
T φ ) φ ∈ M satisfies full inductionscheme, so it has an elementary end extension to a model ( N, X ′ T φ ) φ ∈ M . Asin the proof of Lemma 28, from the predicates X ′ φ , we can obtain a predicate T ′ such that ( N, T ′ ) | = CT − ↾ M + INT + SRP. (One can check with a directelementarity argument that the construction preserves INT.) In particular, N is recursively saturated. Observe that even if ( N, X ′ T φ ) φ ∈ M is recursivelysaturated in the expanded language (it can taken to be so), ( N, T ′ ) need notbe, hence Lemma 41 need not apply. However, by a resplendence argument,we can show that we can find ( N ′ , T ′′ ) so that: • The model ( N ′ , M, T, T ′′ ) is recursively saturated. • M ≺ e N ′ . • T ⊂ T ′′ . • ( N ′ , T ′′ , M ) | = CT − ↾ M + INT + SRP. • ( M, T ) | = CT.More precisely, by Paris–Friedman Theorem 7, M and N are isomorphic.Therefore, in N there is a predicate T ∗ such that ( N, T ∗ ) ≃ ( M, T ) . Let f bean isomorphism between these models. The structure ( N, T ∗ , M, T, f, T ′ ) ( N, T ∗ ) usingadditional predicates I, T I , g, e T : • I is an elementary initial segment of N . • g : ( N, T ∗ ) → ( I, T I ) is an isomorphism. • ( N, e T ) | = CT − ↾ I + INT + SRP . • e T ⊃ T I .In order to see that the theory is consistent, identify I with M , T I with T , g with f , and e T with T ′ . By resplendence, it can be realised by interpreting I, T I , g, e T as relations in N in such a way that the obtained model is recur-sively saturated. Since ( I, T I ) is isomorphic with ( M, T ) , the latter modelhas an end extension ( N ′ , T ′′ ) with the desired properties.We could hope that we could build ω -like models of CT − by takingchains of recursively saturated models of CT − + INT. Unfortunately, thereis a serious obstruction to this strategy: a union of recursively saturatedmodels need not be recursively saturated and at this point we do not see aclear strategy to circumvent this problem.Another possible strategy which one could consider is to show that if M (cid:22) e N and T ⊂ T ′ such that ( M, T ) | = CT − + INT, ( N, T ′ ) | = CT − ↾ M + INT, then T ′ can be extended to a predicate T ′′ ⊃ T ′ such that ( N, T ′′ ) | = CT − + INT. I.e., one could hope that we can get rid of the resplendenceargument in the above proof. However, we unfortunately know that thisis in general impossible without further assumptions. Indeed, there existcountable models
M, N and predicates
T, T ′ such that: • M (cid:22) e N . • ( M, T ) | = CT − + SRP + INT. • ( N, T ′ ) | = CT − ↾ M + SRP + INT. • T ⊂ T ′ ,in which T ′ cannot be further extended to a predicate T ′′ satisfying CT − .The proof of this fact will appear in [14].23et us make one last remark: the example given by Smith shows howa model can fail to have an end-extension because of how its truth predi-cate looks like and it has nothing to do with the structure of the underly-ing arithmetical model. However, quite surprisingly if ( M, T ) is a model ofCT − , possibly uncountable, then M has an elementary end-extension to amodel N which then can be expanded to a model of CT − . This has beenobserved by Albert Visser. In the proof, we use the following result, orig-inally proved essentially in [20]. A (hopefully more perspicuous) proof ofthis exact statement will appear in [14]. Theorem 42.
