Computability Theory of Hyperarithmetical Sets

Home , Analytical hierarchy, Arithmetical hierarchy

WESTFALISCHE¨ WILHELMS-UNIVERSITAT¨ MUNSTER¨ INSTITUT FUR¨ MATHEMATISCHE LOGIK UND GRUNDLAGENFORSCHUNG

Lecture by

Wolfram Pohlers

Summer–Term 1996

Typeset by Martina Pfeifer

Preface

This text contains the somewhat extended material of a series of lectures given at the University of Münster. The aim of the course is to give an introduction to “higher” computability theory and to provide background material for the following courses in proof theory. The prerequisites for the course are some basic facts about computable functions and mathematical logic. Some emphasis has been put on the notion of generalized inductive definitions. Whenever it seemed to be opportune we tried to obtain “classical” results by using generalized inductive definitions. I am indebted to Dipl. Math. INGO LEPPER for the revising and supplementing the original text.

M¨unster, September 1999 WOLFRAM POHLERS 2 Contents

Contents

1 Computable Functionals and Relations ...... 5 1.1FunctionalsandRelations...... 5 1.2TheNormal–formTheorem...... 10 1.3 Computability relativized ...... 13

2 Degrees ...... 15 2.1 m–Degrees...... 15 2.2 TURING–Reducibility ...... 16 2.3 TURING–Degrees...... 18

3 The Arithmetical Hierarchy ...... 25 3.1TheJumpoperatorrevisited...... 25 3.2TheArithmeticalHierarchy...... 27 3.3TheLimitsoftheArithmeticalHierarchy...... 33

4 The Analytical Hierarchy ...... 35 4.1Secondorderarithmetic...... 35 4.2Analyticalrelations...... 36

5 The Theory of Countable Ordinals ...... 41 5.1Ordinalsasorder–types...... 41 5.2Trees...... 46 5.3RecursiveOrdinals...... 51 5.4 KLEENE’sOrdinalNotations...... 54

6 Generalized Inductive Definitions ...... 61 6.1Clausesandoperators...... 61 6.2 The stages of an inductive definition ...... 63 6.3 Arithmetically definable inductive definitions ...... 65 6.4Thestagecomparisontheorem...... 70

1 CK 7 Inductive Deﬁnitions, Π1–sets and the ordinal ω1 ...... 75 1 7.1 Π1–sets vs. inductive sets ...... 75 7.2 The inductive closure ordinal of N ...... 78 0 7.3 Σ1–inductive deﬁnitions and semi–decidable sets ...... 83 1 7.4 Some properties of Π1–andrelatedpredicates...... 84 7.5BasisTheorems...... 90 7.6 The complexity of KLEENE’s O ...... 93

3 Contents

8 The Hyperarithmetical Hierarchy ...... 97 8.1Hyperarithmeticalsets...... 97 8.2Hyperarithmeticalfunctions...... 102 8.3 The hyperarithmetical quantiﬁer theorem ...... 107

9 Appendix ...... 111 1 O 9.1 A Π1–path through ...... 111

Bibliography ...... 113

Index ...... 114

4 1. Computable Functionals and Relations

1.1 Functionals and Relations

Let NN := α α: N −→ N be the space of all functions from the natural numbers into the natural numbers. In this lecture we will deal with the spaces n Nm,n := Nm × (NN) .

The elements of this space will be denoted by lower case Gothic letters such as a, b, c, a1, a2 ...

1.1.1 Deﬁnition 1.) Let D ⊆ Nm,n.An(m, n)-ary partial functional is a map F : D −→ N . We denote this by m,n F : N −→ p N . The set D is the domain of F – denoted by dom(F ). If dom(F )=Nm,n we call F a total functional. 2.) An (m, n)-ary relation is a set R ⊆ Nm,n. We use the notations a ∈ R and R(a) synonymously to denote that a belongs to R. To distinguish notions from Ordinary Computation Theory (OCT) (or Classical Recursion Theory as it used to be called) from Hyperarithmetical Computation Theory (HCT) we refer to (m, 0)-ary functionals as m-ary functions and to (m, 0)-ary relations as m-ary predicates. We use the common notations of OCT freely. E.g., hx1,...,xnidenotes the primitive–recursive coding function, (x)i its decoding and Seq the primitive–recursive set of sequence codes. m,n For a =(x1,...,xm,α1,...,αn)∈N and k ∈ N we put

a(k):=(x1,...,xm,α1(k),...,αn(k)), where hi if k =0 α(k):= hα(0),...,α(l)i if k = l +1 denotes the course of values of α below k. We refer to a(k) as the course of values of the tuple a below k. If a is as above, ~y =(y1,...,yk)and β~ =(β1,...,βl)we put

(a,~y,β~):=(x1,...,xm,y1,...,yk,α1,...,αm,β1,...,βl).

1.1.2 Deﬁnition An (m, n)-ary relation R is semi–decidable (often also called semi–recursive or recursively enumerable) if there is a semi–recursive (which can be regarded as synonymous to recursively enumerable) m + n-ary predicate PR such that

a ∈ R ⇔ (∃x)PR(a(x)).

1.1.3 Discussion The deﬁnition of a semi-decidable relation meets the intuition of a “positively decidable” relation. We show that there is an algorithm which conﬁrms a ∈ R.SincePR(a(x)) is semi–recursive in the sense of OCT there is a decidable predicate, say Q, such that

5 1. Computable Functionals and Relations

a ∈ R ⇔ (∃x)PR(a(x)) ⇔ (∃x)(∃y)Q(a(x),y).

Now we decide Q(a((n)0, (n)1) for n =0,1,.... This algorithm terminates if a ∈ R but will give no information in case that a ∈/ R.

m,n m,n 1.1.4 Deﬁnition Let F, G: N −→ p N .Fora∈N we put F (a) ' G(a):⇔(a∈/dom(F ) ∧ a ∈/ dom(G)) ∨ (a ∈ dom(F ) ∩ dom(G) ∧ F (a)=G(a)).

Sometimes it is helpful to consider partial functionals as maps from Nm,n into N ∪ {↑}. If we put F (a) if a ∈ dom(F ) F˜(a): ' ↑ otherwise then we get F (a) ' G(a) ⇔ F˜(a)=G˜(a). (1.1)

m,n 1.1.5 Deﬁnition Let F : N −→ p N . We call F partial–computable if its graph GF := (a,y) F(a)'y is semi-decidable. We call F computable if F is partial–computable and total.

1.1.6 Discussion The deﬁnition of a partial–computable functional meets the intuition of a positively computable functional. We indicate that there is an algorithm for F which terminates and yields F (a) in case that a ∈ dom(F ).SinceGF is semi-decidable we get as in 1.1.3 a decidable predicate Q such that F (a) ' x ⇔ (∃z)(∃y)Q(a(z),y,x).

Again we decide Q(a((n)0), (n)1, (n)2) for n =0,1,...andpicktheﬁrstsuchn.ThenF(a)= (n)2.IfFis computable, then it is total, and so this algorithm will always terminate. We are now ready to study the closure properties of semi-decidable relations. It will turn out that most of the closure properties are just liftings of the closure properties of semi-decidable predicates.

1.1.7 Theorem The semi-decidable relations are closed under • the positive boolean operations ∧ and ∨; • bounded quantiﬁcation on natural numbers; • unbounded ∃–quantiﬁcation over N and NN; • substitution with computable functionals.

Proof: The only case which is new in comparison to OCT is the closure under second order quantification, i.e. quantifiers ranging over NN. However, we will also give two examples for the more simple cases, e.g. closure under ∧ and bounded ∀-quantification. We have

R(a) ∧ Q(a) ⇔ (∃x)PR (a(x)) ∧ (∃y)PQ(a(y))

⇔ (∃u)[PR(a(u)(u)0) ∧ PQ(a(u)(u)1)] 6 1.1. Functionals and Relations which shows that R ∧ Q is semi-decidable. For bounded ∀-quantiﬁcation we have

(∀x

(∃α)R(a,α) ⇔ (∃α)(∃x)PR(a(x), α(x))

⇔ (∃s)(∃x)[Seq(s) ∧ lh(s)=x∧PR(a(x),s)]. Hence (∃α)R(a,α)is semi-decidable. We call the relation (∃x)R(a,x) the N– or ﬁrst order projection of R(a,x) while (∃α)R(a,α) is the NN– or second order projection of R(a,α). The motivation for this terminology becomes clear from Figure 1.1.1.

N resp. NN

Q N(m,n)

Figure 1.1.1: The N–resp.NN–projection of a relation

1.1.8 Deﬁnition The characteristic functional of an (m, n)–ary relation R is given by 0 if a ∈ R χ (a):= R 1 otherwise.

Let us make some of the conventions explicit which we have been already using. Quantifiers of the form (Qx), (Qy), ...whose bound variables are indicated by lower case Ro- man letters are first order, i.e. quantifiers ranging over N. To emphasize the first order of those quantifiers we sometimes (very rarely) will write (∃0x) or (∀0x). Quantifiers of the form (Qα), (Qβ),...whosebound variables are indicated by lower case Greek letters are second order, i.e. quantifiers ranging over NN. To emphasize the second order of those quantifiers we sometimes will write (∃1α) or (∀1α). Sometimes we want to quantify over subsets of N, i.e. over N2, the set of characteristic functions. ∗ ∗ ∗ This will be denoted by (Qα ), (Qβ ), (Qα1),...

7 1. Computable Functionals and Relations

1.1.9 Deﬁnition Let G be an (m +1,n)–ary functional. The (unbounded) search operator µ turns G into a (m, n)–ary functional (µG) which is deﬁned by (µG)(a) ' y :⇔ G(a,y)'0∧(∀u

Sub(G, H1,...,Hn)(a) ' G(H1(a),...,Hn(a))

1.1.10 Theorem The partial–computable functionals are closed under unbounded search - and hence also under bounded search - and substitution.

Proof: Having in mind the closure properties of semi-decidable relations the ﬁrst claim follows by looking at (1.2). The second claim follows from

Sub(G, H1,...,Hn)(a) ' y ⇔

(∃x1) ...(∃xn)[H1(a) ' x1 ∧ ... ∧ Hn(a)'xn ∧G(x1,...,xn)'y].

The possibilities for substitution, however, are not exhausted by the substitution operator. If H is an (m +1,n)–ary functional and G an (m, n +1)–ary functional then we may try to deﬁne F (a) ' G(a,λx.H(a,x)). (1.3) The problem is that (1.3) is only deﬁned if λx. H(a,x) is total. The following lemma shows how this can be handled.

1.1.11 Lemma (Substitution Lemma) Let G be an (m, n +1)–ary and H an (m +1,n)–ary partial–computable functional. Then there is a partial–computable functional F such that F (a) ' G(a,λx.H(a,x)) for all a for which λx. H(a,x) is total.

Proof: We have semi-decidable predicates PG and PH such that

G(a,α)'u ⇔ (∃z)PG(a(z),α(z),u) (i) and

H(a,x)'v ⇔ (∃y)PH(a(y),x,v). Using (i) we ﬁnd a decidable predicate Q such that G(a,α)'u ⇔ (∃z)(∃x)Q(a(z), α(z),u,x). We put

F (a):'(µw. Q(a((w)0), λx. H(a,x)((w)0), (w)1, (w)2))1. Then a ∈ dom(F ) if λx. H(a,x) is total and (a,λx.H(a,x)) ∈ dom(G). Hence dom(λa.G(a,λx.H(a,x))) ⊆ dom(F )

8 1.1. Functionals and Relations but observe that the inclusion may well be proper. However, we have G(a,λx.H(a,x)) = F(a) for all (a,λx.H(a,x)) ∈ dom(λa.G(a,λx.H(a,x))). We still have to show that F is partial– computable. Checking the graph of F we get F (a) ' a ⇔ (∃s)(∃w)[Seq(s) ∧ Seq(w) ∧ lh(w)=3 ∧lh(s)=w

∧(∀i

∧Q(a((w)0),s(w)0,(w)1,(w)2)∧ (w)1 =a

∧(∀j

1.1.12 Lemma The partial–computable functionals are closed under deﬁnition by cases: Let G1,...,Gn be partial–computable and R1,...,Rn pairwise disjoint semi–decidable relations and  G1(a) if R1(a) F (a):' . . . . Gn(a) if Rn(a) Then F is partial–computable. Proof: We have

F (a) ' y ⇔ (R1 (a) ∧ G1 (a) ' y ) ∨ ... ∨ (Rn(a)∧ Gn(a)'y) which shows that F possesses a semi–decidable graph. The simplest example of a functional is the application functional which is deﬁned by App(α, n):'α(n).

1.1.13 Theorem The application functional is a (1, 1)–ary computable functional. Proof: Since α is total App is total, too. For its graph we get

App(α, n) ' y ⇔ (∃z)[n

1.1.14 Deﬁnition A relation R ⊆ Nm,n is decidable if its characteristic functional is computable. All closure properties of decidable (i.e. recursive) predicates can be lifted to decidable relations. Therefore we state the following theorem without proof.

1.1.15 Theorem The decidable relations are closed under: • all boolean operations, i.e. ¬, ∧, ∨; • bounded quantiﬁcation; • substitution with computable functionals.

9 1. Computable Functionals and Relations

However, as a consequence of Lemma 1.1.11, we get the following additional closure property.

1.1.16 Theorem Let P be an (m, n +1)–ary decidable relation and H be an (m +1,n)–ary computable functional. Then the relation R := a P (a,λx.H(x, a)) is decidable.

Proof: We get

χR (a) ' χP (a,λx.H(x, a)) and the right hand is a computable functional by Lemma 1.1.11 because λx. H(x, a) is total. In OCT we classify the semi–decidable predicates as N–projections of decidable predicates. This too can be lifted to semi–decidable relations.

1.1.17 Theorem An (m, n)–ary relation R is semi–decidable iff there is an (m +1,n)–ary decidable relation Q such that R(a) ⇔ (∃z)Q(a,z), i.e. the semi–decidable relations are exactly the N–projections of the decidable relations.

Proof: Let R be semi–decidable. Then

R(a) ⇔ (∃z)PR(a(z)) ⇔ (∃z)(∃u)Q˜(a(z),u) for some decidable predicate Q˜.Deﬁne Q:= (a,u) Q˜(a((u)0), (u)1) . Then R(a) ⇔ (∃z)Q(a,z) and Q is obviously decidable.

1.1.18 Theorem Let R be an (m +1,n)–ary decidable relation and deﬁne F (a):'µw. R(a,w). Then F is an (m, n)–ary partial–computable functional.

Proof: We have F (a) ' y ⇔ (∃w)[R(a,y) ∧(∀u

1.2 The Normal–form Theorem

One of the most important theorems of OCT is KLEENE’s Normal–form Theorem. The aim of this section is to lift this theorem to HCT. Recall that in OCT we deﬁned We as the domain of a partial–computable function with index e. These domains are exactly the semi–decidable predicates. Thus We e ∈ Ind(P ) enumerates all semi–decidable predicates where Ind(P ) is

10 1.2. The Normal–form Theorem the set of indices of partial–computable functions. We use this enumeration to obtain an indexing of semi–computable functionals. Let R be an (m, n)–ary semi–decidable relation. Then there is an e ∈ Ind(P ) such that ⇔ ∃ m+n R(a) ( z)We (a(z)) (1.4) ⇔ (∃z)(∃u)Tm+n(e, a(z),u) where Tm+n denotes the KLEENE predicate. For a semi–computable (m, n)–ary functional we get from (1.4) F (a) ' y ⇔ (∃z)(∃u)Tm+n+1(e, a(z),y,u). Therefore we deﬁne m,n m+n+1 T := (e, a,w) T (e, a((w)0), (w)1, (w)2) . Then Tm,n is an (m +2,n)–ary decidable relation for which we get m,n F (a) ' (µw. T (e, a,w))1. Therefore we have the following theorem.

1.2.1 Theorem (Normal–form Theorem) There is an (m +2,n)–ary decidable relation Tm,n and a computable (even primitive–recursive) function U such that for all semi–computable (m, n)– ary functionals F there is an e ∈ N with F (a) ' U(µw. Tm,n(e, a,w)).

We agree about the notation {e}m,n(a):'U(µw. Tm,n(e, a,w)) and call e an index for F .

1.2.2 Theorem The functional Φm,n(a,e):'{e}m,n(a) is a partial–computable functional which is universal for the class of (m, n)–ary partial–computable functionals.

Proof: The Normal–form Theorem entails the universality of the functional Φm,n. To show its partial–computability we check its graph. Φm,n(a,e)'y ⇔{e}m,n(a) ' y ⇔ (∃w)[Tm,n(e, a,w)∧ (∀u

We refer to Theorem 1.1.17 to obtain also a Normal–form Theorem for semi–decidable relations. In a ﬁrst step we prove the following theorem.

1.2.3 Theorem A relation is semi–decidable iff it is the domain of a partial–computable functional.

Proof: Using the Normal–form Theorem we get a ∈ dom(F ) ⇔ (∃w)Tm,n(e, a,w) showing that the domains of partial–computable functionals are semi–decidable. For the opposite direction let R be (m, n)–ary semi–decidable. By Theorem 1.1.17 we get a decidable relation Q such that

11 1. Computable Functionals and Relations

R(a) ⇔ (∃z)Q(a,z). Deﬁne F (a):'µz . Q(a,z). Then F is partial–computable by Theorem 1.1.18 and we have dom(F )= a (∃z)Q(a,z) =R.

Now we deﬁne m,n { }m,n ∃ m,n We := dom( e )= a ( w)T (e, a,w) .

m,n 1.2.4 Theorem The collection of (m, n)–ary semi–decidable relations is enumerated by We , i.e. ⊆Nm,n m,n ∈ N R R is semi–decidable = We e .

m,n If R = We we call e an index for R. m The canonical next step is to lift the Sn –Theorem from OCT.

m,n m,n 1.2.5 Theorem ( Sk –Theorem) There is a k +1–ary primitive–recursive function Sk such that { }m+k,n '{ m,n }m,n e (a,y1,...,yk) Sk (e, y1,...,yk) (a) (1.5) and ∈ m+k,n ⇔ ∈ m,n (a,y1,...,yk) We a W m,n . (1.6) Sk (e,y1,...,yk) Proof: We get {e} m+k,n(a,~y)'U(µw. Tm+k,n(e, ~y, a,w)) m+k+n+1 ' U(µw. T (e, ~y, a((w)0), (w)1, (w)2)) ' m+n+1 m+n+1 U(µw. T (Sk (e, ~y), a((w)0, (w)1, (w)2))) ' m,n m+n+1 U(µw. T (Sk (e, ~y), a,w)) '{ m+n+1 }m,n Sk (e, ~y) (a) and we put m,n m+n+1 Sk (e, ~y):=Sk (e, ~y) m+n+1 where Sk is the function of OCT. m+k,n { }m+k,n Since We = dom( e ) we obtain (1.6) immediately from (1.5). m,n The immediate consequence of the Sk –Theorem is — as usual — the Recursion Theorem.

1.2.6 Theorem (Recursion Theorem) Let G be an (m +1,n)–ary partial–computable functional. Then there is an e such that {e}m,n(a) ' G(a,e).

Proof: We mimick the usual proof. Deﬁne ' m,n H(a,x): G(a,S1 (x, x)).

Then H is partial–computable by Theorem 1.1.10. Let e0 be an index for H and deﬁne

12 1.3. Computability relativized

m,n e := S1 (e0,e0). Then { }m,n '{ m,n }m,n e (a) S1 (e0,e0) (a) m+1,n '{e0} (a,e0) ' ' m,n H(a,e0) G(a,S1 (e0,e0)) ' G(a,e).

As an application of the Recursion Theorem we show the closure of the partial–computable functionals under the Recursion Operator. The Recursion Operator turns an (m, n)–ary functional G and an (m +2,n)–ary functional H into the (m +1,n)–ary functional Rec(G, H) which is deﬁned by G(a) if x =0 Rec(G, H)(a,x)' H(a,y,z) if x = y +1and Rec(G, H)(a,y)'z.

1.2.7 Theorem The partial–computable as well as the computable functionals are closed under the Recursion Operator. Proof: Let G and H be functionals of suitable arity. Deﬁne G(a) if x =0 F (a,x,e)' H(a,y,{e}m+1,n(a,y)) if x = y +1. Then F is partial–computable. Using the Recursion Theorem we obtain an index e such that {e}m+1,n(a,x)'F(a,x,e). Deﬁning E := {e}m+1,n we obtain G(a) if x =0 E(a,x)' H(a,y,E(a,y)) if x = y +1 by induction on x. Hence E = Rec(G, H). If moreover G and H are total, we get (∀a)(∀x)(∃y)[E(a,x) 'y] by induction on x.

1.3 Computability relativized

If F is an (1, 1)–ary partial–computable functional and α ∈ NN a given function then we may try to compute the function λx. F (α, x).SinceFis partial–computable we have

F (α, x) ' y ⇔ (∃w)Q(α((w)0), (w)1,x,y) for some decidable predicate Q. Deciding Q(α((w)0), (w)1,x,(w)2) for w =0,1,2,... and picking the least such w yields an algorithm for λx. F (α, x) which asks for at most finitely many values of α. That means that a machine, e.g. a TURING–machine, could compute λx. F (α, x) asking an oracle for the function α within finite time. In this situation we say that the function λx. F (α, x) is computable relatively to α. Generalizing this to functionals leads to the following definition.

m,n 1.3.1 Deﬁnition A functional F : N −→ p N is partial–computable in a given function α if there is an (m, n +1)–ary partial–computable functional G such that

13 1. Computable Functionals and Relations

F (a) ' G(a,α). We call F computable in α if F is partial–computable in α and total. The functional F is (partial– ) computable in a set A ⊆ N if F is (partial–)computable in its characteristic function χA.

1.3.2 Deﬁnition A relation R ⊆ Nm,n is semi–decidable in a function α ∈ NN if R is the domain of a functional which is partial–computable in α. We call R decidable in α if its characteristic functional χR is computable in α. A relation R is (semi–)decidable in a set A ⊆ N if R is (semi–)decidable in its characteristic function χA. The computability of functionals and relations carries over to the relativized case. We put Tα,m,n := (e, a,w) Tm,n+1(e, a,α,w) TA,m,n := TχA,m,n {e}α,m,n := λa.U(µw. Tα,m,n (e, a,w)) {e}A,m,n := {e}χA,m,n ΦA,m,n := λea. {e}A(a) α,m,n { }α,m,n A,m,n χA,m,n We := dom( e ) and We := We . m,n To complete this section we reformulate the Normal–form Theorem, the Sk –Theorem and the Recursion Theorem for the relativized case.

