Admissible extensions of subtheories of second order arithmetic

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Introduction
In this paper we study admissible extensions of several theories T of reverse mathematics. The idea is that in such an extension the structure M = (N, S, ∈) of the natural numbers and collection of sets of natural numbers S has to obey the axioms of T while simultaneously one also has a set-theoretic world with transfinite levels erected on top of M governed by the axioms of Kripke-Platek set theory, KP.
In some respects, the admissible extension of T can be viewed as a proof-theoretic analog of Barwise's admissible cover of an arbitrary model of set theory (in his book "Admissible Sets and Structures"). However, by contrast, the admissible extension of T is usually not a conservative extension of T . Owing to the interplay of T and KP, either theory's axioms may force new sets of naturals to exists which in turn may engender yet new sets of naturals on account of the axioms of the other. This approach will be studied in detail and paradigmatically by combining Π 1 1 comprehension on the natural numbers with Kripke-Platek set theory, the respective theory being called KP + (Π 1 1 -CA * ). In the next two sections we present the syntactic machinery of this system and then turn to its Tait-style reformulation, which is convenient for the later proof-theoretic analysis. For this proof-theoretic analysis we make use of a more or less standard ordinal notation system, following Buchholz [3], that includes the Bachmann-Howard ordinal BH and the collapsing functions needed later.
Section 5 introduces the semi-formal system RS * in which KP + (Π 1 1 -CA * ) will be embedded. For the subsequent analysis of RS * the paper uses a novel type of ordinal analysis, expanding that for KP to the higher set-theoretic universe while at the same time treating the world of subsets of N as an unanalyzed class-sized urelement structure.
Based on these results we turn to the reduction of KP+ (Π 1 1 -CA * ) to Π 1 1 -CA+ TI (<BH) in Section 6. Main ingredients here are the notions of α-suitable trees -extending Simpson's suitable trees in [18] -and a specific truth definition based on those.
The final section shows how the previous results can be extended to other systems of reverse mathematics such as ATR 0 and BI.
2 KP and (Π 1 1 -CA * ) Let L ∈ be the usual language of set theory with ∈ as its only non-logical relation symbol plus set constants ∅ (for the empty set) and ω (for the first infinite ordinal). L set 2 is the extension of L ∈ by countably many unary relation variables.
The set terms of L set 2 are the set constants and the set variables. The atomic formulas of L set 2 are the expressions of the form (a ∈ b), (a / ∈ b), U (a), and ¬U (a), where a, b are set terms and U is a relation variable. The formulas of L set 2 are built up from these atomic formulas by means of the propositional connectives ∨, ∧, the bounded set quantifiers (∃x ∈ a), (∀x ∈ a), the unbounded set quantifiers ∃x, ∀x, and the relation quantifiers ∃X, ∀X. We use as metavariables (possibly with subscripts): • u, v, w, x, y, z for set variables, • a, b, c for set terms, • U,V,W,X,Y,Z for relation variables, • A, B, C, D for formulas.
Observe that all L set 2 formulas are in negation normal form. We assume classical logic throughout this article. Therefore, the negation ¬A of an L set 2 formula A is defined via de Morgan's laws and the law of double negation. Furthermore, (A → B) is defined as (¬A ∨ B) and (A ↔ B) as ((A → B) ∧ (B → A)).
To simplify the notation we often omit parentheses if there is no danger of confusion. Equality is not a basic symbol but defined for sets and relations: Since relations are supposed to range over subcollections of ω -see axiom (Sub ω) below -this definition of the equality of relations makes sense.
The vector notation a will be used to denote finite sequences of L set 2 terms. A (a) results from A by restricting all unbounded set quantifiers to a; relation quantifiers QX are not affected by this restriction. The ∆ 0 , Σ, Π, Σ n , and Π n formulas of L set 2 are defined as usual where relation variables are permitted as parameters. The bounded formulas are all those L set 2 formulas which do not contain unbounded set quantifiers; they may contain relation quantifiers.
(Empty set) (∀x ∈ ∅)(x = x). The formulas D in the schemas (Equality), (∆ 0 -Sep), and (∆ 0 -Col) are ∆ 0 whereas the formula A in (∈-Ind) ranges over arbitrary formulas of L set 2 . It is easy to see that the theory KP is a conservative extension of the usual first order formalization of Kripke-Platek set theory with infinity.
In this article we are interested in the theory that is obtained from KP by adding the following form of From (∆ 0 -Sep) -and here it is crucial that relations are permitted as parameters -we immediately obtain that the intersection of any relation with ω is extensionally equal to a subset of ω. On the other hand, in view of (Π 1 1 -CA * ) we also know that for every subset of ω there exists a relation with the same elements.
Let L 2 be the usual language of second order arithmetic as presented, for example, in Simpson [18]. A natural translation of L 2 into L set 2 is defined as follows: The number variables of L 2 are interpreted in L set 2 as ranging over ω and the set variables of L 2 are replaced in L set 2 by relation variables. The other symbols and connectives of L 2 are dealt with as in [18,VII.3.9].
It is also clear that our comprehension schema (Π 1 1 -CA * ) implies the usual form of Π 1 1 comprehension of second order arithmetic modulo its natural translation into L set 2 .
The basic idea of a Tait-style is that it derives finite sets of L set 2 formulas rather than individual L set 2 formulas. The intended meaning of such a set is the disjunction of its elements. This reformulation is for technical reasons only and simplifies the proof of Theorem 21.
The Tait-style axioms of KP . .
The Tait-style Inference rules Of course, it is demanded that in (∀) and (b∀) the eigenvariable u must not occur in the conclusion; the same is the case for the variable U in (∀ 2 ). We say that Γ is Tait-style devivable from KP+(Π 1 1 -CA * ) iff there exists a finite sequence of finite sets of L set 2 formulas Θ 0 , . . . , Θ h such that Θ h is the set Γ and for any i = 0, . . . , h one of the following two conditions is satisfied: In this case we write KP + (Π 1 1 -CA * ) ⊢ h Γ and say that Γ has a proof of length h. It is an easy exercise to show that a formula A is provable in one of the usual Hilbert-style formalizations of