Let ( M, T ) | = CT − . Then there exists a T ′ and a nonstandard c ∈ M such that ( M, T ′ ) | = CT ↾ [0 , c ] . Theorem 43 (Visser) . Suppose that ( M, T ) | = CT − . Then there exists an ele-mentary end-extension M (cid:22) e N and a T ′ such that ( N, T ′ ) | = CT − .Sketch of a proof. Let ( M, T ) | = CT − and let ( M, T ) | = CT ↾ [0 , c ] with c non-standard which exists by Theorem 42.Consider the L PA ∪ { T } -definable set Θ containing the compositionalaxioms of CT − and all arithmetical sentences φ , possibly nonstandard, suchthat φ ∈ T . Within PA, we can formalise the Enayat–Visser conservativityproof for CT − over PA and show that Θ is consistent. More precisely, forevery n , (the straightforward arithmetisation of) the following assertion isprovable in PA (see [4], Lemma 4.3 which is formulated for the language ofarithmetic but generalises to other theories with full induction extendingPA): ( ∗ ) If K | = I ∆ + exp is any ∆ n model, then there exists a ∆ n +1 model ( N, T ′ ) | = Th and K is an L PA -elementary submodel of N .where Th denotes the compositional axioms for the truth predicate fromCT − . Using cut-elimination one checks that (the set of sentences satisfying) T is consistent and then applies ( ∗ ) to a ∆ definable model K of T (viewedas a set of sentences). We can fix ( N, T ′ ) given by the above claim. The original argument was slightly different, since it did not use the results of [4]. Weare grateful for his permission to include here the proof of this unpublished result. Actually, in order to prove the result below, we only need to have a predicate T ′ whichsatisfies uniform Tarski biconditionals for standard formulae and full induction scheme forthe extended language. This is exactly what Theorem 4.1 in [20] gives us. However, wewanted to use a formulation more in line with the notation of this paper. Different approaches to the formalisation of the conservativity of CT − within PA werepresented in [19] and [6]. ( N, T ′ ) satisfies CT − , N is an end-extension of M and for everystandard formula φ ( x , . . . , x n ) and elements a , . . . , a n ∈ M , N | = φ ( a , . . . , a n ) iff ( M, T ) | = T φ ( a , . . . , a n ) iff M | = φ ( a , . . . , a n ) . This guarantees that M (cid:22) N which concludes the proof. Note that in the above theorem, we do not make any assumptions onthe cardinality of M . For countable models, the theorem may be provedin a much easier way, since by Lachlan’s Theorem [18], for every model ( M, T ) | = CT − , the underlying arithmetical model M is recursively sat-urated. By a theorem of Friedman, every countable recursively saturatedmodel M has an elementary end extension to another such model N . Thismodel, in turn, can be expanded to a model of CT − by resplendence ofcountable recursively saturated models, since CT − is conservative over PAby Theorem 17. In this article, we made use of some facts which relied on modification of theEnayat–Visser proof. The required changes are rather straightforward, andtherefore we moved the proofs to the Appendix. We tried to make the pre-sentation reasonably self-contained, but the reader might want to consultthe original paper [6].Let us begin with a proof of Lemma 29. We restate it for the convenienceof the reader.
Lemma 44.
