Ordinal Analysis

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Ordinal analysis Toshiyasu Arai December 30, 2018

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This is a draft. Comments are welcome.

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Contents 1 Introduction 1.1 Sequent calculi . . . . . 1.2 Arithmetic . . . . . . . . 1.3 Proof-theoretic ordinals 1.4 Kripke-Platek set theory 1.5 Notes . . . . . . . . . .

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2 Calculi for classical logic 2.1 Completeness of cut-free fragments . 2.1.1 Canonical proof search . . . . 2.1.2 Applied calculi . . . . . . . . 2.1.3 Predicative second order logic 2.1.4 ω-logic . . . . . . . . . . . . . 2.2 Consequences of cut-elimination . . . 2.2.1 Midsequent theorem . . . . . 2.2.2 Interpolation theorem . . . . 2.2.3 Herbrand’s theorems . . . . . 2.2.4 Epsilon theorems . . . . . . . 2.2.5 Parikh’s theorem . . . . . . . 2.2.6 Primitive recursion . . . . . . 2.3 Exercises . . . . . . . . . . . . . . . 2.4 Notes . . . . . . . . . . . . . . . . .

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3 Cut-elimination with depths 3.1 Classical logic . . . . . . . . . . . . . 3.1.1 Predicative second order logic 3.1.2 Propositional logic . . . . . . 3.2 Fixed point logic . . . . . . . . . . . 3.3 Intuitionistic logic . . . . . . . . . . 3.3.1 Intutionistic fixed point logic 3.4 Epsilon calculi . . . . . . . . . . . . 3.5 Exercises . . . . . . . . . . . . . . . 3.6 Notes . . . . . . . . . . . . . . . . .

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CONTENTS

4 Epsilon numbers 4.1 Provability of transfinite induction . . . . . . . . . . . . . . . 4.2 Unprovability of transfinite induction . . . . . . . . . . . . . . 4.2.1 Bounding order types of provably well-founded orders 4.2.2 Embeddings . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 2-Consistency . . . . . . . . . . . . . . . . . . . . . . . 4.3 Consistency strengths calibrated with ordinals . . . . . . . . . 4.3.1 Consistency proof . . . . . . . . . . . . . . . . . . . . 4.3.2 Π2 -analyses . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Intuitionistic fixed point theories . . . . . . . . . . . . . . . . 4.4.1 Finitary analysis of intuitionistic fixed points theories 4.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 Iterations 5.1 Binary Veblen functions and club sets . . . . . . . . . . . . . 5.2 Axiom schemata in second order arithmetic . . . . . . . . . . 5.3 Predicative cut elimination . . . . . . . . . . . . . . . . . . . 5.4 Classical fixed point theories . . . . . . . . . . . . . . . . . . 5.5 Proof-theoretic strengths of the well-ordering principles . . . 5.5.1 Derivative and countable coded ω-models . . . . . . . 5.5.2 Elimination of the inference for well-ordering principle 5.5.3 Proof-theoretic ordinals of the well-ordering principles 5.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6 Collapsings 6.1 Theories IDν , Π11 -CA and KPν . . . . . . . . . . . . . . . . . . . . 6.2 Higher ordinals with collapsing functions . . . . . . . . . . . . . . 6.2.1 Recursive notation system T (Ων ) of ordinals . . . . . . . 6.3 Well-foundedness proof . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Impredicative cut elimination . . . . . . . . . . . . . . . . . . . . 6.4.1 Operator controlled derivations . . . . . . . . . . . . . . . 6.4.2 How the method of the operator controlled derivations works? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.3 Ω-rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.4 Interpretations . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Ordinal analysis of Kripke-Platek set theory KPν . . . . . . . . . 6.5.1 Ramified set theory . . . . . . . . . . . . . . . . . . . . . 6.5.2 Embedding . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Applications of operator controlled derivations . . . . . . . . . . 6.6.1 A relativized ordinal analysis . . . . . . . . . . . . . . . . 6.6.2 Reducing Πn+1 -reflections to iterated Πn -reflections . . . 6.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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CONTENTS A Answers to Exercises A.1 Chapter 2 . . . . . A.2 Chapter 3 . . . . . A.3 Chapter 4 . . . . . A.4 Chapter 5 . . . . . A.5 Chapter 6 . . . . .