Ordinal notations
In the next sections we establish the upper proof-theoretic bound of the theory KP + (Π 1 1 -CA * ). Our method of choice is the ordinal analysis of KP + (Π 1 1 -CA * ) via a system RS * of ramified set theory. And in order to build up this system and to control the derivations in RS * we work with specific ordinal notations. The following outline is based on ideas going back to Buchholz [3].
Definition 2. The set of ordinals C(α, β) and the ordinals ψ(α) are defined for all ordinals α and β by induction on α.
The following lemma summarizes some key properties of the sets C(α, β) and the function ψ. For its proof see Buchholz [3].
We write ε Ω+1 for the least ordinal α > Ω such that ω α = α. Its collapse BH := ψ(ε Ω+1 ) is called the Bachmann-Howard ordinal. This number gained importance in proof theory since it is the proof-theoretic ordinal of the theory ID 1 of one positive inductive definition and of Kripke-Platek set theory KP; see, for example, Buchholz and Pohlers [6], Jäger [11], and Pohlers [13]. 5 The semi-formal ramified system RS * In this section we introduce the semi-formal proof system RS * of ramified set theory. We begin with extending our language L set 2 to the language L ⋆ and then present the axioms and rules of inference of RS * . Afterwards we turn to operator controlled derivations and some basic properties of RS * . We show that KP + (Π 1 1 -CA * ) can be embedded into RS * and prove cut elimination and collapsing for RS * . Henceforth, all ordinals used in this section on the metalevel range over the set C(ε Ω+1 , 0) if not stated otherwise.

The language L ⋆
The basic idea is to extend the language L set 2 to the language L ⋆ by adding unary relation symbols M α and new quantifiers ∃x α / ∀x α for all α < Ω. The quantifiers Qx α are supposed to range over M α , and later, see Subsection 6.1, an atomic formula M α (a) will be interpreted as stating that a can be coded by a so-called α-tree.   if Ω ≤ rk (F ).
To be precise: If F is a formula of RS * , then |F | collects the levels of the relations symbols M α and of the quantifiers Qx α occurring in F plus possibly the number 0.
There are several collections of L ⋆ formulas that will play an important role later.
1. D is the collection of all L ⋆ formulas in which unbounded set quantifiers Qx and relation quantifiers QX do not occur.
2. S is the closure of D under the propositional connectives, quantifiers Qx α , bounded quantifiers (Qx ∈ r), and unbounded existential set quantifiers.
3. S 0 is the subclass of S that contains all Σ formulas of L set 2 that are not ∆ 0 .
4. B consists of all L ⋆ formulas that do not contain unbounded set quantifiers Qx.
Some important properties of the ranks of RS * formulas are summarized in the following lemma. Its proof is straightforward and will be omitted.