Suppose that ( M, T, I ) | = CT − ↾ I + Coll + SRP is a countable modelrecursively saturated in the extended language with I an initial segment, possiblyempty. Then there exists T ′ ⊃ T such that ( M, T ′ ) | = CT − + SRP and, more-over, the model ( M, T, X T ′ φ ) φ ∈ M satisfies full collection scheme, where the family { X T ′ φ } φ ∈ M is the disintegration of T ′ .Proof. Let ( M, T, I ) | = CT − ↾ I + Coll + SRP. We will find an extension: ( M, T, I ) ⊂ ( M ∗ , T ∗ , I ∗ , T ′ ) such that More information on Arithmetised Completeness and the fact that inner models in mod-els of PA give rise to end-extensions may be found in [11], Section 13.2. This can be seen as follows: by resplendence, M has an initial segment I (cid:22) M suchthat I is also recursively saturated. By Paris–Friedman Theorem 7, I ≃ M . Since M isisomorphic to I and I has an elementary recursively saturated end-extension, the same istrue for M . ( M, T, I ) (cid:22) ( M ∗ , T ∗ , I ∗ ) is an elementary extension,2. ( M ∗ , T ′ ) | = CT − + SRP,3. T ∗ ⊆ T ′ and4. for every φ , . . . , φ n ∈ Form L PA ( M ∗ ) , the predicates X T ′ φ i satisfy fullcollection jointly with T ∗ .Since ( M, T, I ) is resplendent, this will conclude our proof. Note that thepredicates X T ′ φ are not present in the language, but point 4 in the above listcan be expressed in terms of the predicate T ′ alone by quantifying univer-sally over the formulae φ , . . . , φ n . (Which is important, since otherwise theresplendence argument would not be valid.)We will construct an ω -chain of models ( M j , T j , I j , X jφ , S j ) φ ∈ M j − usingauxiliary predicates X jφ such that for any k , the chain { ( M j , T j , I j , X jφ ) } j ≥ k,φ ∈ M k − is elementary. Finally, we will set S M j = M ∗ , S I j = I ∗ , S T j = T ∗ .The predicates S j will be satisfaction predicates compositional for formu-lae from the model M j − if j > . Finally, we will set: T ′ = { φ ∈ Sent L PA ( M ∗ ) | ∃ j φ ∈ Sent L PA ( M j ) ∧ ( φ, ∅ ) ∈ S j +1 } . At the initial step, we set M = M, I = I, T = T ′ = T . By convention M − = ∅ . At each step, we inductively take ( M j +1 , T j +1 , I j +1 , X j +1 φ , S j +1 ) φ ∈ M j to be the model of the theory Θ j expanded with extra predicate consistingof the following axioms: • ElDiag ( M j , T j , I j , X jφ ) φ ∈ M j − . • (Compositional axioms) Comp ( φ ) , for φ ∈ Form L PA ( M j ) which statethat S j +1 behaves compositionally with respect to φ . Shortly, we willgive a more precise definition. • ( T ′ contains T ) ∀ xT j +1 ( x ) → S j +1 ( x, ∅ ) . • (The sequence S j stabilises) S j +1 ( φ, α ) , where φ ∈ M j − and ( φ, α ) ∈ S j . • (The definition of X j +1 φ ) X j +1 φ ( α ) ≡ α ∈ Asn ( φ ) ∧ S j +1 ( φ, α ) , φ ∈ Form L PA ( M j ) . • All the instances of the collection scheme in the language with arith-metical symbols and the predicates T j +1 , X j +1 φ , where φ ∈ Form L PA ( M j ) .26 (Extensionality Axiom) ∀ φ ∈ Form L PA ∀ α ∈ Asn ( φ ) S j +1 ( φ, α ) ≡ S j +1 ( φ [ α ] , ∅ ) . • (Structural Regularity Axiom) ∀ φ, ψ ∈ Sent L PA (cid:16) φ ≈ ψ → S j +1 ( φ, ∅ ) ≡ S j +1 ( ψ, ∅ ) (cid:17) . Recall that the structural equivalence relation φ ≈ ψ was introduced in Def-inition 14. Notice one important (but admittedly subtle) difference betweenthis formulation and the original proof of [6]. In the original version, Enayatand Visser required that the constructed chain be elementary only in thesignature of the base language. Here, we additionally require elementar-ity with respect to the predicates X jφ . . Let us explain a bit what actuallyhappens.In the j -th step, we introduce a predicate S j +1 which is compositionalfor the formulae from the current model M j . Together with this model, weintroduce a family of predicates X j +1 φ which are the disintegration of thepredicate S j +1 , but defined only for formulae in the current model M j . No-tice that by elementarity, we actually require that T j +1 behaves like T j , I j +1 behaves like I j , and crucially, X j +1 φ behaves like X jφ whenever φ ∈ M j − . Inother words: we really do require more regularity that in the usual Enayat–Visser construction. For instance, if φ ( v ) ∈ Form L PA ( M j ) has only one freevariable, and x is the smallest element satisfying φ under S j +1 , (the smallestelement such that S j +1 ( φ, α ) holds, where α ( v ) = x ), then by elementar-ity requirement for X j +1 φ , x will stay the smallest such element throughoutthe whole construction. On the other hand, in the original construction ofEnayat–Visser, we essentially only require that S j +1 ( φ, α ) still holds in thelater stages. This is enough to guarantee that the compositional conditionshold in the final model, but would not suffice to guarantee that the disinte-gration of the final model satisfies full collection. To this end, we need someelementarity, and we introduce the predicates X jφ to conveniently describewhat amount of elementarity is needed. One last remark for the scrupulousreader: from the strict reading of our notation, it follows that there are lotsof predicates X jc , where c is not a formula. We keep them, as we do notwant to overload our notation, but they are harmless to the construction.The compositional axioms Comp ( φ ) are defined as the conjunction ofthe following formulae: • ∀ s, t ∈ Term L PA ∀ α ∈ Asn ( φ ) (cid:16) φ = ( s = t ) → (cid:0) S j +1 ( s = t, α ) ≡ s α = t α (cid:1)(cid:17) . ∀ ψ ∈ Form L PA ∀ α ∈ Asn ( φ ) (cid:16) φ = ¬ ψ → (cid:0) S j +1 ( φ, α ) ≡ ¬ S j +1 ( ψ, α ) (cid:1)(cid:17) . • ∀ ψ, η ∈ Form L PA ∀ α ∈ Asn φ (cid:16) φ = ψ ∨ η → (cid:0) S j +1 ( φ, α ) ≡ S j +1 ( ψ, α ) ∨ S j +1 ( η, α ) (cid:1)(cid:17) . • ∀ ψ ∈ Form L PA ∀ v ∈ Var ∀ α ∈ Asn ( φ ) (cid:16) φ = ( ∃ vψ ) → (cid:0) S j +1 ( φ, α ) ≡∃ β ∼ v αS j +1 ( ψ, β ) (cid:1)(cid:17) .Note that we officially work in a language without conjunction or universalquantifiers. This choice is simply for convenience and does not affect ourresults.By direct and simple verification, we check that if our construction canbe performed, the resulting model ( M ∗ , T ∗ , I ∗ , T ′ ) satisfies our requirements,i.e.: • ( M ∗ , T ∗ , I ∗ ) | = ElDiag ( M, T, I ) . • ( M, T ′ ) | = CT − . • T ∗ ⊆ T ′ . • Full collection scheme holds for the arithmetical language expandedwith the predicates T ∗ , X T ′ φ , where φ ∈ Form L PA ( M ∗ ) (where T ′ φ arethe disintegration of T ′ ).In order to verify the last item, notice that every such collection axiom con-tains only finitely many predicates X T ′ φ . A model with finitely many suchpredicates is a union of the elementary chain of models containing the pred-icates X jφ . In our construction, we guaranteed that collection scheme holdsfor the arithmetical language expanded with finitely many such predicatesand the predicates T j corresponding to T ∗ . Therefore, full collection schemeholds by elementarity.So the only thing which we need to check is whether our constructioncan indeed be performed. In other words, we need to verify whether thetheories Θ j can be inductively shown to be consistent. This will be provedin a separate lemma. Lemma 45.
All theories Θ j defined in the above proof of Lemma 29 are consistent. In order to deal with the regularity axioms in the following proof, it willbe handy to have some extra notation. Let φ, ψ ∈ Form L PA , α ∈ Asn ( φ ) , β ∈ ( ψ ) . We say that ( φ, α ) , ( ψ, β ) are structurally equivalent iff φ [ α ] ≈ ψ [ β ] ,i.e., the sentences φ [ α ] and ψ [ β ] are structurally equivalent. We will alsodenote this relation ( φ, α ) ≈ ( ψ, β ) . Recall that we introduced this notion inDefinition 16. Recall that we call φ and ψ structurally similar iff they havethe same template b φ (see Definition 11.) If φ and ψ differ only by renamingbounded variables without making any free variable bounded, we say that φ and ψ are α -similar and denote it with φ ≃ ψ . We are extremely sorry forthe amount of notation we need to introduce which has deceptively similarmeaning. After these preliminary remarks, we can proceed to the proof. Proof of Lemma 45.