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Chapter 1

Introduction Ordinal analysis is a field in proof theory, in which ordinals play a central role. Let us expound how theories relate to ordinals. Let T be a computable theory comprising a small portion of arithmetic, where by a computable theory we mean the set of axioms in the theory is computable under a suitable encoding of syntax. We are supposing that the theory T is Π11 -sound. Namely for every Π11 -sentence A on natural numbers, or in an equivalent form, if T proves the sentence A, then A is supposed to be true in the standard model N. On the other side, we see easily that the cut-free fragment of the ω-logic with the following ω-rule is Π11 -complete. ···

Γ, A(n) · · · (n ∈ N) Γ, ∀x A(x)

Furthermore for every true Π11 -sentence A, there exists a computable and cutfree derivation dA of A in ω-logic. This means that there exists a computable, and well-founded tree tree(A) ⊂ 0 is the least ordinal which is closed under the operations. For example, when T is the first-order arithmetic PA, two operations on ordinals are typically considered: the ordinal ω and the exponential function α 7→ 2α . The first epsilon number ε0 is the least ordinal closed under these two. 7

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CHAPTER 1. INTRODUCTION

Which operations on ordinals are associated with theories? This depends not only on theories, but also on the methods how the theories are expressed and analyzed. The main approaches in ordinal analysis are cut-elimination in sequent calculi (Gentzen, Sch¨ utte), and epsilon substitution method (Hilbert, Ackermann). In this monograph we focus on the first, which has been the dominant one in ordinal analysis. The associated operations on ordinals correspond to, or express a structural complexity of procedures in cut-elimination. We transform derivations to eliminate cut inferences in sequent calculi, and the operations are closely related to the transformations. Γ, ¬C C, Γ (cut) Γ These operations warrant the executability of the procedures. Specifically let us assume that the associated operations on ordinals are well-defined, by which we mean that an operation yields a well-ordering from well-orderings. Then we see that a cut-elimination procedure with these operations gives us a cut-free derivation. Once a cut-free derivation with a subformula property is gained, we can conclude that the end-formula is true by transfinite induction on depths of the cut-free derivations, where the depth is bounded by the least ordinal closed under the associated operations. Thus we comprehend what structure is involved in derivations of the theory T through the associated operations on ordinals as long as we are concerned with cut-elimination. Moreover the Π11 -soundness of T follows from the welldefinedness of the operations. In this way we obtain a deep understanding of T in terms of the ordinal analysis of T . The ordinal |T | is called proof-theoretic ordinal of the theory T , and it can be alternatively defined by the supremum of the order types of computable, strict partial orders such that T proves its well-foundedness. Although the definition of the ordinal |T | concerns only with provable Π11 -sentences, it turns out in almost all cases that |T | characterizes some proof-theoretic limitations of T . For example, let f be a computable total function. We see that T proves the fact that f is a total function iff f is defined by a transfinite recursion on ordinals up to an ordinal< |T |. Note that the fact is written in a Π02 -sentence. By the equivalence we could yield an independent Π02 -statement. Let us mention of the contents of the monograph. In this chapter some prerequisites are explained such as one-sided sequent calculi, recursion-theoretic prerequisites, and the definition of the proof-theoretic ordinal. Our base theory, the elementary (recursive) arithmetic EA, is introduced, and the Kripke-Platek set theory KP is also defined. These materials are scattered in textbooks on logic. Though this monograph concerns mainly with ordinal analysis through cutelimination in sequent calculi, chapters 2 and 3 are intended to be an introduction to proof theory. The reason for such an introduction is due to the fact that the ordinal analysis is based on proof theoretic analyses of finite derivations.