Axioms and rules of inference of RS *
For technical reasons we shall use a Tait-style sequent calculus as proof system for RS * . This means that we derive finite sets of L ⋆ formulas rather than individual L ⋆ formulas. In the following the Greek capital letters Γ, Θ, Λ (possibly with subscripts) act as metavariables for finite sets of L ⋆ formulas. Also, we write (for example) Γ, F 1 , . . . , F n for Γ∪{F 1 , . . . , F n }; similarly for expressions such as Γ, Θ, F . If Γ is the set {F 1 , . . . , F k } we define such that |Γ| collects the parameter sets of the formulas in Γ and Γ ∨ is the disjunction of its elements. Clearly, Γ ∨ reflects the inteded meaning of Γ.
Please observe that the main formulas of all axioms belong to B. This will be important in the later subsections when it comes to the ordinal analysis of the derivations in RS * .

Inference rules of RS
If F is from S, we write F β for the result of replacing each unbounded existential quantifier ∃x by ∃x β . This must not be confused with F (a) where each unbounded set quantifier Qx in F is replaced by (Qx ∈ a).
The meaning of the rules (∨) -(S 0 -Ref) should be self-explaining. Rule (BC) will be needed in connection with the boundedness and collapsing results in subsection 5.7.

Derivation operators
The general theory of derivation operators and operator controlled derivations has been introduced in Buchholz [4]. In the following we adapt his general approach to the more specific (and simpler) situation with which we have to deal here. is called a derivation operator d-operator for short) iff it is a closure operator and satisfies the following conditions for all X, Y ∈ Pow (On): (iv) If α has Cantor normal form ω α 1 + . . . + ω α k , then α ∈ H(X) ⇐⇒ α 1 , . . . , α k ∈ H(X).
These requirements ensure that every d-operator H is monotone, inclusive, and idempotent. Every H(X) is closed under + as well as ξ → ω ξ , and under the decomposition of its members into their Cantor normal form components.
let H be a d-operator. Then we define for all finite sets of ordinals m the operators by setting for all X ∈ Pow (On): In this case K is called an extension of H. The following observation is immediate from these definitions.

Lemma 7.
If H is a d-operator, then we have for all finite sets of ordinals m, n: (1) H[m] is a d-operator and an extension of H.  Now we turn to specific operators H σ . They will play a central role in the collapsing process of RS * derivations; see Theorem 24. Definition 8. We define, for all d-operators H, the operators by setting for all X ⊆ On: In the following lemmas we collect those properties of these operators that will be needed later. For their proofs we refer to Buchholz [4], in particular Lemma 4.6 and Lemma 4.7.
Lemma 9. We have for all ordinals σ, τ and all X ⊆ On: (1) H σ is a d-operator.
Lemma 10. Let m be a finite set of ordinals and σ be an ordinal such that the following conditions are satisfied: Then we have for α := σ + ω Ω+α and β := σ + ω Ω+β : Convention. From now on the letter H will be used as a metavariable that ranges over d-operators. Likewise, if each f 1 , f 2 , . . . , f k is an ordinal, a formula or a finite set of formulas we define Inference rules of RS * with ordinal assignments

Operator controlled derivations in RS
This concludes the list of the rules of inference of RS * with their respective ordinal assignments.
The following lemmas collect a few basic properties of the system RS * . They are proved by straightforward induction on α.
This boundedness lemma is one of the main ingredients of impredicative cut elimination and will play a central role in the proof of the collapsing theorem in Subsection 5.7. However, before dealing with cut elimination we turn to some embedding results.

Embedding
In a next step we show that KP + (Π 1 1 -CA * ) can be embedded into RS * , and as it will turn out, that finite derivations in KP + (Π 1 1 -CA * ) translate into uniform infinitary derivations in RS * . We begin with showing (TnD).

Lemma 15. For any H and any F we have that
We proceed by induction on rk (F ). If F is from B, then ¬F, F is an axiom, and we are done. Now suppose that F is of the form ∃xG[x] and observe that, for any β < Ω, the rank of the formula M β (a) ∧ G[a] is independent of a. Therefore, we can set and immediately obtain β < ρ β # ρ < ρ # ρ. Moreover,

Thus the induction hypothesis yields
and an application of (∃) gives us This is so for all β < Ω and all a. Therefore, by means of (∀) we obtain as desired. The other cases are similar.
) and all H we have that ) we obtain from the previous lemma that for all β ≤ α. By applying (∃) and (∀ α ) our assertion follows.