1.3.3 Theorem (Relativized Normal–form Theorem) For any (m, n)–ary functional which is partial–computable in α there is an index e such that F (a) '{e}α,m,n(a). The functional Φα(e, a) is universal for the (m, n)–ary functionals which are partial–computable in α. For any (m, n)–ary relation which is semi–decidable in α there is an index e such that α,m,n R = We . To emphasize the relativized meaning we often talk about α–indices or A–indices, respectively.

m,n 1.3.4 Theorem (Relativized Sk –Theorem) There is an k +1–ary primitive–recursive func- m,n tion Sk such that { }α,m+k,n '{ m,n }α e (a,y1,...,yn) Sk (e, y1,...,yn) (a) and ∈ α,m+k,n ⇔ ∈ α,m,n (a,y1,...,yn) We a W m,n . Sk (e,y1,...,yn)

m,n 1.3.5 Theorem (Relativized Recursion Theorem) Let G: N −→ p N be partial–computable in α. Then there is an index e such that {e}α,m,n(a) ' G(a,e).

14 2. Degrees

This chapter will contain a brief introduction to Degree Theory. In Degree Theory we aim at classifying sets according to the difﬁculty of their decision problem. Two sets belong to the same degree if the solution of the decision problem for one set entails the solution of the decision problem for the other set and vice versa. There are different reducibility relations which are regarded in Computability Theory. Here we will only regard two of them. A quite narrow one — m–Reducibility — and the most general one — TURING–Reducibility.

2.1 m–Degrees

2.1.1 Deﬁnition Let A, B ⊆ N. We say that A is many–one reducible to B, m–reducible to B for short, if there is a computable function, say f, such that x ∈ A ⇔ f(x) ∈ B.

This will be denoted by A ≤m B. In case that the reducing function f is one–one, we talk about one–one Reducibility or 1–Reducibility and denote this by A ≤1 B.

2.1.2 Discussion If A ≤m B or A ≤1 B we obviously can reduce the decision problem for A to that of B. To decide x ∈ A we compute f(x), which is possible because of the computability of f and then decide f(x) ∈ B. There are some simple observations about m–Reducibility.

2.1.3 Lemma The relation ≤m is reﬂexive and transitive. If A ≤m B then also ¬A ≤m ¬B where ¬A := x ∈ N x/∈A denotes the complement of the set A.

Proof: We have A ≤ m A via the identity. If A ≤m B via f and B ≤m C via g then A ≤m C via g ◦ f. If A ≤m B via f we get x ∈ A ⇔ f(x) ∈ B which implies also x/∈A ⇔ f(x)∈/B.

Therefore we also have ¬A ≤m ¬B via f.

2.1.4 Deﬁnition We put

A ≡m B :⇔ A ≤m B ∧ B ≤m A and conclude from Lemma 2.1.3 that ≡m is an equivalence relation. Its equivalence classes are called m–degrees.By degm(A):= B⊆N A≡m B we denote the m–degree of A.

15 2. Degrees

We will not study the theory of m–degrees in this lecture. However, since we need m–degrees sometimes we decided to introduce them. Without proof we mention that Pow(N) together with ≤m is an upper semi–lattice. It is a known result of OCT that for all decidable sets A/∈{∅,N} we have K 6≤m A where K := x (∃w)T(x, x, w) . Just two simple facts about m–Reducibility.

2.1.5 Theorem 1) If B is decidable in α and A ≤m B then A is also decidable in α. 2) If A ≤m B and B is semi–decidable in α then A is also semi–decidable in α.

Proof: 1) If A ≤m B via f then

χA = χB ◦ f. 2) A = x f(x) ∈ B is semi–decidable in α since these sets are closed under substitution with computable functions.

2.2 TURING–Reducibility

The most general reduction of the decision problem is given by TURING–Reducibility.

2.2.1 Deﬁnition We say that a set A is decidable in B if χA is computable in χB. This is denoted by

A ≤T B or brieﬂy A ≤ B if there is no danger of confusion. Synonymously we say that A is TURING– reducible to B. We put

A ≡T B :⇔ A ≤T B ∧ B ≤T A.

2.2.2 Discussion If A ≤T B and we want to decide x ∈ A we compute χA(x). Thisiscom- putable in B. If we assume that the decision problem for B is solved, we can use a decision procedure for B in the computation of χA(x). Therefore the decision problem for A is reduced to that of B. This is, however, a reduction in a much weaker sense than m–Reducibility. So we have obviously

A ≤m B ⇒ A ≤T B while the opposite direction is not true in general. In this sense ≤T is a coarser relation than ≤m.

2.2.3 Theorem The relation ≤T is reﬂexive and transitive. Therefore the relation ≡T is an equivalence relation on Pow(N).

Proof: Because of χA(x)=App(χA,x)we see that A is decidable in A. Hence ≤T is reﬂexive. If A ≤T B and B ≤T C we have computable functionals F and G such that

χA(x)=F(χB,x) and

χB(x)=G(χC,x). So we get

χA(y) ' F (λx. G(χC ,x),y).

16 2.2. TURING–Reducibility

Since G is total λx. G(α, x) is total for any α and we get by the Substitution Lemma (Lemma 1.1.11) that λαy . F (λx. G(α, x),y) is a computable functional, say H.Butthen

χA(y)=H(χC,y) which shows that A ≤T C.

2.2.4 Theorem (POST’s Theorem) AsetA⊆Nis decidable in B ⊆ N iff both A and ¬A are semi–decidable in B.

Proof: If A is decidable in B then both A and ¬A are decidable in B. Hence also semi–decidable in B. This gives the easy direction. For the opposite direction assume that both A and ¬A are semi–decidable in B. Then we get indices e1 and e2 such that B,1,0 A = x (∃z)T (e1,x,z) (i) and B,1,0 ¬A = x (∃z)T (e2,x,z) . (ii) Put

B,1,0 B,1,0 f(x):'µz . [T (e1,x,z)∨T (e2,x,z)]. Then f is partial–computable in B and we get from (i) and (ii) that f is also total. So f is computable in B and we have B,1,0 A = x ∈ N T (e1,x,f(x)) which shows by Theorem 1.1.15 that A is decidable in B.

2.2.5 Remark We formulated POST’s theorem for sets in order to have it ﬁt into this section. The proof, however, shows that it is true also for arbitrary relations.

2.2.6 Lemma Let A be semi–decidable in B and B ≤T C.ThenAis semi–decidable in C.

Proof: We have an index e such that A = x ∈ N (∃z)TB,1,0(e, x, z) 1,1 = x ∈ N (∃z)T (e, χB,x,z) .

Because of B ≤T C there is an index e0 such that C,1,0 χB = {e0} C,1,0 = λx. U(µw. T (e0,x,w)) (i) 1,1 = λx. U(µw. T (e0,χC,x,w)). Hence 1,1 1,1 A = x ∈ N (∃z)T (e, λx. U(µw. T (e0,χC,x,w))),x,z 1,1 1,1 (ii) =dom(µz . T (e, λx. U(µw. T (e0,χC,x,w)))).

Since χB is total we get by (i) and the Substitution Lemma (Lemma 1.1.11) that the functional in the last line of (ii) is partial–computable in C. Hence A is semi–decidable in C.

17 2. Degrees

2.3 TURING–Degrees

We say that two sets A, B ⊆ N are TURING–equivalent iff A ≡T B. The class degT (A)= B B≡T A forms the TURING–degree (or just degree)ofA. We will denote degrees by lower case bold Roman letters, e.g., a, b, c, a1,.... For degrees a, bwedeﬁne

a≤b ⇔ (∃A∈a)(∃B ∈ b)[A ≤T B]. (2.1) It follows from Theorem 2.2.3 that (2.1) is independent of the choice of A and B. We put a < b :⇔ a ≤ b ∧ a =6 b. There is a minimal degree

0 := degT (∅) which contains exactly the decidable sets. To show that for any degree a there is a strictly bigger degree a0 we introduce the jump operator which is deﬁned by ∃ A,1,0 ∈ A,1,0 j(A):= x ( w)T (x, x, w) = x x Wx for A ⊆ N. We call j(A) the jump of A. For a degree a we introduce 0 a := degT (j(A)) for some A ∈ a. (2.2) We will show later (cf. Theorem 3.1.1) that

A ≤T B ⇒ j(A) ≤T j(B). 0 Therefore a in (2.2) is well–deﬁned. We will moreover see that A ≤m j(A). Hence also A ≤T j(A) and j(A) is obviously semi–decidable in A. We have, however, the following fact.

2.3.1 Theorem The jump j(A) is not decidable in A.

Proof: Towards a proof by reductio ad absurdum assume j(A) ≤T A.Then¬j(A)≤T Awhich entails that there is an index e such that ¬ A,1,0 j(A)=We . Hence ∈ ⇔ ∈ A,1,0 ⇔ ∈ e/j(A) e We e j(A). A contradiction. As an immediate corollary of Theorem 2.3.1 we get

2.3.2 Theorem For any degree a we have a < a0.

Now the canonical questions arise • Are the degrees linearly ordered by ≤? • Are there degrees between a and a0?

These questions have already been asked by E. POST in 1944. It lasted until 1954 before they could be answered independently by R. FRIEDBERG and A. MUCHNIK. They proved the following theorem.

18 2.3. TURING–Degrees

2.3.3 Theorem (FRIEDBERG,MUCHNIK) There are semi–decidable sets A, B which are incomparable with respect to ≤T , i.e. we have neither A ≤T B nor B ≤T A.

Proof: Before we start proving the theorem let us discuss it brieﬂy. The proof will show that the theorem also holds in relativized form. It is just for simpler notations that we omitted the relativization. We have ∅≤T Cfor any set C and – as we will see soon – A ≤T j(∅) for any semi–decidable set A. Thus if A and B are semi–decidable and incomparable we get the picture shown in Figure 2.3.1 where the arrows represent ≤T . This shows that Theorem 2.3.3 in fact answers both questions.

Figure 2.3.1: Two incomparable semi–decidable sets

The degrees are not linearly ordered and there are degrees between a and a0. To prepare the technical part of the proof we start with a few heuristic remarks. Since we have D ≤T C for any decidable set D and any set C none of the sets A and B, which we are going to construct, must be decidable. Since we aim at A T B as well as B T A we have to ensure B A that χA =6 {e} for all e and also χB =6 {e} for all e,i.e. B (∀e)(∃y)[χA(y) 6' {e} (y)] (i) and

A (∀e)(∃y)[χB (y) 6' {e} (y)]. (ii) To obtain (i) and (ii) it suffices to construct a function F which satisfies (∀e)[F (2e) ∈ A ⇔{e}B,1,0(F(2e)) ' 1] (iii) and (∀e)[F (2e +1)∈B ⇔{e}A,1,0(F (2e +1))'1]. (iv) The function F , however, must not be computable. To see that assume that F is computable satisfying (iii) and (iv). We define 1 if x/∈B f(x, y) ' 0 if x ∈ B. Then f is computable in B and we obtain an index e such that f = {e}B.