In the proof we assume that j > . The case j = 0 ishandled in a similar fashion with a slightly simpler argument. Fix a model ( M j , T j , I j , S j , X jφ ) φ ∈ M j − | = Θ j . We will argue by compactness that thetheory Θ j +1 defined using this model is consistent. (The model determinesthe theory Θ j +1 via its elementary diagram.)Fix any finite Θ ⊂ Θ j +1 . There are only finitely many φ ∈ Form L PA ( M j ) either occurring under the predicate S j +1 or as an index of the predicate X j +1 φ . Let us enumerate these formulae as φ , . . . , φ n . It is enough to findan interpretation of the predicate S j +1 in the model ( M j , T j , I j ) such that: • T j ⊂ S j +1 . • S j +1 respects regularity axioms. • S j +1 respects compositional conditions on formulae φ , . . . , φ n . • S j +1 together with T j satisfies full collection scheme. • S j +1 ( φ, α ) ≡ S j ( φ, α ) holds whenever φ ∈ M j − and φ is among φ , . . . , φ n .Notice that the last item guarantees both that stabilisation condition andelementarity for the language with the predicates X φ hold.Consider the equivalence classes [ φ i ] ∼ of φ , . . . , φ n . Consider the fol-lowing relation ⊳ : [ φ ] ⊳ [ ψ ] iff there exist φ ′ ∈ [ φ ] and ψ ′ ∈ [ ψ ] such that φ ′ is a direct subformula of ψ ′ . One can check that E , the transitive closure of ⊳ , is a partial order on classes (since it is a transitive closure of some binaryrelation, it is enough to check that no loops can occur, which is obvious).We define the predicate S j +1 by induction on E . If [ φ ] is minimal withrespect to this ordering, we consider two cases: either [ φ ] ∩ M j − = ∅ or not.If the former holds, we set: ∀ ψ ∈ [ φ ] ∀ α ∈ Asn ( ψ ) ¬ S j +1 ( ψ, α ) . φ defines an empty set under the satisfaction predicate. In the lattercase, since [ φ ] ∩ M j − is nonempty, and the template b φ is definable with aparameter in M j − , it must also be in M j − by elementarity. Notice that forany ψ ∈ [ φ ] , there exists ¯ s ∈ TermSeq L PA such that ψ ≃ b φ (¯ s ) . Now, for any α ∈ Asn ( φ ) , ψ [ α ] is also an element of [ φ ] , so there exists the unique ¯ t suchthat ψ [ α ] ≃ b φ (¯ t ) and, consequently, there is (the unique) β ∈ Asn ( b φ ) suchthat ψ [ α ] ≈ b φ [ β ] (namely, β equal to the sequence of values ¯ t ◦ ). We set: S j +1 ( ψ, α ) ≡ S j ( b φ, β ) . Finally, if φ ∼ φ i for some i ≤ n and [ φ ] is not minimal in the ordering E , we inductively define the behaviour of S j +1 so that the compositionalconditions are satisfied. For instance, if φ = ∃ vψ , we set S j +1 ( φ, α ) ≡ ∃ β ∼ v αS j +1 ( ψ, β ) . We add to S j +1 all pairs ( φ, α ) such that φ ∈ Form L PA ( M j ) , α ∈ Asn ( φ ) , and T j ( φ [ α ]) holds. Having defined S j +1 , we set X j +1 φ ( α ) ≡ X jφ ( α ) for φ ∈ M j − and define the sets X j +1 φ so as the definition-axiom of X j +1 φ is satisfied for φ ∈ M j \ M j − .We have to check that our requirements are satisfied. Let us notice that S j +1 satisfies full collection scheme, since it is arithmetically definable inthe predicates T j and X jφ , where φ ∈ M j − ∩ { c φ , . . . , c φ n } . These predicatessatisfy jointly full collection by assumption. This means that X j +1 φ for φ ∈ M j defined using S j +1 also satisfy collection.Since by induction hypothesis S j satisfied regularity and composition-ality axioms, we check that S j +1 agrees with S j on formulae from M j − .Hence the defined structure satisfies ElDiag ( M j , T j , I j , X jφ ) φ ∈ Θ ′ where Θ ′ isthe finite set of φ such that X φ occurred in the analysed finite theory Θ .The obtained structure satisfies compositional axioms, since