9 There are two topics in chapter 2. First, the completeness of cut-free fragments is proved through a canonical proof-search method, or refutation method. The sequent calculus for the classical first-order logic is best seen as a refutation procedure. Inference rules are regarded as searching step of a counter-model of the endsequent when we turn inferences upside down. Besides the classical first-order logic, the same method is employed to applied logic calculi, predicative second-order logic, and the ω-logic. Since the calculi with the cut inference is sound, this yields the cut-elimination theorem (Hauptsatz). It is convenient for us to have cut-free derivations for a further analysis. Second, some consequences of the cut-elimination are presented. Herbrand’s theorems for formulas in prenex normal form is proved through the midsequent theorem. Moreover witnesses for existential quantifiers are extracted from cut-free derivations. An example of the witnessing theorem is the Parikh’s theorem stating that if a Σ1 -formula is provable in a bounded theory, then there exists a term bounding the unbounded existential quantifier in the Σ1 -formula. The theorem is applied to bounded arithmetic and employed to characterize proof-theoretically the class of rudimentary functions in set theory. The second example of the witnessing theorem is on primitive recursion. In theories with induction schema for existential formulas, only a primitive recursive function is proved to be total. The Parsons-Mints-Takeuti’s theorem is the prominent example: provably computable functions in a fragment IΣ1 of arithmetic are primitive recursive functions. The proof of the theorem is based on a witnessing argument due to S. Buss, and is modified to obtain a proof-theoretic proof of a theorem due to H. Friedman, which states that the second-order arithmetic WKL0 for the weak K¨onig’s lemma is Π02 -conservative over IΣ1 . A proof-theoretic proof of the fact was first obtained by W. Sieg. Our proof deviate from Sieg’s in replacing the weak K¨onig’s lemma by an equivalent reduction property. Furthermore the witnessing argument is employed to show a theorem due to M. Rathjen, which states that only the primitive recursive set functions in the sense of JensenKarp are provably total Σ1 -functions in a restricted Kripke-Platek set theory with Σ1 -induction schema. The proof of the cut-elimination theorem via a canonical proof search provides us no quantitative information. In chapter 3 cut-elimination procedures are presented, by which we obtain a bound of the depth of the resulting cutfree derivations. The bound is given by an iteration of exponential functions of the depth of the given derivation. Logics dealt in the chapter are classical logic, intuitionistic logic, and fixed point logics in which a predicate constant Iφ together with inference rules are added for positive formula φ. The inference rules are for the axiom ∀⃗x[φ(Iφ , ⃗x) ↔ Iφ (⃗x)] stating that Iφ is a fixed point of φ. When the underlying logic is intuitionistic, φ is a strictly positive formula. For the cut-elimination in the classical fixed point logic, an asymmetric interpretation is employed to replace the fixed point Iφ by stages Iφn of inductive definitions. The replacement is done according to contexts. As contrasted with the classical logic, the cut inferences from derivations in the intuitionistic fixed logic are eliminated quickly through a parallel elimination procedure. The chapter 4 is of central importance in ordinal analysis. The target theo-