Lemma 17 (∈-induction). For every formula F [u] and all H we have
Proof. We set and prove in a first step by induction on α. If |α| = 0, then simply apply Axiom (4). If α is a limit, then the assertion is immediate from the induction hypothesis by means of inference rule (¬M ). Now suppose α = β + 1.
In view of Axiom (10) we also have Therefore, a cut yields Using (∨) and (b∀) we can continue with On the other hand, the previous lemma also implies that From the previous two assertions we obtain via (∧) and from the latter via Axiom (1) and (∧). Thus (∃) and the definition of G lead to Therefore, (*) is proved.
The rest is simple. The rule (∨) applied to (*) yields for all α < Ω and all a. Hence we are in the position to apply (∀) and deduce From this our assertion follows by applying (∨).
Lemma 18. We have for all H, all a, b, and all α, β: .
It α = α 1 , . . . , α k and a = a 1 , . . . , a k , then we write ¬M α ( a ) for the set Now we turn to ∆ 0 separation and our form of Π 1 1 class comprehension. Again, they are basically given by the axioms. However, for (∆ 0 -Sep) a cut is needed.
Proof. The first assertion is obtained from Axiom (13) and Lemma 16 by a cut. The second is a direct consequence of Axiom (14).
Lemma 20 (S 0 reflection). Let A[ u ] be a formula from S 0 whose free set variables are from the list u. Then we have for all H, all α, and all a that Proof. In view of Lemma 15 we have for ρ := rk (A[ a]) # rk (A[ a]). Since A[ u] is a proper Σ formula, we know that Ω ≤ ρ. Now we apply (S 0 -Ref) and obtain Thus an application of (∨) yields our assertion.
Theorem 21 (Embedding). Let Γ[u 1 , . . . , u k ] be a finite set of L set 2 formulas whose free set variables are exactly those indicated and suppose that Then there exist m, n < ω such that for all H, all α = α 1 , . . . , α k , and all a = a 1 , . . . , a k .
where v is a fresh set variable not occurring in Thus the induction hypothesis provides us with m 0 , n 0 < ω such that for all α, β and all a, b. Now we apply (∨) and then (∀) and obtain for m := m 0 +1 and arbitrary α and a.  (7) and (∧) we thus obtain (where β is 1 or ω + 1) and thus, by (∃), for m := m 0 + 1 and arbitrary α and a.
(ii) b is the variable u i , 1 ≤ i ≤ n. Then the induction hypothesis yields ω Ω+m 0 for suitable m 0 , n 0 < ω, all α and all a. Now Axiom (1) and (∧) give us and from there we proceed as in (i).
(iii) b is a set variable not occurring in Γ[ u ]. Then the induction hypothesis supplies us with m 0 , n 0 < ω such that for all α and all a. Now we make first use of Axiom (5) and a cut and then of Axiom (5) and (∧) to derive from where we can proceed again as in (i).
In all three cases we obtain for m := m 0 + 1 and arbitrary α and a. Since ∃xA[x, a ] belongs to Γ[ a ] this finishes (∃). All other cases are straightforward from the induction hypothesis or can be treated in the same vein.

Predicative cut elimination
The rules of inference of RS * can be divided into two classes: (i) In all rules except (S 0 -Ref) the principal formula is more complex than the respective minor formulas. We may, therefore, consider these rules as predicative rules. (ii) The rule (S 0 -Ref), on the other hand, transforms (in the general case) a formula of rank greater than Ω into one of rank Ω. This is a sort of impredicativity and we consider (S 0 -Ref) as an impredicative inference.
Also remember that the principal formulas of the axioms of RS * belong to B and, therefore, have rank less than Ω. These formulas are an obstacle for cut elimination below Ω. However, above Ω we can follow the pattern of the usual (predicative) cut elimination.
In this section we sketch how all cuts with cut formulas of ranks greater than Ω can be eliminated by standard methods as presented, for example, in Schütte [17] or Buchholz [4].
and assume that the rank ρ of F is greater than Ω. Then we have: Proof. By straightforward induction on β.
Theorem 23 (Predicative cut elimination). We have the following two elimination results, where the second is an immediate consequence of the first. For all H, Γ, α, and all n < ω: Ω+1 , Γ.
Proof. The proof of the first assertion is standard. For details see, for example, Buchholz [4]. The second assertion is an immediate consequence of the first.