19 2. Degrees

m Using the relativized S1 –Theorem this yields {S(e, x)}B(y) ' 1 ⇔ x/∈B for all x ∈ N. Hence x/∈B ⇔{S(e, x)}B(F (2 · S(e, x)) ' 1 ⇔ F (2 · S(e, x)) ∈ A by (iii). Since A is semi–decidable and F computable ¬B is semi–decidable. Hence B is decidable by POST’s Theorem. This, however, is impossible as we have seen above. The problem is to construct F in such a way that F does not become computable. This cannot be simple because any construction of a non–computable function is close to conﬂict with CHURCH’s Thesis. The basic idea is to approximate A, B and F stepwise by An, Bn and

λx. F (n, x) such that χAn , χBn and λx. F (n, x) are computable. In step n we compute either 1,1 ∧ yn := µw < n. T (e, F (n, 2e),χBn,w) U(w)=1 (v) or 1,1 ∧ yn := µw < n. T (e, F (n, 2e +1),χAn,w) U(w)=1 (vi) according to the shape of n which also determines e in an effective way. Whenever yn =6 n we put in the ﬁrst case F (n, 2e) into An+1 or — in the second case — F (n, 2e +1)in Bn+1.The obvious problem now is that at a later point m>n, where a larger portion Am of A (or Bm of B) is known, the computation may change. Therefore we give F (n +1,x)a value above yn to ensure that the computations in (v) and (vi) will not be changed. The index n in F (n, 2e) is therefore the priority with which F (n, 2e) has to be put into A (or F (n, 2e +1)into B). Once we have reached the highest priority n we may put F (x):=F(n, x). Of course we need to prove that such highest priorities exists. Though certainly still vague, we hope that these remarks will be helpful in the following technical part of the proof. We put 2e if x =2e A := ∅,B:= ∅ and F (0,x):= (vii) 0 0 2e if x =2e+1.

Assume that An, Bn and F (n, x) are deﬁned for all x. We distinguish the following cases:

1) (n)0 =2efor some e ∈ N. Then we compute 2,0 ∧ ∧ ∈ yn := µw < n. T (e, F (n, 2e), χBn ((w)0), (w)1) U((w)1)=1 F(n, 2e) / An . (viii)

If yn = n we put

An+1 := An,Bn+1 := Bn and F (n +1,x):=F(n, x). (ix) Otherwise we deﬁne

An+1 := An ∪{F(n, 2e)},Bn+1 := Bn (x) and F(n, x) if x ≤ 2e or x ≡ 0 mod 2 F (n +1,x):= (xi) F (n, x) · 3yn if 2e

2) (n)0 =2e+1for some e ∈ N. Again we compute

20 2.3. TURING–Degrees yn := 2,0 ∧ ∧ ∈ (xii) µw < n. T (e, F (n, 2e +1),χAn((w)0), (w)1) U((w)1)=1 F(n, 2e +1) /Bn .

If yn = n we put

An+1 := An,Bn+1 := Bn,F(n+1,x):=F(n, x) (xiii) and otherwise

An+1 := An,Bn+1 := Bn ∪{F(n, 2e +1)} (xiv) and F(n, x) if x ≤ 2e +1or x ≡ 1 mod 2 F (n +1,x):= (xv) F (n, x) · 3yn if 2e +1

One should observe that in (vii) through (xv) we deﬁne the functions λnx. χAn (x), λnx. χBn (x) and λnx. F (n, x) simultaneously by the Recursion Theorem. Hence all these functions are partial–computable. It follows by induction on n that all these functions are also total. By construction we have F (n, x) ≤ F (n +1,x) which yields m ≤ n ⇒ F (m, x) ≤ F (n, x) by induction on n. Similarly we get

m ≤ n ⇒ Am ⊆ An ∧ Bm ⊆ Bn. The essential step is to show: Vx := n F (n, x) =6 F (n +1,x) is ﬁnite.

The proof is by induction on x.Fory

We construct a one–one mapping from Vx into V .Letn∈Vx.Ifxis even then by (xi) and (xv) there is an xn

F (m, xm) ∈ Bm+1 ⊆ Bn 63 F (n, xn) for x even or

F (m, xm) ∈ Am+1 ⊆ An 63 F (n, xn) for x odd, respectively. Hence F (m, xm) =6 F (n, xn) and

n 7→ F (n, xn) is a one–one map from Vx into V . Therefore Vx is ﬁnite. We deﬁne [ [ A := An; B := Bn n∈N n∈N and

21 2. Degrees

F (x):=F(n, x) if (∀m)[m ≥ n ⇒ F (m, x)=F(n, x)]. (xvi) Now we prove {e}A(F (2e +1))'1 ⇒ F(2e +1)∈B (xvii) {e}B(F (2e)) ' 1 ⇒ F (2e) ∈ A. Both lines of (xvii) are proved analogously. We show the ﬁrst. From {e}A(F (2e +1))'1 we get for some w ∈ N 2,0 ∧ T (e, F (2e +1),χA((w)0), (w)1) U((w)1)=1. (xviii) There are inﬁnitely many n ∈ N such that

(n)0 =2e+1. We choose n so big that

F (2e) ∈ An+1 \ An which implies

F (2e)=F(n, 2(n)0).

According to (xvi) and the ﬁrst line in (xxii) this yields e =(n)0. Hence by (viii) 2,0 ∧ T (e, F (n, 2e), χBn ((yn)0), (yn)1) U((yn)1)=1 (xxiii) and yn

χB ((yn)0)=χBn((yn)0) (xxiv) we get {e}B(F (2e)) ' 1 from (xxiii). Towards a contradiction assume 6 χB ((yn)0) = χBn ((yn)0).

Then there is a z<(yn)0

22 2.3. TURING–Degrees

6 χB(z) = χBn (z).

But then z ∈ B \ Bn which shows that there is an m>nsuch that

z ∈ Bm+1 \ Bm. Hence z = F (m, 2f +1)for some f ∈ N.If2e<2f+1we get by (xi) z = F (m, 2f +1)≥F(n+1,2f+1)=F(n, 2f +1)·3yn which contradicts z2f+1, however, we get by (xv) F (m +1,2e)=F(m, 2e) · 3yn >F(m, 2e)=F(n, 2e)=F(2e) contradicting the deﬁnition of F (2e) in (xvi). Hence (xxiv).

23 2. Degrees

24 3. The Arithmetical Hierarchy

3.1 The Jump operator revisited

The jump operator ∈N ∃ A,1,0 ∈ A,1,0 j(A):= x ( w)T (x, x, w) = x x Wx (3.1) is introduced in Section 2.3. We are going to study its properties more profoundly in this section. It follows from (3.1) and Theorem 2.3.1 that j(A) is semi–decidable but not decidable in A.The following theorem strengthens that.

3.1.1 Theorem 1) A set A ⊆ N is semi-decidable in B iff A ≤m j(B). 2) We have A ≤T B iff j(A) ≤m j(B). Both claims hold uniformly in A, i.e. an index of the m–reducing computable function f can be computed from a B–index for the set A.

Proof: 1) Deﬁne B B,1,0 K := (x, y) (y)1 ∈ W 0 (y)0 B,1,0 = (x, y) (∃w)T ((y)0, (y)1,w) .

B B The predicate K0 is semi–decidable in B.Lete0be an index for K0 .Thenweget (y) ∈WB,1,0 ⇔ (x, y) ∈ KB 1 (y)0 0 ⇔ ∈ B,2,0 (x, y) We0 (i) B,1,0 ⇔ x ∈ W 2,0 . S1 (e0,y) B,1,0 If A is semi–decidable in B we have an index e for A,i.e.A=We , and deﬁne a function f by 2,0 h i f(x):=S1 (e0, e, x ). Then f is computable and an index for f can be computed from e. According to (i) we get ∈ ⇔ ∈ B,1,0 f(x) j(B) f(x) Wf(x) ⇔ ∈ B,1,0 f(x) W 2,0 h i S1 (e0, e,x ) ⇔ ∈ B,1,0 x We ⇔ x ∈ A which shows that A ≤m j(B) via f. For the opposite direction we assume A ≤m j(B) via f.Butthen x∈A ⇔ (∃w)TB,1,0(f(x),f(x),w) which shows immediately that A is semi–decidable in B. 2) We start with the “if”–direction. Since j(A) is semi–decidable in A we get from A ≤T B by Lemma 2.2.6 that j(A) is also semi–decidable in B. Hence j(A) ≤m j(B) by 1). To obtain also the uniformity we need to know that a B–index of j(A) can be computed from a B–index of A, i.e. we need a computable function, say h, with

25 3. The Arithmetical Hierarchy

{ }B,1,0 ⇒ B,1,0 χA = e j(A)=Wh(e) . Though simple a rigid proof is quite tedious and depends heavily on our special deﬁnition of indices. Therefore we restrict ourselves to a rough sketch. We have x ∈ j(A) ⇔ (∃w)T1,1(x, χ ,x,w) A (ii) ⇔ (∃w)T1,1(x, λy . {e}B(y),x,w). The function g := µw. T1,1(x, λy . {e}B(y),x,w) is obvious partial–computable in B and its index depends only on e. This dependence is effective which means that there is a computable (even primitive–recursive) function, say h, such that h(e) is a B–index for g. Hence by (ii) { }B B,1,0 j(A)=dom( h(e) )=Wh(e) .

For the “only–if”–direction assume j(A) ≤m j(B).SinceAas well as ¬A are semi–decidable in A we get

A ≤m j(A) ≤m j(B) and also

¬A ≤m j(A) ≤m j(B) by part 1). By the transitivity of ≤m and part 1) this implies that A and ¬A are both semi– decidable in B.UsingPOST’s Theorem (Theorem 2.2.6) we obtain A ≤T B.

3.1.2 Deﬁnition The n–th jump of a set A ⊆ N is deﬁned by A(0) := A A(n+1) := j(A(n)).

3.1.3 Lemma We have (n) (k) n ≤ k ⇒ A ≤m A (3.2) and (n) (n) A ≤m B ⇒ A ≤m B . (3.3) Claim (3.2) holds uniformly in n and k, i.e. an index for the reducing function can be computed from n and k, while claim (3.3) holds uniformly in n and the index of the function which reduces A to B.

Proof: We show (3.2) by induction on k. The claim is obvious for k = n.Fork=l+1>nwe have (n) (l) A ≤m A (l) (l+1) via {f(n, l)} by the induction hypothesis. By part 1) of Theorem 3.1.1 we have A ≤m A (n) (l+1) via some function g. Hence A ≤m A . To show also the uniformity we observe that g does not depend on A.Since ∈ 1,1 j(B)= x (x, χB ) Wx we see that there is an e ∈ N such that B,1,0 j(B)=We holds for any B ⊆ N. So, according to Theorem 3.1.1, g depends only on the constant e and the index of the reducing function g ◦{f(n, l)} can be computed from n and e.

26 3.2. The Arithmetical Hierarchy

We prove (3.3) by induction on n.Forn=0we have A ≤m B by hypothesis. In the successor (n) (n) (n+1) (n+1) case we have A ≤m B by the induction hypothesis and obtain A ≤m B by Theorem 3.1.1 2). The index of the reducing function is computed from a B(n)–index of A(n) (n) (n) which in turn depends on the reducing function for A ≤m B . This function, however, can by induction hypothesis be computed from an index of the reducing function for A ≤m B.

3.2 The Arithmetical Hierarchy

The Arithmetical Hierarchy classifies the subsets of N which can be defined arithmetically. The most obvious classification is according to the complexity of the defining formula. Therefore we introduce first a classification of the arithmetical formulas.

3.2.1 Definition Let ϕ be a formula in the language of arithmetic, i.e. the only non–logical sym- bols occurring in ϕ are constants for natural numbers, for primitive–recursive functions and of predicates which can be decided primitive–recursively. In an arithmetical formula all quantifiers are supposed to range over individuals, i.e. we are in first order, however, we allow free function variables ξ,η, ξ1,.... 0 We say that ϕ is a ∆0–formula, if ϕ contains at most bounded quantifiers. 0 0 ≡ ∃ We say that ϕ is Σ1,ifthereisa∆0–formula ψ(x) such that ϕ ( x)ψ(x). 0 ¬ 0 Dually ϕ is Π1 if ϕ is Σ1. 0 0 ≡ ∃ Aformulaϕis in Σn+1 if there is a formula ψ(x) in Πn such that ϕ ( x)ψ(x). 0 ¬ ∃0 Dually ϕ is Πn+1 if ϕ is n+1.