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CHAPTER 1. INTRODUCTION

ries in the chapter is theories with induction schemata, i.e., complete induction schema, transfinite induction schema and foundation schema in set theory. Let PA + TI(< Λ) denote an extension of the first-order arithmetic PA with transfinite induction schema for a standard and computable well-order < of type Λ. Assuming that the ordinal Λ is an epsilon number, the proof-theoretic ordinal of the theory PA + TI(< Λ) is shown to be equal to the ordinal Λ. First the fact is proved by eliminating cut inferences from derivations of transfinite induction TI(≺) for a computable strict partial order ≺ in ω-logic. We see that the ordinal Λ is an upper bound of the depths of the resulting cut-free derivations. On the other side each cut-free derivation of TI(≺) is seen to express the wellfoundedness of ≺ directly. This means that the depth of the cut-free derivation is nearly equal to the order type of ≺. Moreover it is easy to embed finitary proofs in the theory PA + TI(< Λ) into infinitary derivations with cut inferences with depth< Λ. Thus we get an upper bound Λ of the proof-theoretic ordinal of PA + TI(< Λ), which is at the same time the least upper bound. By analyzing cut-free infinitary derivations, we show over a weak arithmetic that each Π03 -theorem in PA + TI(< Λ) follows from a weakly descending chain principle up to an ordinal< Λ. To obtain a finer analysis, i.e., not just the proof-theoretic ordinal, but a characterization of consequences in lower complexity such as Π0i (i = 1, 2), it is convenient for us to analyze finitary proofs directly following Gentzen’s consistency proof. We show over a weak arithmetic that each Π02 -theorem follows from a descending chain principle, and each Π01 -theorem follows from an inference rule for a transfinite induction, each of which is up to an ordinal< Λ. For example the class of provably computable functions in PA + TI(< Λ) is shown to be equal to the class R Fn+1 (x) by (y) (y+1) 2 induction on y. Fn+1 (x) > Fn+1 (x) is seen from 1.2.4.1. Lemma 1.2.5 For each primitive recursive function f (⃗x), there exists an n for which f (⃗x) ≤ Fn (max{⃗x, 2}) holds for any ⃗x = x1 , . . . , xk . Proof. We show the lemma by induction on the definitions of primitive recursive functions f . For the initial functions f (⃗x) ≡ 0, f (⃗x) = xi , f (⃗x) = xi + 1, f (⃗x) ≤ F0 (max{⃗x, 2}) holds. Let f (⃗x) = h(g1 (⃗x), . . . , gm (⃗x)) by Composition. By IH with Proposition 1.2.4.4 we have an n such that gi (⃗x) ≤ Fn (max{⃗x, 2}) for any ⃗x and any i ≤ m. Moreover h(y1 , . . . , ym ) ≤ Fn (max{y1 , . . . , ym , 2}) for any yi . Then Propositions 1.2.4.3 and 1.2.4.1 yield f (⃗x) ≤ Fn (max{g1 (⃗x), . . . , gm (⃗x), 2}) ≤ Fn (Fn (max{⃗x, 2})) ≤ Fn+1 (max{⃗x, 2}). Finally let f (⃗x, 0) = g(⃗x) and f (⃗x, y + 1) = h(⃗x, y, f (⃗x, y)) by Primitive Recursion. By IH with Proposition 1.2.4.4 we have an n such that g(⃗x) ≤ Fn (max{⃗x, 2}) and h(⃗x, y, z) ≤ Fn (max{⃗x, y, z, 2}) for any ⃗x and any y, z. By (y+1) (max{⃗x, y, 2}). From this we induction on y we show that f (⃗x, y) ≤ Fn (z+1) (y+1) (z + 1) = (z) ≤ Fn see that for z = max{⃗x, y, 2} ≥ 2, f (⃗x, y) ≤ Fn (2) Fn+1 (z + 1) ≤ Fn+1 (z) ≤ Fn+2 (z). (1) When y = 0, f (⃗x, 0) = g(⃗x) ≤ Fn (max{⃗x, 2}) = Fn (max{⃗x, 2}). For y > 0, f (⃗x, y) = h(⃗x, y − 1, f (⃗x, y − 1)) ≤ Fn (max{⃗x, y − 1, f (⃗x, y − 1), 2}) ≤ (y+1) (y) (max{⃗x, y, 2}) by PropoFn (max{⃗x, y − 1, Fn (max{⃗x, y − 1, 2}), 2}) ≤ Fn sitions 1.2.4.3 and 1.2.4.1. 2 Corollary 1.2.6 E ω = PRIM. Proof. E ω ⊂ PRIM is obvious. To see PRIM ⊂ E ω inductively, let f be a function defined from g, h ∈ E n+1 by Primitive Recursion. By Lemma 1.2.5 we can assume that f (⃗x) ≤ Fn (max{⃗x, 2}). Proposition 1.2.3.4 yields f ∈ E n+1 . 2 Definition 1.2.7 The elementary (recursive) arithmetic EA for E 3 is a fragment of the first-order arithmetic in the language L(EA) = {=, ≤, 0, S, +, ·, exp}, where exp(x) = 2x is the function symbol for the exponential. EA4 is another fragment in an expanded language L(EA4 ) = L(EA) ∪ {tower}, where tower(x) = 2x is the function symbol for the tower of exponentials. L(PRA) = {=, ≤}∪{f : f ∈ PRIM} denotes a language for primitive recursive arithmetic PRA having function symbols f for each primitive recursive function f ∈ PRIM.