regularityand compositionality was satisfied on S j by induction hypothesis and on T j by elementarity and the assumption that T satisfies CT − ↾ I + SRP. Compo-sitional axioms are satisfied on formulae which are not structurally similarto the ones in M j − directly by our construction.Finally, we check that the regularity axioms are satisfied. We prove thisby induction on the order E . If [ φ ] is minimal and [ φ ] ∩ M j − = ∅ , thenstructural regularity and extensionality conditions follow by the inductionhypothesis on S j and the definition of S j +1 . These properties follow directlyby definition for the minimal [ φ ] such that [ φ ] ∩ M j − is empty. If [ φ ] is not30inimal, then we check that they are preserved by extending S j +1 compo-sitionally. This is very simple for the negation and disjunction case, so letus check that they preserved in the step for the existential quantifier.Suppose that S j +1 satisfies structural regularity and extensionality forformulae in [ ψ ] . Let φ = ∃ vψ .In order to verify the extensionality condition, we want to check that S j +1 ( φ, α ) holds iff S j +1 ( φ [ α ] , ∅ ) holds. By compositionality and the induc-tion hypothesis, we have the following equivalences: S j +1 ( φ, α ) ≡ ∃ β ∼ v α S j +1 ( ψ, β ) ≡ ∃ β ∼ v α S j +1 ( ψ [ β ] , ∅ ) ≡ ∃ x S j +1 ( ψ [ α ] , {h v, x i} ) ≡ S j +1 ( φ [ α ] , ∅ ) . Notice that {h v, x i} is an assignment which sends v to x , so S j +1 ( ψ [ α ] , {h v, x i} ) makes sense. Observe that ψ [ α ] is a formula with only one free variable v ,the rest of free variables in ψ being filled out by the numerals α ( w ) .We verify structural regularity in a similar manner. Let us assume that φ ≈ φ ′ and that φ = ∃ vψ. Then φ ′ = ∃ wψ ′ and there exist sequences ¯ t, ¯ s ∈ ClTermSeq L PA ( M j ) with the same values such that φ ≃ b φ (¯ t ) , φ ′ ≃ b φ (¯ s ) . Suppose that S j +1 ( φ, ∅ ) holds. By symmetry, it is enough to show that S j +1 ( φ ′ , ∅ ) holds as well.Since S j +1 ( φ, ∅ ) holds, by compositionality there exists α ∼ v ∅ such that S j +1 ( ψ, α ) holds (where α is an assignment with the domain either equalto { v } or empty). By extensionality, S j +1 ( ψ [ α ] , ∅ ) holds. Let β be an assign-ment such that β ( w ) = α ( v ) . It is enough to show that ψ [ α ] ≈ ψ ′ [ β ] , sincethen S j +1 ( φ ′ , ∅ ) follows by structural regularity and compositionality.Let us check that ψ [ α ] ≈ ψ ′ [ β ] . Let α ( v ) = β ( w ) = c . Let ¯ t ′ = ¯ t ⌢ h c i , ¯ s ′ = ¯ s ⌢ h c i . There exists a variable u and a formula η such that b φ = ∃ uη and both b ψ and b ψ ′ are equal to b η . We see that ψ [ α ] ≃ η (¯ t ′ ) and ψ ′ [ β ] ≃ η (¯ s ′ ) where ¯ s ′ and ¯ t ′ have the same values. On the other hand, there existsequences of terms ¯ s ′′ and ¯ t ′′ with the same values such that η (¯ t ′ ) ≃ b η (¯ t ′′ ) and η ( s ′ ) ≃ b η (¯ s ′′ ) . Since b η = b ψ = b ψ ′ , by definition of structural equivalencethis implies ψ [ α ] ≈ ψ ′ [ β ] thus concluding the proof of Lemma 45.Let us notice that in the above proof, collection was preserved, since theinterpretations of S j +1 for finitely many new formulae were arithmetically31efined in finitely many predicates X φ and T j . We could run a very similarargument in order to obtain a truth predicate satisfying internal inductionINT, assuming that it was satisfied by our initial T , thus proving Lemma41 and Theorem 23. Since the argument in that case is essentially the samewith no non-trivial modifications required, we omit it. Acknowledgements
This research was supported by an NCN OPUS grant 2017/27/B/HS1/01830,"Truth theories and their strength."