3.2.2 Remark In the above deﬁnition we assume that the language of arithmetic is given as a TAIT–language (cf. [4]), i.e. a language containing =6 as basic symbol in which ¬ϕ is deﬁned by ¬(s = t):≡s=6 t ¬(s=6 t):≡s=t

¬(Rt1,...,tn):≡(¬R)t1,...,tn

¬((¬R)t1,...,tn):≡Rt1,...,tn ¬(ϕ ∧ψ):≡¬ϕ∨¬ψ ¬(ϕ∨ψ):≡¬ϕ∧¬ψ ¬(∀x)ϕ(x):≡(∃x)¬ϕ(x) ¬(∃x)ϕ(x):≡(∀x)¬ϕ(x) where ¬R is a relation constant whose interpretation is the complement of the interpretation of R. We obviously have ∈ 0 ⇔ ≡ ∃ ∀ ϕ Σn ϕ ( x1)( x2) ...(Qxn)ψ(~x) and ∈ 0 ⇔ ≡ ∀ ∃ ϕ Πn ϕ ( x1)( x2) ...(Qxn)ψ(~x) 0 ∃ ∀ ∀ ∃ where ψ(~x) is a ∆0–formula and ( x1)( x2) ...(Qxn)as well as ( x1)( x2) ...(Qxn)are al- ternating strings of N–quantiﬁers.

m,n 3.2.3 Deﬁnition A relation R ⊆ N is deﬁnable with parameters β1,...,βl by a formula

ϕ(ξ1,...,ξl,x1,...,xn,η1,...,ηm) if ϕ possesses only the indicated free variables and

27 3. The Arithmetical Hierarchy

m,n R = a ∈ N N |= ϕ[β1,...,βl,a] .

0 0 3.2.4 Definition 1) A relation is Σn[A] if it is definable with parameter χA by a Σn–formula. 0 Πn[A]-relations are defined analogously. 0 0 0 2) A relation is ∆n[A] if it is both, Σn[A] and Πn[A]. 0 ∅ 0 ∅ 0 ∅ 0 0 0 Instead of Σn[ ], Πn[ ],and∆n[ ]we write Σn, Πn and ∆n. 0 ∈ N 3) A relation is called arithmetical (in A) if it is in ∆n[A] for some n . To unify notations we put 1 1 1 ∆0[A]:=Π0[A]:=Σ0[A]:= R Ris arithmetical in A .

0 3.2.5 Theorem 1) The ∆0–predicates are exactly the primitive–recursively decidable predicates. 0 2) The Σ1–relations are exactly the semi–decidable relations.

⊆ N 0 3.2.6 Theorem (POST) 1) A relation R is semi–decidable in a set A iff R is Σ1[A]. ⊆ N 0 2) A relation is decidable in a set A iffitis∆1[A]. The proofs of Theorems 3.2.5 and 3.2.6 are obvious from our previous knowledge.

3.2.7 Deﬁnition Let F denote one of the complexity classes introduced in Deﬁnition 3.2.4. We say that a partial functional is an F–functional iff its graph belongs to F.

0 0 3.2.8 Lemma Any total Σn[A]–functional is already in ∆n[A]. 0 ∧ ∃0 Proof: The proof needs already the closure of Σn under and –quantiﬁcation. Let F be a total 0 Σn[A]–functional. Then

¬GF (a,y) ⇔ F(a)6' y ⇔ (∃z)[F (a) ' z ∧ z =6 y]. 0 ∈ 0 Which shows that both, the graph of F and its complement, are in Σn[A]. Hence GF ∆n[A].

We list the closure properties of these newly introduced relation–classes in the table shown in Figure 3.2.1. The positive closure properties, i.e. those which carry a “yes”, are shown by induc- 0 tion on k. We already proved them for the case k =1with the exception of the closure of Σ1 under ∃1–quantiﬁcation. However, we want to postpone this property because it does not carry over to k>1. So assume that we have the positive closure properties for k.LetR1and R2 be 0 Σk+1–relations. Then we have

R1(a) ⇔ (∃x)Q1(a,x) and

R2(a) ⇔ (∃y)Q2(a,y) ∈ 0 for Qi Πk. Hence ∧ R1(a) ∨ R2(a) ⇔ ⇔ 0 and the expression in square–brackets is Πk by the induction hypothesis. 0 ∃0 The closure of Σk+1–relations under –quantiﬁcation follows by contraction of quantiﬁers, i.e. by

(Qx)(Qy)R(a,x,y) ⇔ (Qu)R(a,(u)0,(u)1). (3.4)

28 3.2. The Arithmetical Hierarchy

¬ ∨ ∧ ∃< ∀< ∃0 ∀0 ∃1 ∀1 Substitution Relation–class with

primitive–recursive yes yes yes yes yes no no no no primitive–recursive functionals computable ∆0–relations yes yes yes yes yes no no no no 1 functionals computable Σ0–relations no yes yes yes yes yes no yes no 1 functionals computable Π0–relations no yes yes yes yes no yes no yes 1 functionals 0 ∆0 –relations yes yes yes yes yes no no no no total Σk+1- k+1 functionals 0 Σ0 –relations no yes yes yes yes yes no no no total Σk+1- k+1 functionals 0 Π0 –relations no yes yes yes yes no yes no no total Σk+1- k+1 functionals yes yes yes yes yes yes yes no no total arithmetical arithmetical functionals

Figure 3.2.1: Closure Properties of Relation–Classes

∀ 0 So we are left with bounded –quantiﬁcation and substitution with total Σk+1–functionals. As- sume P (a) ⇔ (∀x

⇔ (∃s)[Seq(s) ∧ lh(s)=n∧(∀x

0 0 for a Σk+1–(Πk+1–)relation R we get P (a) ⇔ (∃z)[F (a) ' z ∧ R(a,z)] ⇔ (∀z)[F (a) ' z ⇒ R(a,z)]. 0 ∃0 Applying Lemma 3.2.8 – which is possible since we know that Σk+1 is closed under –quantification ∧ 0 0 and , we get that P is in Σk+1 or Πk+1, respectively. 0 0 0 The closure properties for ∆k–relations follow by combining those of Πk–andΣk–relations (we still regard only the positive closure properties) and the shown (positive) closure properties for 0 arithmetical relations follow from those of ∆n–relations. 0 ∃1 0 It remains to show that Σ1 is closed under –quantification. For a Σ1–relation R we get (∃α)R(a,α) ⇔ (∃α)(∃y)Q(a(y), α(y)) ⇔ (∃s)[Seq(s) ∧ lh(s)=y∧Q(a(y),s)] where Q is a semi–decidable predicate. But then the expression in square–brackets is also semi– decidable which implies that a (∃α)R(a,α) is a semi–decidable relation. By duality we 0 ∀1 obtain that Π1 is closed under –quantification. Observe that the closure under second order 29 3. The Arithmetical Hierarchy quantifiers cannot be lifted to the higher levels of the hierarchy. As soon as we have a quantifier string of the form (∃α)(∀x)q(α, x) it is obvious that we cannot replace α by a finite sequence. A rigid proof will be in the Analytical Hierarchy Theorem in the next chapter. Let ϕ(x1,...,xm,ξ1,...,ξn) be any first order formula in the language of arithmetic. Then ϕ(~x, ξ~) is logically equivalent to a formula in prenex form and we may use the quantifier contraction (3.4) to see that N |=(∀~x)(∀ξ~)[ϕ(~x, ξ~) ⇐⇒ ψ ( ~x, ξ~)] for a formula ψ which is either in Σ0 or Π0 for some k ∈ N.Then k k R:= a N |= ϕ[a]

0 is ∆k+1, i.e. arithmetical. We put this into a lemma.

3.2.9 Lemma Let ϕ(x1,...,xm,ξ1,...,ξn) be a ﬁrst order formula in the language of arithmetic. Then the relation R := a ∈ Nm,n N |= ϕ[a] is arithmetical.

We are now going to investigate the connection of the arithmetical hierarchy to the jump hierarchy introduced in Deﬁnition 3.1.2.

0 (k) 3.2.10 Theorem A relation R is Σk+1[A] iff it is semi–decidable in A . Proof: We prove the theorem by induction on k. We begin with the “if”–direction. For k =0 ∈ 0 this is Theorem 3.2.6 1). For the induction step assume that R Σk+2[A].Then a∈R ⇔ (∃x)P(a,x) 0 ¬ ∈ 0 ¬ (k) for a Πk+1[A]–relation P .Then P Σk+1[A] and P is semi–decidable in A by induction (k+1) hypothesis. By Theorem 3.1.1 1) this implies ¬P ≤T A and, since P ≤T ¬P , P ≤T A(k+1).SinceRis the N–projection of the relation P which is decidable in A(k+1) we get by the relativization of Theorem 1.1.17 that R is semi–decidable in A(k+1). For the “only if” direction let R be semi–decidable in A(k+1).SinceA(k+1) is semi–decidable in A(k) we get (k+1) ∈ 0 (k+1) A Σk+1[A] by induction hypothesis. Let e be an A –index for R. Then we obtain ⇔ ∈ A(k+1),m,n R(a) a We ⇔ ∃ m,n+1 ( w)T (e, a,χA(k+1) ,w) ⇔ ∃ m+n+2 ( w)T (e, a((w)0), χA(k+1) ((w)0), (w)1, (w)2) (i) ⇔ (∃s)(∃w)[Seq(s) ∧ lh(s)=(w)0 ∧ ∀ ∧ m+n+2 ( i<(w)0)(χA(k+1) (i)=(s)i) T (e, a((w)0),s,(w)1,(w)2)]. 0 The predicates Seq, T , = etc. are all in ∆0. So we only have to check the complexity of χA(k+1) (i)=y. Because of ⇔ ∧ ∈ (k+1) ∨ ∧ ∈ (k+1) χA(k+1) (i)=y (y=0 x A ) (y =1 x/A ) (k+1) ∈ 0 and the fact that A Σk+1[A] we get ∈ 0 (i, y) χA(k+1) (i)=y ∆k+2[A]. (ii) ∈ 0 But (i) together with (ii) show R Σk+2[A]. As a consequence of Theorem 3.2.10 we get the following generalization of POST’s theorem.

30 3.2. The Arithmetical Hierarchy

0 (k) 3.2.11 Theorem A relation R is in ∆k+1[A] iff R is decidable in A . ∈ 0 ¬ (k) Proof: We have R ∆k+1[A] iff R and R are semi–decidable in A by Theorem 3.2.10. By POST’s theorem this holds if and only if R is decidable in A(k). The next theorem will help us in conﬁrming also the negative closure properties listed in Fig- ure 3.2.1.

3.2.12 Lemma We have (k+1) ∈ 0 A / ∆k+1[A] for all k ∈ N.

Proof: We have shown in Theorem 2.3.1 that j(M) 6≤T M for any set M. By Theorem 3.2.11, (k+1) ∈ 0 however, this means A / ∆k+1[A].