Collapsing
We begin this subsection by showing that specific operator controlled derivations of finite sets of formulas from S ∪ B in which all cut formulas have ranks ≤ Ω can be collapsed into derivations of depth and cut rank less than Ω. This technique -called collapsing technique -is a corner stone of impredicative proof theory. See Buchholz [4] for more on general collapsings. Together with the results of the previous subsection this collapsing theorem will then lead to our main theorem about the S ∪ B fragment of RS * derivations and the Σ formulas provable in KP + (Π 1 1 -CA * ). For the collapsing theorem we work with the derivation operators H σ introduced in Definition 8.
Now we distinguish cases according to the last inference of H σ [Γ] α Ω+1 Γ and note that this cannot be (∀). In the following we confine our attention to the interesting cases; all others can be dealt with in a similar manner.  (1) and (2).
(ii) The last inference was (∀ τ ). Then Γ contains a formula of the form ∀x τ F [x] and we have with α ξ < α for all ξ ≤ τ and all a. We know τ ∈ H σ [Γ](∅). Lemma 7, Lemma 9(5), and (1) yield Hence (3) gives us and by the induction hypothesis we obtain that and, therefore, always for all ξ ≤ τ and all a.
(iv) The last inference was (S 0 -Ref). Then there exist an F ∈ S 0 and an α 0 < α such that ∃zF (z) belongs to Γ and The induction hypothesis immediately yields Since F ∈ S 0 , we are in the position to make use of Lemma 14 and obtain

Finite bounds for the lengths of formulas in infinitary derivations
Clearly, every proof P in the theory KP + (Π 1 1 -CA * ) is finite and, therefore, there exists a natural number p such that every formula in P has a length less than p. We sketch now how this bound p carries over to the "proof-theoretically treated" derivation stemming from P. First we introduce a suitable definition of length of an L ⋆ formula.
Definition 25. The length ℓ(F ) of an L ⋆ formula F is inductively defined as follows: Based on this definition we now introduce a refined notion of derivability in RS * . Given a natural number p we let H α ρ, p Γ be defined as H α ρ Γ in Definition 11 with the additional requirements that ℓ(F ) < p for every element F of Γ.
The proofs of the following three theorems do not raise any questions of principle. We simply follow the original proofs and check by a tedious case to case analysis that the additional requirements are satisfied. In the infinitary derivations new formulas may occur; however, their lengths are always bounded by the lengths of formulas in the finite derivation in KP + (Π 1 1 -CA * ).
Theorem 26 (Refined embedding). Let Γ[u 1 , . . . , u k ] be a finite set of L set 2 formulas whose free set variables are exactly those indicated and suppose that such that ℓ(F ) < p for all formulas occurring in this proof. Then there exist m, n < ω such that for all H, all α = α 1 , . . . , α k , and all a = a 1 , . . . , a k .
6 Reduction to Π 1 1 -CA + TI (<BH) The goal of this section is to interpret certain RS * -derivations of length and cut rank < Ω in the theory Π 1 1 -CA + TI (<BH). Here Π 1 1 -CA stands for the usual system of second order arithmetic with Π 1 1 comprehension and full induction on the natural numbers; it is formulated in the language L 2 . This is a familiar theory in proof theory and, therefore, we refrain from saying more about this theory. If you are interested in all details you may consult, for example Simpson [18].
In the sequel -when arguing in the framework of second order arithmetic -we assume an arithmetization of the notation system C(ε Ω+1 , 0) such that all relevant ordinal sets, functions and relations become primitive recursive. Also, we will identify the ordinal notations with their arithmetical codes. In addition, we write ≺ for the primitive recursive relation defined on C(ε Ω+1 , 0) ∩ Ω by Recall that (the codes for) the elements of C(ε Ω+1 ∩ Ω denote exactly all ordinals less than the Bachmann-Howard ordinal BH. In the context of L 2 quantifiers ∃α, ∀β range over the latter set. By TI (<BH) we mean the scheme of transfinite induction over all initial segments (indexed externally) below the Bachmann-Howard ordinal BH. More precisely, TI (<BH) is the collection of all formulas where F[u] is any formula of L 2 and n is any natural number (or rather the n th numeral).

α-suitable trees
To interpret set theory in Π 1 1 -CA + TI (<BH) we use well-founded trees, also called suitable trees. We will mostly follow the terminology and presentation in Simpson [18,VII.3].