1.2. ARITHMETIC

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The defining axioms of function symbols are: x + 0 = x, x + S(y) = S(x + y), x · 0 = 0, x · S(y) = x · y + x, exp(0) = 1 := S(0), exp(S(x)) = exp(x) · 2, (2 := S(1)), tower(0) = 0, tower(S(x)) = exp(tower(x)). If f is defined from h, g1 , . . . , gm by Composition, then f(⃗x) = h(g1 (⃗x), . . . , gm (⃗x)) is its defining axiom. If f is defined from g, h by Primitive Recursion, then f(⃗x, 0) = g(⃗x) and f(⃗x, S(y)) = h(⃗x, y, f(⃗x, y)) are its defining axioms. The axioms for the discrete order ≤ are 0 ≤ x, x ≤ S(y) ↔ x ≤ y∨x = S(y), x ≤ x, x ≤ y ≤ z → x ≤ z, x ≤ y ∨ y ≤ x, x ≤ y ≤ x → x = y. In the following L is a language with a binary relation symbol ≤. 1. A formula in L is said to be bounded if every quantifier occurring in it is bounded by a term in L, ∃x ≤ t, ∀x ≤ t, where ∃x ≤ t A(x) :≡ (∃x(x ≤ t ∧ A(x)) and similarly for bounded universal quantifier. 2. ∆0 (L) = Σ0 (L) denotes the class of bounded formulas in the language L. 3. Σn (L) denotes the class of Σn -formulas in L, each of which is of the form ∃x1 ∀x2 · · · Qxn B with B ∈ ∆0 (L) and Q = ∀ when n is even, and Q = ∃ else. 4. Let {=, ≤, 0, S} ⊂ L ⊂ L(PRA). IΣn (L) denotes a fragment of firstorder arithmetic in the language L. Its axioms are the Peano axioms ∀x(S(x) ̸= 0), ∀x, y(S(x) = S(y) → x = y), the defining axioms for function symbols f for f ∈ L, the equality axioms, the axioms for the discrete order ≤, and the complete induction schema for Σn (L)-formulas A: A(0) ∧ ∀x(A(x) → A(S(x))) → ∀x A(x). The ∪ Peano arithmetic or the first-order arithmetic 3PA(L) := IΣω (L) := n 0 and classes {f : f ∈ E 3 } ⊂ L. Proposition 1.2.8 Let 0 < n ≤ ω. Then for any languages L, L′ such that {f : f ∈ E 3 } ⊂ L ∩ L′ and L ∪ L′ ⊂ L(PRA), IΣn (L) = IΣn (L′ ) holds. Proof. This is seen from the fact that each primitive recursive function is Σ1 (L)-definable in IΣ1 (L) for L = {f : f ∈ E 3 }. 2 Proposition 1.2.9 1. The partial truth definition for Σ0 (L(EA))-formulas is definable by a Σ0 (L(EA4 ))-formula provably in the arithmetic EA4 . 2. For each n > 0 a partial truth definition for Σn (L(EA))-formulas is definable by a Σn (L(EA))-formula Trn provably in the arithmetic EA.