3.2.13 Theorem (Arithmetical Hierarchy Theorem) We have 0 $ 0 1) ∆k+1[A] Σk+1[A] 0 $ 0 2) ∆k+1[A] Πk+1[A] 0 ∪ 0 $ 0 3) Σk+1[A] Πk+1[A] ∆k+2[A]. Proof: All inclusions are obvious by definition. It remains to show that these inclusions are proper. According to Lemma 3.2.12 we have (k+1) ∈ 0 (k+1) ∈ 0 A / ∆k+1[A] but A Σk+1[A] by Theorem 3.2.10. This proves 1) and 2) is an immediate consequence of 1). To prove 3) regard the “effective union” of A(k+1) and ¬A(k+1) which is given by B := 2x x ∈ A(k+1) ∪ 2x +1 x/∈A(k+1) . ∈ 0 (k+1) ≤ ¬ (k+1) ≤ Then B ∆k+2[A] and A m B via λx. 2x as well as A m B via λx. 2x +1. ∈ 0 ∈ 0 Hence neither B Σk+1[A] nor B Πk+1[A] because any of both assumptions would lead to (k+1) ∈ 0 A ∆k+1[A] which contradicts Lemma 3.2.12. 0 It follows from the Arithmetical Hierarchy Theorem that Σk[A] cannot be closed under negation ∀0 0 ∃0 and –quantification. Dually Πk[A] cannot be closed under negation and –quantification. Since any first order quantifier (Qx)[...x...] can be replaced by a second order quantifier (Qα)[...α(0) ...] 0 ∀1 0 we see that Σk[A] cannot be closed under –quantifiers and dually that Πk[A] cannot be closed ∃1 0 ∃1 under –quantifiers. So the only open items in Figure 3.2.1 are closure of Σk[A] and – 0 ∀1 quantifiers and Πk[A] and –quantifier for k>1. We have to postpone that until the next chapter. Up to now we get a picture of the Arithmetical Hierarchy as shown in Figure 3.2.2. Let us recall the notion of an universal relation.

3.2.14 Deﬁnition Let R be a collection of (m, n)–ary relations. An (m +1,n)–ary relation U is universal for R if for any R ∈ R there is an e ∈ N such that

31 3. The Arithmetical Hierarchy

1 ∆0

0 0 Σ3 Π3

00 0 0 ∆3

0 0 Σ2 Π2

0 0 0 ∆2

0 0 Σ1 Π1 0 ∆1 0

Figure 3.2.2: The Arithmetical Hierarchy

R(a) ⇔ U(a,e). A collection K of relations is a universal class if for any (m, n) there is an (m+1,n)–ary relation U m,n ∈ K which is universal for the (m, n)–ary relations in K. K is an acceptable universal class if K is a universal class and there are k +1–ary computable m,n functions Sk such that m+k,n ⇔ m,n m,n U (a,y1,...,yk,e) U (a, Sk (e, y1,...,yk)). If K is a universal class and R(a) ⇔ U m,n(a,e) we call e an K–index for R.

0 We have already seen that the class of Σ1[A]–relations, i.e. the class of relations which are semi– decidable in A, is an acceptable universal class. This can be lifted to all levels of the Arithmetical Hierarchy.

0 0 3.2.15 Theorem The classes of Σk[A] and Πk[A] are acceptable universal. 0 Proof: By Theorem 3.2.10 we get for an (m, n)–ary Σk+1[A]–relation R ⇔ ∈ A(k),m,n R(a) a We . Putting

0 (k) Σk+1[A],m,n ∈ A ,m,n U := (a,e) a We 0 deﬁnes a universal relation for the (m, n)–ary relations in Σk+1[A]. The acceptability, however, m,n is an immediate consequence of the relativized Sk –Theorem. 32 3.3. The Limits of the Arithmetical Hierarchy

0 Πk+1[A],m,n 0 By dualization we get universal relations U for the (m, n)–ary Πk+1[A]–relations.

3.3 The Limits of the Arithmetical Hierarchy

The collection of all arithmetically deﬁnable sets forms a countable set. This shows that we are far from having characterized all subsets of N. We will indicate that we are even still far from having characterized all deﬁnable subsets of N. Put (ω) ((x)1) A := x (x)0 ∈ A .

(ω) (n) (ω) We may regard A as “effective” union of all A . EffectiveS because for any x ∈ A we can (n) by computing (x)1 effectively determine to which member of A the element (x)0 belongs. Clearly any effective union has to be pairwise disjunct. Because of the effectiveness of A(ω) we get x ∈ A(n) ⇔hx, ni∈A(ω) which shows (n) (ω) A ≤m A (3.5) for any n.

3.3.1 Theorem The set A(ω) is not arithmetical in A. (ω) ∈ 0 (ω) ≤ Proof: Towards an indirect proof assume that A ∆k[A] for some k. Hence A T (k) ≤ (k+1) ≤ (ω) (k) ≡ (k+1) (k+1) ∈ 0 A T A T A which implies A T A .ButthenA ∆k+1[A] by Theorem 3.2.11 which contradicts Lemma 3.2.12. Building A(ω) means to iterate the jump operator arbitrarily finitely often. But when we have A(ω) we can go on building j(A(ω)),j(j(A(ω))) ...Such an infinite iteration of jumps, however, needs a theory of ordinals, which we postpone until Chapter 5. First we want to extend the hierarchy by allowing second order formulas in the defining formulas of relations.

33 3. The Arithmetical Hierarchy

34 4. The Analytical Hierarchy

4.1 Second order arithmetic

In order to extend the arithmetical hierarchy we extend the hierarchy of arithmetical formulas. We are going to allow quantifiers on functions which are supposed to range over NN. We briefly denote by N |= ϕ that the sentence ϕ is valid in the standard interpretation. Let us start with a classification of the second order arithmetical formulas according to their second order quantifier–complexity.

1 ≡ ∀ 0 4.1.1 Deﬁnition Aformulaϕis a Π1–formula if ϕ ( α)ψ(α) and ψ(α) is Σ1. 1 ¬ 1 Dually a formula ϕ is Σ1 iff ϕ is Π1. 1 ≡ ∀ 1 Aformulaϕis Πk+1 iff ϕ ( α)ψ(α) and ψ(α) is Σk. 1 ¬ 1 Dually ϕ is Σk+1 iff ϕ is Πk+1. Again we get ∈ 1 ⇔ ≡ ∀ ∃ ˘ ϕ Πk ϕ ( α1)( α2) ...(Qαk)(Qx)ψ(α,~ x) and ∈ 1 ⇔ ≡ ∃ ∀ ˘ ϕ Σk ϕ ( α1)( α2) ...(Qαk)(Qx)ψ(α,~ x) where ψ(α,~ x) is quantiﬁer free. 1 1 A formula is analytic if it is Σn or Πn for some n. We introduce the abbreviation

(α)x := λu. α(hx, ui). (4.1)

4.1.2 Lemma For any formula ϕ in the language of second order arithmetic and Q ∈{∀,∃} we have (Qx)ϕ(x) ⇔ (Qα)ϕ(α(0)) (4.2)

(Qα)(Qβ)ϕ(α, β) ⇔ (Qγ)ϕ((γ)0, (γ)1) (4.3)

(∀x)(∃α)ϕ(x, α) ⇔ (∃β)(∀y)ϕ(y,(β)y) (4.4)

(∃x)(∀α)ϕ(x, α) ⇔ (∀β)(∃y)ϕ(y,(β)y) (4.5)

Proof: Claim (4.2) holds obviously and (4.3) becomes clear by putting α((x) ) if (x) =0 γ(x):= 1 0 β((x)1) if (x)0 =1. The direction from right to left in (4.4) holds for logical reasons. For the opposite direction assume (∀x)(∃α)ϕ(x, α) and choose αx for every x ∈ N. Deﬁning

β(y):=α(y)0((y)1) 35 4. The Analytical Hierarchy we get

(∀y)ϕ(y,(β)y). Hence

(∃β)(∀y)ϕ(y,(β)y). Equation (4.5) follows from (4.4) by taking negations. We observe that every formula in the language of Second Order Arithmetic is logically equivalent to some formula in prenex form. Using Lemma 4.1.2 it becomes equivalent to some formula of the form

(∀α1)(∃α2) ...(Qαn)(Q˘x)ϕ(α1,...,αn,x) or

(∃α1)(∀α2) ...(Q˘αn)(Qx)ϕ(α1,...,αn,x) where ϕ(α1,...,αn,x)is quantiﬁer free and Q˘ denotes the quantiﬁer which is dual to Q. Hence every formula in the language of Second Order Arithmetic is equivalent to some analytical formula.

4.2 Analytical relations

1 1 4.2.1 Deﬁnition 1) A relation is Πn[A](Σn[A]) iff it is deﬁnable with parameter χA by some 1 1 1 1 1 ∅ 1 ∅ Πn–(Σn–)formula. Again we write Πk and Σk instead of Πk[ ] and Σk[ ]. 1 1 1 2) A relation is ∆k[A] iff it is both Σk[A] and Πk[A]. 1 1 3) A relation is analytical (in A)ifitisin∆k (∆k[A])forsomek.

A picture of the analytical hierarchy is given in Figure 4.2.1.

4.2.2 Remark By the considerations in the end of the previous section we get that a relation is deﬁnable in second order arithmetic iff it is analytical.

To obtain the closure properties of analytical relations we begin with the lowest level.

1 4.2.3 Lemma The relations in Π1[A] are closed under • the positive boolean operations ∧, ∨ • all N–quantiﬁcations •∀1–quantiﬁcations • 1 substitution with total functionals having Π1[A]–graphs

Proof: Let

P1(a) ⇔ (∀α)(∃y)Q1(α, y, a) and

P2(a) ⇔ (∀β)(∃z)Q2(β,z,a) 1 be Π1–relations. Then 36 4.2. Analytical relations

1 1 Π1 Σ1 1 ∆1 0ω 1 ∆0

0 0 Σ3 Π3

00 0 0 ∆3

0 0 Σ2 Π2

0 0 0 ∆2

0 0 Σ1 Π1 0 ∆1 0

Figure 4.2.1: The Analytical Hierarchy

∧ ∧ P1(a) ∨ P2(a) ⇔ (∀α)(∃y)Q1(α, y, a) ∨ (∀β)(∃z)Q2(β,z,a) ∧ ⇔ (∀α)(∀β)(∃y)(∃z)[Q1(α, y, a) ∨ Q2(β,z,a)] ∧ ⇔ (∀γ)(∃u)[Q1((γ)0, (u)0, a) ∨ Q2((γ)1, (u)1, a)] . This gives the closure under positive boolean operations. Closure under ∀1–quantification follows from the quantifier contractions (4.3); closure under ∀0–quantification is obtained by converting it into a ∀1–quantifier according to (4.2) and then using quantifier contraction; closure under ∃0–quantification follows from the choice–principle (4.5). Let’s turn to closure under substitution. For a total functional F we get F (a) =6 y ⇔ (∃x)[F(a)'x∧x=6 y] ⇔ (∀x)[F(a)'x⇒x=6 y] ∈ 1 1 1 1 which shows that GF ∆1[A] for Π1[A]–andforΣ1[A]–functionals F (hereweusethatΣ1[A] 1 1 has the dual closure properties of Π1[A]). For a Π1[A]–relation P we obtain P (a,F(a)) ⇔ (∃y)[F(a)'y ∧P(a,y)] (4.6) ⇔ (∀y)[F(a)'y ⇒P(a,y)] . 37 4. The Analytical Hierarchy

1 1 1 This shows that Π1[A] (as well as Σ1[A]) is closed under substitution with total Π1[A]–functionals.

By dualization we obtain from Lemma 4.2.3.