For a function
Remark 31. If T is a tree and σ ∈ T , we put and note that if T is suitable tree then so is T σ .
Suitable trees furnish a way of talking about sets in the language of second order arithmetic. Two such trees are considered to be coding the same set if there is a bisimulation between them.
In particular, X is an equivalence relation T .
Definition 35. In order to model the set-theoretic naturals and ω via suitable trees we introduce trees n * for n ∈ N. Let Note that n * ∈ * ω * and m * ∈ * n * whenever m < n. Since the sequences in n * , and thus in ω * , are strictly descending sequences of naturals it is clear that n * and ω * are suitable trees. Moreover, there is a map f : ω * → ω+1 such that To see this, let f ( ) = ω and f ( n 1 , . . . , n r ) = n r . This means that ω * is an ω+1-tree in the sense of Definition 36 below.
The natural translation of the set-theoretic language into that of second order arithmetic, L 2 , proceeds by letting quantifiers range over suitable trees and interpreting ∈ and = as ∈ * and = * , respectively. In [18,Theorem VII.3.22] it is shown that this yields an interpretation of the axioms of ATR set 0 in ATR 0 . In what follows it is our goal to interpret the richer language of RS * in L 2 . For the predicates M α and the quantifiers ∀x α and ∃x α we use the notion of an α-tree.
Proof. For a tree T , define a function g with domain {β : β ≺ α} by arithmetical transfinite recursion as follows Note that g is uniquely determined by T . One can then show that: T is an α-tree ⇔ T is the image of g.
If T is an α-tree witnessed by a function f : T → {β : β ≺ α}, then one shows, using ≺-induction on f (σ), that σ ∈ g(f (σ)) for all σ ∈ T , hence T is the image of g.
Conversely, if T is the image of g, then a function f : T → {β : β ≺ α} witnessing the α-treeness of T is obtained by letting f (σ) be the least β ≺ α such that σ ∈ g(β).
The next task, which presents itself, is to translate the formulas of L ⋆ that belong to the collection B into the language of second order arithmetic, L 2 .
Definition 39. For convenience assume that we have injections x → T x and X → T X from set variables x and relation variables X of L * to second order variables of L 2 in such a way that T x is always different from any T X . The latter provides a translation of the terms of L * except for the constants ∅ and ω. These can just be translated as { } and ω * , respectively (see Definition 35).
Since the relation ∈ * of Definition 32 has only be defined for trees, let us agree that henceforth a formula S ∈ * T will be considered a shorthand for S, T trees ∧ S ∈ * T .
The translation * from B to L 2 is then effected as follows.
4. (M α (a)) * is T a is an α-tree. (¬M α (a)) * is T a is not an α-tree. and respectively. Here F * [T X ] stands for (F [X]) * , noting that * replaces X by T X .
In order to show that the collapsed derivations of Theorem 28 prove true statements when subjected to the interpretation of the previous, we need to show that the rule (BC) preserves truth. This will mainly be a consequence of Σ 1 1 -AC .
with all free variables indicated, and ℓ F be the length of the latter formula as a string of symbols. Let be the results of replacing the unbounded existential set quantifiers ∃x in the formula by ∃x α and ∃x ∈ a, respectively.
Arguing in the theory Π 1 1 -CA + TI (α+ω), we have that whenever T 0 , . . . , T q are α-trees and Q 0 , . . . , Q r are trees such that Q i ⊆ * ω * (so actually ω+1-trees), and the * -translation of F α [x 0 , . . . , x q , U 0 , . . . , U r ] with x i and U j replaced by T i and Q j , respectively, is true, then the * -translation of again with x i replaced by T i and U j replaced by Q j , holds true as well.
Proof. We proceed by (meta) induction on ℓ F . The most interesting case arises when this formula starts with a bounded universal quantifier, i.e., if it is of the form So assume that T 0 , . . . , T r are α-trees and Q 0 , . . . , Q r are trees such that Q i ⊆ * ω * and holds, where G α 0 is the * -translation of F α 0 . Inductively we then have where G  (1) is ∆ 1 1 . As a result, we may apply Σ 1 1 -AC (which is provable in Π 1 1 -CA 0 ) to conclude that there exists a set Z such that for all n ∈ T 0 , where Z (n) = {k : (n, k) ∈ Z}.
Using Σ 1 1 -AC again, we can also single out a sequence of functions is an α+ℓ F 0 -tree whenever n ∈ T 0 and k ∈ Z (n) . Now define a tree S by By design, S is a tree and, moreover, S is an α+ℓ F -tree as witnessed by the function f : S → α+ℓ F defined by As a consequence of (2) and the fact that F [x 0 , . . . , x q , U 0 , . . . , U r ] is a Σ-formula, we infer that (∀ n ∈ T 0 )G S 0 [T 0 , . . . , T q , T n 0 , Q 0 , . . . , Q r ] and thus, as desired, the truth of the * -translation of ∃y α+ℓ F F (y) (x 0 , . . . , x q , U 0 , . . . , U r ) with the substitutions x i → T i and U j → Q j follows.