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CHAPTER 1. INTRODUCTION

Proof. Note that EA is strong enough to encode finite sequences of natural numbers, cf. Proposition 1.2.3.3. Then an arithmetization of syntax and a partial truth definition are carried out in EA, cf. [77]. Proposition 1.2.9.1 is seen from the fact that each E 3 -function f (x1 , . . . , xn ) is bounded by a function 2c (max{x1 , . . . , xn }) for a c, and there is an E 4 -function sending G¨odel numbers ⌈t⌉ of closed terms t in the language L(EA) to their values tN in the standard model N. 2 Incompleteness As said above, the language is assumed to be countable. We assume tacitly a suitable encoding of the syntax, such as terms, formulas, sequents. ⌈A⌉ is then the code (G¨odel numebr) of formulas A. Furthermore we are concerned only with formal (first-order or second-order) theories T , which are elementary recursive. This means that the set of coeds of axioms in T is elementary recursive. Proposition 1.2.10 (Craig’s trick) Any r.e.(=recursively enumerable) theory is axiomatizable by an elementary recursive set of sentences. Proof. Let T h be an r.e. theory. We can assume that T h ̸= ∅. Pick an elementary recursive function f such that rng(f ) = T h, i.e., ∀x(x ∈ T h ↔ ∃n(f (n) = x)). Namely let e be a number such that ∀x(x ∈ T h ↔ ∃y T (e, x, y)) for the Kleene’s T -predicate. Pick an a ∈ { T h. Then for inverses (n)i (i = 0, 1) of a bi(n)0 if T (e, (n)0 , (n)1 ) jective pairing function, let f (n) = . Since each of T a otherwise and (n)i is elementary recursive, so is f . Moreover it is clear that rng(f ) = T h. Now let T h′ = {A(n+1) : ⌈A⌉ = f (n)}, where for formulas A, A(1) ≡ A and A(n+1) ≡ (A ∧ A(n) ). Then T h′ axiomatizes the theory T h, and we see easily that T h′ is elementary recursive. 2 Let PrT (y) :≡ ∃xPrfT (x, y) be a standard Σ1 -provability predicate for an elementary recursive theory T ⊃ EA, where PrfT (x, y) is an elementary recursive relation codifying ‘x is a code of a proof a formula y in T ’. Assume that PrT (y) enjoys the L¨ob’s derivability conditions, D1: T ⊢ A ⇒ EA ⊢ PrT (⌈A⌉) for formulas A, D2: EA ⊢ PrT (⌈A → B⌉) → PrT (⌈A⌉) → PrT (⌈B⌉) for formulas A, B, and D3: EA ⊢ PrT (⌈A⌉) → PrT (⌈PrT (⌈A⌉)⌉) for formulas A. Let Φ be a class of sentences. RfnΦ (T ) denotes the local reflection schema for T with respect to sentences in Φ: RfnΦ (T ) = {PrT (⌈A⌉) → A : A ∈ Φ}. RFNΦ (T ) denotes the uniform reflection schema for T with respect to sentences in Φ: RFNΦ (T ) = ∀⌈A⌉ ∈ Sntn [PrT (⌈A⌉) → T rn (⌈A⌉)]