1 4.2.4 Lemma The Σ1[A]–relations are closed under • the positive boolean operations ∧, ∨ • all N–quantifications •∃1–quantifications • 1 substitution with total Π1[A]–functionals 1 The ∆1[A]–relations are closed under • all boolean operations • all N–quantifications • 1 substitution with total Π1[A]–functionals. Using induction on k in the same way as we did it in the case of the arithmetical hierarchy gives the positive closure properties of the levels of the analytical hierarchy as displayed in Figure 4.2.2. 1 1 To answer the obvious question whether the Πk[A] and Σk[A] form a proper hierarchy we check

acceptable Relation–class ¬ ∨ ∧ ∃< ∀< ∃0 ∀0 ∃1 ∀1 Substitution universal with

primitive– yes yes yes yes yes no no no no no primitive–recursive recursive functions computable ∆0 yes yes yes yes yes no no no no no 1 functionals computable Σ0 no yes yes yes yes yes no yes no yes 1 functionals computable Π0 no yes yes yes yes no yes no yes yes 1 functionals 0 ∆0 yes yes yes yes yes no no no no no total Σk+1- k+1 functionals 0 Σ0 no yes yes yes yes yes no no no yes total Σk+1- k+1 functionals 0 Π0 no yes yes yes yes no yes no no yes total Σk+1- k+1 functionals 1 yes yes yes yes yes yes yes no no no total arithmetical– ∆0 functionals 1 Π1 no yes yes yes yes yes yes no yes yes total Πk+1– k+1 functionals 1 Σ1 no yes yes yes yes yes yes yes no yes total Πk+1– k+1 functionals 1 ∆1 no yes yes yes yes yes yes no no no total Πk+1– k+1 functionals analytical yes yes yes yes yes yes yes yes yes no total analytical– functionals

Figure 4.2.2: Closure Properties of Relation–Classes the universality of these classes.

38 4.2. Analytical relations

1 1 4.2.5 Theorem The classes Πk+1[A] and Σk+1[A] are acceptable universal. ∈ 1 Proof: For P Π1[A] we have P (a) ⇔ (∀α)(∃x)R(a,α,x) ∃ 0 0 such that (a,α) ( x)R(a,α,x) is Σ1[A]. By the universality of Σ1[A] we therefore get an index e such that ∃ ⇔ ∈ A,m,n+1 ( x)R(a,α,x) (a,α) We . We deﬁne

1 Π1 [A],m,n ∀ ∈ A,m,n+1 U := (e, a) ( α) (a,α) We . (4.7)

1 Π1[A],m,n Π1[A],m,n 1 ∈ 1 Then U is by construction universal for Π1[A]. We usually write a Ue instead Π1[A],m,n of (e, a) ∈ U 1 . To see that it is also acceptable we do the following computation Π1[A],m+k,n ( ,y ,...,y )∈U 1 ⇔ (∀α) ( ,α,~y)∈WA,m+k,n+1 a 1 k e h a e i ⇔ ∀ ∈ A,m,n ( α) (a,α) W m,n+1 Sk (e,~y) 1 Π1[A],m,n ⇔ a ∈ U m,n+1 . Sk (e,~y) ∈ 1 ⇔¬ ∈ 1 Since R Σ1[A] R Π1[A]we may put

1 Σ1[A],m,n ∃ ∈ A,m,n+1 U := (e, a) ( α) (a,α) / We (4.8)

1 Σ1[A] 1 and obtain by duality that U is acceptable universal for Σ1[A]. Using induction on k we can lift Theorem 4.2.5 to all levels of the analytical hierarchy. We just put h i Π1 [A],m,n Σ1[A],m,n+1 U k+1 := (e, a) (∀α) (a,α)∈U k (4.9) and h i Σ1 [A],m,n Π1[A],m,n+1 U k+1 := (e, a) (∃α) (a,α)∈U k . (4.10)

Turning this into a theorem we get:

1 1 4.2.6 Theorem The classes Πk+1[A] as well as the classes Σk+1[A] are acceptable universal Π1 [A] Σ1 [A] for all k. The universal predicates U k+1 and U k+1 are deﬁned in (4.6) through (4.10). Putting

1 1 ∆k+1[A],m,n { ∧ ∧ ∈ Σk+1[A],m,n U := (e, a) Seq(e) h lh(e)=2 ((e)0, a) U i Σ1 [A],m,n Π1 [A],m,n ∧ (∀b) ((e)0, b) ∈ U k+1 ⇔ ((e)1, b) ∈ U k+1 }

1 1 ∆k+1[A],m,n 1 we get also indices for ∆k+1[A]–relations. Note, however, that U is not a ∆k+1[A]– relation. To show that the analytical hierarchy does not collapse we need the following lemma.

A A ∈ 1 \ 1 4.2.7 Lemma For all k>0there is a relation Kk such that Kk Πk[A] Σk[A]. 1 ⊆ Proof: We use a diagonalization argument. Towards an indirect proof we assume that Πk[A] 1 Σk[A]. Deﬁne

39 4. The Analytical Hierarchy

Σ1[A],1,0 A ∈ k Kk := x x/Ux A ∈ 1 ⊆ 1 1 A then Kk Πk[A] Σk[A].Letebe a Σk[A]–index for Kk .Then Σ1[A],1,0 ∈ A ⇔ ∈ k ⇔ ∈ A e Kk e Ue e/Kk, which is absurd.

4.2.8 Theorem For all k we have • 1 $ 1 ∆k+1[A] Πk+1[A] • 1 $ 1 ∆k+1[A] Σk+1[A] • 1 ∪ 1 $ 1 Πk[A] Σk[A] ∆k+1[A]. A ∈ 1 \ 1 ¬ A ∈ 1 \ 1 Proof: By Lemma 4.2.7 we have Kk+1 Πk+1[A] ∆k+1 [A]or Kk+1 Σk+1 [A] ∆k+1[A], ∈ A ∪ ∈ A ∈ respectively. For k>0we put B := 2e e Kk 2e +1 e/Kk and get B 1 A ≤ ¬ A ≤ ∈ 1 ∈ 1 ∆k+1[A], but since Kk m B as well as Kk m B neither B Πk[A] nor B Σk[A] is possible. It remains the case of k =0. We have already seen that (ω) ((x)1) A := x (x)0 ∈ A

1 (ω) is not arithmetical. We give ∆1[A]–deﬁnition of A . First we describe the jump–hierarchy. We deﬁne

JHA(α):⇔(∀n)(∀x)[(α(x) =06 ⇒ Seq(x) ∧ lh(x)=2) ∧α(hn, xi) ≤ 1 ∧ (α(h0,xi)=0 ⇔ x∈A)

∧(α(hn+1,xi)=0 ⇔ (∃x)(∃u)[Seq(s) ∧ lh(s)=(u)0 2,0 ∧(∀j<(u)0)((s)j = α(hn, ji) ∧ T (x, x, s, (u)1))])] Then we prove JH(α) ∧ JH(β) ⇒ (∀n)(∀x)[α(hn, xi)=β(hn, xi)] by induction on n. Hence JH(α) ∧ JH(β) ⇒ α = β and we see that x α(hn +1,xi)=0 = x (∃u)T1,1(x, x, x α(hn, xi)=0 ,u) =j( x α(hn, xi)=0 ). Therefore we obtain (ω) n ∈ A ⇔ (∃α)[JHA(α) ∧ α(n)=0]

⇔ (∀α)[JHA(α) ⇒ α(n)=0]. (ω) 1 Since JHA(α) is arithmetical in A we see that A is ∆1[A]. By Theorem 3.3.1, however, (ω) ∈ 1 ∪ 1 A / Π0[A] Σ0[A].

40 5. The Theory of Countable Ordinals

In Chapter 3 we succeeded in characterizing the Arithmetical Hierarchy by iteration of the jump operator. We indicated that iterating the jump infinitely often leads outside the Arithmetical Hier- archy. We even proved that already its ω–fold iteration ∅(ω) is outside the arithmetically definable sets. We are going to study the effects of transfinite iterations of various operators. To prepare that we need an introduction to the theory of transfinite numbers, i.e. ordinals. It has become common to regard ordinals set–theoretical, i.e. as sets which are well–ordered by the membership relation ∈. In presence of the axiom of foundation any hereditarily transitive set is already an ordinal. This is probably the easiest way to introduce ordinals. However, since we can restrict ourselves to countable ordinals and don’t want to require too much pre–knowledge in Set Theory, we are going to develop the theory of countable ordinals in a more old fashioned way. This should be profitable even for somebody who already knows set–theoretical ordinals.

5.1 Ordinals as order–types

5.1.1 Definition Let N be some set. 1) Let R ⊆ N × N be a binary predicate. For binary predicates we sometimes prefer the infix notation, i.e. we write xRyinstead of (x, y) ∈ R or R(x, y).Wedefine field(R)= x (∃y)[xRy∨yRx] (5.1) and call field(R) the field of the predicate R. We put

xR=6 y :⇔ xRy∧x=6 y (5.2) and call R=6 the strict predicate associated with R. In case that R is irreﬂexive,i.e.if • (∀x∈ﬁeld(R)) [¬(xRx)],

R and R=6 are the same predicates. 2) A predicate ⊆N×Nis a partial ordering if is reflexive, anti–symmetrical and transitive,i.e.if • (∀x∈field()) [x x] • (∀x ∈ field())(∀y ∈ field()) [x y ∧ y x ⇒ x = y] and • (∀x ∈ field())(∀y ∈ field())(∀z ∈ field()) [x y ∧ y z ⇒ x z]. If field() ⊆ N we talk about a countable partial ordering. 3) A predicate ⊆N×Nis an ordering if is a partial ordering which is linear, i.e. if • (∀x ∈ field())(∀y ∈ field()) [x y ∨ y x]. For a partial–ordering we denote its associated strict predicate by ≺. Hence x ≺ y ⇔ x y ∧ x =6 y. We call ≺ a strict partial ordering. Vice versa we can associate a partial ordering x y :⇔ x ≺ y ∨ (x ∈ field(≺) ∧ x = y) (5.3)

41 5. The Theory of Countable Ordinals to every irreflexive and transitive predicate ≺⊆N×N. 4) A predicate R ⊆ N × N is well–founded if every nonempty subset of field(R) possesses a R–least element, i.e. if (∀M)[M ⊆field(R) ∧ M =6 ∅⇒(∃x∈M)(∀y)(yRx⇒y=x∨y/∈M)] . 5) A well–ordering is an ordering which is well–founded. 6) Two orderings 1 and 2 are equivalent if there is a strictly order–preserving map from field(1) onto field(2),i.e.ifwehave onto f:field(1) −→ field(2) such that

(∀x ∈ field(1))(∀y ∈ field(1)) [x ≺1 y ⇒ f(x) ≺2 f(y)] where ≺1 and ≺2 are the corresponding strict orderings as defined in (5.3). By 1≡2 we denote the equivalence of 1 and 2.