Conservativity
It remains to ascertain that KP + (Π 1 1 -CA * ) is conservative over Π 1 1 -CA + TI (<BH) for formulas in the language of second order arithmetic. More precisely, if F is a formula of second order arithmetic (as for instance defined in [18, I.2]) and F 0 denotes its natural translation into the language L set 2 , then we aim to show that The latter being a finite deduction, it follows from what we have etablished in the previous sections that we can determine fixed naturals n, p such that H β η,p F 0 for some derivation operator H and β, η ∈ C(ω n [Ω + 1], 0). This follows from Theorems 26, 27, and 28. To dilate on this, note that because of the finiteness of the deduction in KP + (Π 1 1 -CA * ) we can find this a priori bound ω n [Ω + 1] so that the entire ordinal analysis, commencing with the embedding of this finite deduction into the infinitary proof system, solely uses ordinals from C(ω n [Ω + 1], 0).
In light of the previous, the next step will be to show that H β η,p F 0 entails that F is true, all the while working in our background theory Π 1 1 -CA + TI (<BH). This is where the up to now neglected parameter p comes into its own in that the idea is to employ a formal truth predicate for formulas of length < p. However, it is not possible to just focus on formulas F 0 , where F resides in the language of second order arithmetic, since in showing that H β η,p F 0 implies the truth of F we shall induct on β and the pertaining derivation is usually not cut-free, so we have to take formulas from B of length < p into account.
It is perhaps noteworthy that if H β η,p Γ with Γ a set of formulas in B and β, η < Ω, then all formulas occurring in the derivation must be in B, too, as there can be no cuts in it involving formulas with unbounded set quantifiers since their ranks would be ≥ Ω (see Definition 4 items (7) and (8)).
The translation of the formulas from B into L 2 formulas has been introduced in Definition 39. The purpose of the following definition is to fix an arithmetized truth definition for B-formulas of length < p in Π 1 1 -CA + TI (<BH).
Definition 41. Let Z be a set of naturals such (Z) k is an α-tree for all k. We'd like to engineer a formula T p (x, X) of L 2 such that for all B-formulas F [x 1 , . . . , x r , U 1 , . . . , U s ] of length < p with all free variables exhibited, The formal definition of such a truth predicate is a standard but cumbersome procedure. A place in the literature, where one finds this carried out in detail, is Takeuti [20,CH. 3,19], and another is Troelstra [21, 1.5.4].
Proof. We reason in Π 1 1 -CA + TI (<BH) by induction on β. The axiom cases are obvious. If H β η,p Γ is the result of an inference of a form other than (BC), then this follows immediately from the induction hypothesis applied to the immediate subderivations. Note also that in case of a cut, the cut formulas belong to B and have lengths < p. Observe also that the derivation cannot contain (S 0 -Ref) inferences since β < Ω.
So it remains to deal with the case where the last inference is an instance of (BC). Fortunately, this is what Lemma 40 is really about. The latter shows that if the premise Θ of an instance of (BC) is true under an assignment Z, then so is the conclusion.
Theorem 43. KP + (Π 1 1 -CA * ) is conservative over Π 1 1 -CA + TI (<BH) for formulas of second order arithmetic. More precisely, if F is sentence of second order arithmetic (i.e. L 2 ) and F 0 denotes its natural translation into the language L set 2 , then Proof. Assume KP + (Π 1 1 -CA * ) ⊢ F 0 . As elaborated on before, with the help of the results in subsection 5.8 and Theorem 42, it follows that Thus, in light of (*) and noting that F 0 has no free variables, where F * 0 is the translation of F 0 according to Definition 41. It remains to establish the relationship between F * 0 and F. F * 0 arises from F by translating numerical quantifiers Qn as ranging over the immediate subtrees of ω * and second order quantifiers as ranging over ω+1-trees T such that T ⊆ * ω * . Now, as the naturals with their ordering are isomorphic to the immediate subtrees of ω * ordered via ∈ * and also the collection of sets of natural numbers is naturally isomorphic to the collection of ω+1-trees T such that T ⊆ * ω * , it follows that F * 0 implies F, completing the proof. Anyone insisting on a more formal proof is invited to proceed by induction on the buildup of F.
Remark 44. In this article we do not compute the proof-theoretic analysis of the system Π 1 1 -CA + TI (<BH). However, it seems that the ordinal analysis of Π 1 1 -CA + TI (<BH) simply has to follow that of Π 1 1 -CA with ε 0 replaced by BH. So we conjecture that it is the ordinal ψ 0 (Ω ω · BH) in the terminology of Buchholz and Schütte [7].