References [1] Cezary Cieśliński. Deflationary truth and pathologies.
The Journal ofPhilosophical Logic , 39(3):325–337, 2010.[2] Cezary Cieśliński. Truth, conservativeness and provability.
Mind ,119:409–422, 2010.[3] Cezary Cieśliński.
The Epistemic Lightness of Truth. Deflationism and itsLogic . Cambridge University Press, 2017.[4] Ali Enayat, Mateusz Łełyk, and Bartosz Wcisło. Truth and feasiblereducibility. to appear in Journal of Symbolic Logic .[5] Ali Enayat and Shahram Mohsenipour. Model theory of the regularityand reflection schemes.
Archive for Mathematical Logic , 47:447–464, 2008.[6] Ali Enayat and Albert Visser. New constructions of satisfaction classes.In Theodora Achourioti, Henri Galinon, José Martínez Fernández, andKentaro Fujimoto, editors,
Unifying the Philosophy of Truth , pages 321–325. Springer-Verlag, 2015.[7] Mateusz Łełyk. Axiomatic theories of truth, bounded induction andreflection principles.[8] Torkel Franzen.
Inexhaustibility: an Inexhaustive Treatment . A K Pe-ters/CRC Press, 2004.[9] Petr Hájek and Pavel Pudlák.
Metamathematics of First-Order Arithmetic .Springer-Verlag, 1993. 3210] Volker Halbach.
Axiomatic Theories of Truth . Cambridge UniversityPress, 2011.[11] Richard Kaye.
Models of Peano Arithmetic . Oxford: Clarendon Press,1991.[12] Richard Kaye and Alexander Jones. Truth and collection in nonstan-dard models of PA.
Midlands Logic Seminar .[13] Jerome Keisler.
Model Theory for Infinitary Logic .[14] Roman Kossak and Bartosz Wcisło. Disjunctions with stopping condi-tion.[15] Henryk Kotlarski. Bounded induction and satisfaction classes.
Zeitschrift für matematische Logik und Grundlagen der Mathematik , 32:531–544, 1986.[16] Henryk Kotlarski, Stanisław Krajewski, and Alistair Lachlan. Con-struction of satisfaction classes for nonstandard models.
CanadianMathematical Bulletin , 24:283–93, 1981.[17] Henryk Kotlarski and Zygmunt Ratajczyk. More on induction in thelanguage with a full satisfaction class.
Zeitschrift für mathematische logik ,36:441–454, 1990.[18] Alistair H. Lachlan. Full satisfaction classes and recursive saturation.
Canadian Mathmematical Bulletin , 24:295–297, 1981.[19] Graham Leigh. Conservativity for theories of compositional truth viacut elimination.
The Journal of Symbolic Logic , 80(3):845–865, 2015.[20] Mateusz Łełyk and Bartosz Wcisło. Models of weak theories of truth.
Archive for Mathematical Logic , 56:453–474, 2017.[21] Mateusz Łełyk and Bartosz Wcisło. Notes on bounded induction forthe compositional truth predicate.
The Review of Symbolic Logic , 10:455–480, 2017.[22] Stuart T. Smith. Nonstandard definability.