1.2. ARITHMETIC

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where Sntn denotes the set of codes (G¨odel numbers) of Σn -sentences, and T rn a Σn -formula defining a partial truth definition for Σn -sentences. RFNΣn (T ) is said to be the n-consistency statement of T when n = 1, 2. It is easy to see that RFNΠn+1 (T ) is equivalent (over EA) to RFNΣn (T ) for each n < ω, and RFNΠ1 (T ) is equivalent to the consistency statement CON(T ) :⇔ ¬PrT (⌈⊥⌉), where ⊥ is a false sentence ⊥, e.g., 0 = 1. G¨odel’s second incompleteness theorem states that T ̸⊢ CON(T ) unless T is inconsistent. Moreover 1-consistency RFNΣ1 (T ) is stronger than the consistency CON(T ), i.e., T + RFNΣ1 (T ) ⊢ CON(T ), but T + CON(T ) ̸⊢ RFNΣ1 (T ) unless T + CON(T ) is inconsistent. Moreover T + RFNΣ1 (T ) ⊢ CON(T + CON(T )) is seen from the fact that CON(T ) is a Π1 -sentence. In general the following holds. Theorem 1.2.11 (Kreisel-L´evy Essential unboundedness theorem) Let T ⊃ EA be one of fragments in Definition 1.2.7. 1. For any n ≥ 0 and any Σn+1 -sentence C, T +C ⊢ RfnΠn+1 (T ) ⇒ T ⊢ ¬C. 2. For any n > 0 and any Πn+1 -sentence C, T +C ⊢ RfnΣn+1 (T ) ⇒ T ⊢ ¬C. Proof. We show Theorem 1.2.11.1. Theorem 1.2.11.2 is seen similarly. Assume T + C ⊢ RfnΠn+1 (T ) for a Σn+1 -sentence C. We show T ⊢ ¬C. Let A denote a Πn+1 -sentence such that EA ⊢ A ↔ [C → ¬PrT (⌈C → A⌉)]

(1.1)

Since T + C ⊢ RfnΠn+1 (T ), we have T + C ⊢ PrT (⌈C → A⌉) → C → A. Hence T + C ⊢ ¬A → ¬PrT (⌈C → A⌉). Thus (1.1) yields T + C ⊢ ¬A → A. Therefore T + C ⊢ A and T ⊢ C → A. By the derivability condition (D1) we have T + C ⊢ PrT (⌈C → A⌉). By (1.1) we conclude T + C ⊢ ¬A, and T ⊢ ¬C. 2 Jumps and trees Let T X (e, x, y) be Kleene’s T -predicate with an oracle X , which means that y is the code of a halting computation of a program P with its code e = ⌈P ⌉ under the input x and the oracle X ⊂ N. For the result extracting function U , {e}X (x) ≃ z :⇔ ∃y[T X (e, x, y) ∧ U (y) = z]. {e}X (x) ↓ :⇔ ∃y[{e}X (x) ≃ y]. A set A of natural numbers is computable (recursive) in X if there exists an e such that A = {x ∈ N : {e}X (x) ≃ 0} and ∀x({e}X (x) ↓). Then the jump of the set X , denoted by X ′ is the set {x ∈ N : {x}X (x) ↓}. The jump operation X 7→ X ′ can be iterated. Define X (n) recursively by X (0) = X and X (n+1) = (X (n) )′ . It is well known that a set A of natural numbers is arithmetical in X , i.e., A = {x ∈ N : N |= B[X , x]} for a first-order formula B in the language L(EA) ∪ {X} iff there exists an n < ω such that A is computable in the jump X (n) . RCA0 and WKL0 are two of the big five of the subsystems of second order arithmetic. The axioms of RCA0 are purely universal ones which define

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CHAPTER 1. INTRODUCTION

0, S, +, ·, ≤, the complete induction schema for Σ01 -formulas, and ∆01 -Comprehension Axiom ∆01 -CA: ∀n(A(n) ↔ ¬B(n)) → ∃X∀n(X(n) ↔ A(n)) (A, B ∈ Σ01 ). In ⃗ with a bounded these axioms, by a Σ01 -formula we mean a formula ∃x R(x, ⃗a, X) formula, i.e., ∆01 -formula R possibly having first-order parameters ⃗a, and second⃗ order ones X. WKL0 is obtained from RCA0 by adding the weak K¨ onig’s lemma, i.e., K¨onig’s lemma for binary trees. Definition 1.2.12 1. For strings σ, τ ∈