Extensions
Thus far we have investigated what happens if one adds Kripke-Platek set theory to the subtheory of second order arithmetic based on Π 1 1 -comprehension. There obtains a certain analogy to what Barwise [2] called the Admissible Cover, Cov M , of a basic structure M. In his case, the basic structures M were models of set theory. Cov M is the intersection of all admissible sets which cover M (see [2], Appendix 2.1). 2 In our context, the admissible cover amounts to grafting the theory KP onto a subsystem of second order arithmetic, T . This could be called the proof-theoretic admissible cover of T . There is, however, a crucial difference between Barwise's model-theoretic construction and the proof-theoretic one employed in this paper. In the former the basic structure one starts from remains unchanged in a strong sense when building its admissible cover in that no new subsets of M become available in Cov M (see [2] Appendix Corollary 2.4), whereas in the proof-theoretic case the axioms of the basic theory T will interact with the axioms of the set theory KP, witnessed by the fact that, in general, more theorems of second order arithmetic become deducible in T + KP than in T alone.
It is also interesting to investigate how the proof-theoretic admissible cover plays out in the case of other well-known subsystems of second order arithmetic. It turns out that a certain pattern emerges. Moreover, the techniques developed in this paper, when combined with insights from the literature, suffice to get these additional results.
Definition 45. We will focus on two well-studied theories.
1. ATR 0 with its signature axiom of arithmetical transfinite recursion is the fourth system of the "big" five of reverse mathematics (see [18, V]).
2. The second system we will consider is traditionally called the theory of bar induction, BI, by proof theorists. In [18, VII.2.14] it is denoted by Π with WO(X) expressing that the ordering < X defined by v < X u :⇔ 2 v · 3 u ∈ X is a well ordering.
(ATR * 0 ) and(BI * ) are the axiom schemas -formulated in the language L set 2 -that comprise the natural translations of all instances of arithmetical transfinite recursion and transfinite induction, respectively. If C is a collection of sentences of L 2 and T 1 , T 2 are theories of the language L 2 or L set 2 , we write T 1 ≡ C T 2 to convey that T 1 and T 2 possess the same C-theorems (perhaps modulo the translation of C into L set 2 ).
(ii) By Jäger [11] and the papers cited in the previous line, it is known that KP and BI share the same proof-theoretic ordinal. Moreover, from the ordinal analyses of these two theories it can be inferred that they prove the same Π 1 1 -theorems. The degree of conservativity, though, cannot be improved much beyond this level as ATR 0 has a Π 1 2 -axiomatization and KP does not prove ATR 0 . To see this, first note that KP has a model HYP N which is the intersection of all admissible sets that contain the standard structure N of the naturals as a set (see [2,Theorem 5.9] ). The subsets of N in HYP N are the hyperarithmetical sets. Then we can proceed, for example, in one of the following two ways: -The collection of the hyperarithmetical sets, as apparently proved by Kreisel, constitutes the smallest ω-model of Σ 1 1 -AC, but it cannot be a model of ATR 0 as it would have to have a proper ω-submodel that is again an ω-model of ATR 0 , and thus of Σ 1 1 -AC. The result about ω-models of ATR 0 is due to Quinsey [15, pages 93-96] (for more details see [18] Theorem VIII.6.12 and the Notes for §VIII.6).
-Alternatively, observe that the theory ATR 0 is equivalent to the theory FP 0 which extends ACA 0 by axioms stating that any positive arithmetic operator has a fixed point; see Avigad [1]). But the hyperarithmetical sets do not constitute a model of FP 0 according to Probst [14,II.2.4] and Gregoriades [9].
(iii) By Theorem 46 we have ATR * 0 +KP ≡ Π 1 ∞ ATR 0 +TI (<BH). Since ATR 0 +TI (<BH) is a subtheory of BI and BI * + KP ≡ Π 1 1 KP, it follows that ATR * 0 + KP ≡ Π 1 1 KP. In connection with the previous results we would also like to mention Sato [16]. He states there, among other things, a special case of Theorem 47(ii), namely that the addition of a Π 1 2 theorem of BI to KPω does not increase the consistency strength of the augmented theory. Further interesting work about the relationship between Kripke-Platek set theory and Π 1 1 comprehension on the natural numbers is presented in Simpson [19]. There the interplay between KP and Π 1 1 -CA is studied from a recursion-and model-theoretic perspective. However, it does not provide the exact proof-theoretic strength of KP + (Π 1 1 -CA * ).