For well–founded predicates R we have the principle of transﬁnite induction along R which says

(∀x)[(∀y)(yR=6 x⇒ϕ(y)) ⇒ ϕ(x)] ⇒ (∀x)ϕ(x). (5.4) To realize (5.4) observe that its premise entails x/∈field(R) ⇒ ϕ(x). Thus assume x ¬ϕ(x) ∩ field(R) =6 ∅. Since R is well–founded we find a z ∈ x ¬ϕ(x) ∩field(R) such that (∀y)[yR=6 z ⇒ ϕ(y)]. This, however, implies ϕ(z) by the premise of (5.4). A contradiction. Another important principle is that of transfinite recursion along a well–ordering .LetGbe a total functional. Then there is a functional F satisfying the equation F (a,x)=G(a,λz≺x.F(a,z)) (5.5) where F(a,n) if n ≺ x (λz ≺ x.F(a,z))(n):= 0 otherwise. The principle of transfinite recursion is provable within a framework of Set Theory. We will, however, regard (5.5) as an axiom. But for computable G and decidable we can prove that there is a computable functional F satisfying (5.5). We use the Recursion Theorem to obtain an index e such that {e}m+1,n(a,x)'G(a,λz ≺x.{e}m+1,n(a,z)). Now we show by transfinite induction along that (∀a)(∃y)[{e}m+1,n(a,x)'y]. Putting F := {e}m+1,n we have a computable solution of (5.5). The following lemma is an immediate consequence of the definition of the equivalence of orderings.

5.1.2 Lemma The equivalence of orderings is a reﬂexive, transitive and symmetric relation.

5.1.3 Deﬁnition A countable ordinal is the equivalence class of a countable well–ordering.

We are going to denote ordinals by lower case Greek letters in the end of the alphabet, e.g. σ,τ,ξ,µ,...The order–type of a well–ordering ≺ is the ordinal which is represented by ≺.The

42 5.1. Ordinals as order–types order–type of ≺ is often denoted by otyp(≺). The class of countable ordinals is denoted by On. We want to show that there is a strict well-ordering < on On.Todeﬁne

5.1.4 Deﬁnition Let σ, τ be ordinals. We say that σ is less than τ, written as σ<τ,ifthere are well–orderings σ and τ representing σ and τ, respectively, and a z ∈ ﬁeld(τ ) such that σ ≡τz,i.e.

σ<τ :⇔ (∃σ ∈ σ)(∃τ ∈ τ)(∃z ∈ ﬁeld(τ )) [σ≡τz]. (5.6)

5.1.5 Theorem The relation σ<τdeﬁned in (5.6) is an irreﬂexive well–ordering on the ordinals.

Proof: The proof is easy but a bit lengthy. Therefore we concentrate on the more tricky parts. If 1≡2, 3≡4and 2≡3zfor some z ∈ field(3),weget1≡4f(z)if f is an order– isomorphism between 3 and 4. Hence (5.6) is well–defined. Irreflexivity and transitivity are equally easy to check. The most difficult part is to check linearity. Let 1 and 2 be two well–orderings such that 16≡2. We have to show that there is either a z ∈ field(1) such that 1z ≡2 or a z ∈ field(z2) such that 1≡2z. Putting f(x):=min z ∈ field(2) (∀y ≺ 1x)[f(y)≺2 z] we get a partial function

f: field(1) −→ p field(2) which is order–preserving by definition. By construction dom(f) and rng(f) are segments of 1 and 2, respectively. More precisely 1dom(f) and 2rng(f) are segments. But we will often use the more sloppy way of talking as above. Since 16≡2 either dom(f) or rng(f) has to ≺ ∈ ∈ ≡ be proper. In the first case we get for z := min 1 x field( 1) x/dom (f) that 1 z 2 ≡ ∈ ∈ and in the second 1 2 zfor z := min≺2 x field( 2) x/rng(f) . To see that < is well–founded on On take some M ⊆ On such that M =6 ∅. Assume that M does not possess a <–least element. Pick any σ ∈ M and let be a well–ordering representing σ. Then there is a τ ∈ M such that τ<σ. Therefore we find a z0 ∈ field() such that z0 represents τ. Assuming we already defined the sequence

z0 z1 ... zn such that τi := otyp(zi) ∈ M for i =0,...,nwe find some τn+1 <τn in M and therefore also some zn+1 ≺ zn such that τn+1 = otyp(zn+1). This gives an infinite strictly descending sequence z0 ... zn ...in field() which is impossible because zi i ∈ N ⊆ field() would not have a –least element. We just used the fact that in a well–ordering there are no infinite strictly descending sequences. This is in fact equivalent to being a well–ordering.

43 5. The Theory of Countable Ordinals

5.1.6 Theorem A binary predicate R is well–founded iff there are no inﬁnite R=6 –descending sequences.

Proof: We have just seen that a well–founded predicate does not allow infinite strictly descending sequences. For the opposite direction assume that every R–descending sequence is finite. Towards an indirect proof let M be a nonempty subset of field(R) without R–least element. Then we may choose some z0 ∈ M. Suppose that we already have chosen z0,...,zn ∈M such that

z0 z1 ... zn.

Since M has no R–least element there is an zn+1 R=6 zn and we may thus construct an inﬁnite R=6 –descending sequence. If M ⊆ On is bounded, i.e. if there is some α ∈ On such that (∀ξ ∈ M)[ξ ≤α]then we deﬁne sup M := min η ∈ On (∀ξ ∈ M)[ξ ≤η] . (5.7)

5.1.7 Theorem The class On of countable ordinals is unbounded in the countable ordinals, i.e. for every countable ordinal σ there is a countable ordinal τ such that σ<τ.

Proof: Let σ ∈ On and a well–ordering representing σ.Put x≺0 y :⇔ Seq(x) ∧ Seq(y) ∧ lh(x)=lh(y)=2

∧[(x)0 =0∧(y)0 =0∧(x)1 ≺(y)1)

∨((x)0 =0∧(x)1 ∈ﬁeld(≺) ∧ (y)0 =1∧(y)1 =1)], i.e. we add a single point h1, 1i on top of the well–ordering ≺.Thenweget ≡ 0h1, 1i. Hence σ = otyp() < otyp(0)=:τ. Using Theorem 5.1.7 we deﬁne the successor σ +1:=min ξ ∈ On σ<ξ . (5.8) We put 0:=min On (5.9) and get 0=otyp(∅).

5.1.8 Deﬁnition An ordinal σ is a successor–ordinal if there is an ordinal τ such that σ = τ +1. An ordinal σ is a limit–ordinal if σ =06 and σ is not a successor ordinal. We denote the class of limit ordinals by Lim.

We obtain λ ∈ Lim ⇔ λ =06 ∧(∀ξ<λ)[ξ+1<λ] (5.10) because ξ<λimplies ξ +1 ≤ λand ξ +1 = λis excluded by the deﬁnition of Lim.An equivalent formulation of (5.10) is λ ∈ Lim ⇔ λ =06 ∧(∀ξ<λ)(∃η<λ)[ξ<η]. (5.11)

44 5.1. Ordinals as order–types

Equation (5.11) follows immediately from (5.10) with ξ +1as witness for η and the opposite direction follows because ξ<η<λimplies ξ +1≤η<λand < is transitive on the countable ordinals. We use (5.11) to prove ω := otyp(<) ∈ Lim (5.12) where < stands for the standard ordering of the natural numbers. It is obvious that ω =06 and for σ<ωwe obtain an n ∈ N such that σ = otyp(<n) < otyp(<n +1)<ω. Hence ω ∈ Lim by (5.11) Ordinals <ωare ﬁnite. Finite ordinals are represented by <n for n ∈ N, i.e. by orderings of the form 0 < 1 <...

otyp(x):=otyp(x). (5.13) Then we obtain otyp(x)=sup otyp(y)+1 y≺x . (5.14) To prove (5.13) we observe that σ := sup otyp(y)+1 y≺x ≤ otyp(x). If we assume σ

σ = otyp(y0) < otyp(y0)+1≤σ contradicting that < is irreflexive on On. In a similar way we show otyp()=sup otyp(y)+1 y∈field() . (5.15) Putting σ := sup otyp(y)+1 y∈field() we obviously have σ ≤ otyp(). The assump- tion σ

σ = otyp(y) < otyp(y)+1≤σ. Generalizing (5.14) and (5.15) we define otypR(x):=sup otypR(y)+1 yR=6 x (5.16) and otyp(R):=sup otypR(y)+1 y∈field(R) (5.17) for arbitrary well-founded orderings by transfinite recursion along R. We close this section by examining the complexity of the notions of partial–ordering, ordering and well–ordering in the Analytical Hierarchy. We express a binary predicate R by the characteristic function of its contractions hRi := hx, yi (x, y) ∈ R . We define CF(α):⇔(∀x)[α(x) ≤1 ∧(α(x)=0⇒Seq(x) ∧ lh(x)=2)] PO(α):⇔CF(α) ∧ (∀x)(∀y)[α(hx, yi)=0⇒α(hx, xi)=0∧α(hy,yi)=0] ∧(∀x)(∀y)[α(hx, yi)=0∧α(hy,xi)=0⇒x=y] ∧(∀x)(∀y)(∀z)[α(hx, yi)=0∧α(hy,zi)=0⇒α(hx, zi)=0],

45 5. The Theory of Countable Ordinals

LO(α):⇔PO(α) ∧ (∀x)(∀y)[α(hx, xi)=0∧α(hy,yi)=0 ⇒ α(hx, yi)=0∨α(hy,xi)=0]

WF(α):⇔(∀β∗)[(∀x)(β∗(x)=0⇒α(hx, xi)=0)∧(∃x)[β∗(x)=0] ⇒(∃z)(β∗(z)=0∧(∀y)[α(hy,zi)=0⇒y=z∨β∗(y)=1])] and ﬁnally WO(α):⇔LO(α) ∧ WF(α). Then PO(α) expresses that α is the characteristic function of the contraction of a partial–ordering, LO(α) that α is the characteristic function of the contraction of an ordering, WF(α) expresses that α is the characteristic function of the contraction of a well-founded binary predicate and WO(α) denotes that α is the characteristic function of the contraction of a well–ordering. We moreover have

PO LO 0 WF WO 5.1.9 Theorem The relations (α), (α) are Π1 and the relations (α) and (α) are 1 Π1.

5.2 Trees

An extremely important tool in the investigation of hyperarithmetical set are trees. We are going to introduce trees as sets (of codes of) ﬁnite sequences which are closed under initial segments.

5.2.1 Deﬁnition Let s, t ∈ N. We put

s ⊆ t :⇔ Seq(s) ∧ Seq(t) ∧ lh(s) ≤ lh(t) ∧ (∀i

For any tree B we have hi ∈ B by (5.19). We call hi the root of the tree B. Trees should be visualized as shown in Figure 5.2.1. Notice that writing T(B) as an analytical formula, i.e. T(α):⇔(∀x)[α(x) ≤1 ∧(α(x)=0→Seq(x))] ∧ α(hi)=0 (5.19) ∧(∀x)(∀y)[α(x)=0∧y⊆x→α(y)=0], 0 showsthatitisa(0, 1)–ary Π1–predicate. If s_hxi∈Bthen we also have s ∈ B. We call s_hxi an immediate B–predecessor of s and s the immediate B–successor of s_hxi. A path in atreeBis a subset P ⊆ B which is a linearly ordered by ⊆ and closed under immediate successors. A path through atreeBis a path in B which also satisﬁes s ∈ P ∧ (∃x)[s_hxi∈B] ⇒ (∃x)[s_hxi∈P]. Atreeiswell–founded if it does not contain inﬁnite paths, i.e. if T(B) ∧ (∀β)(∃z) β(z) ∈/ B . Expressing that by an analytical formula we put WT(α):⇔T(α)∧(∀β)(∃z) α(β(z)) = 1 . (5.20) From (5.19) and (5.20) we have the following lemma.

46 5.2. Trees

h0, 0, 0, 0, 0, 1i h0, 0, 0, 0, 1, 3i

h0, 0, 0, 0, 1i h i h0, 0, 0, 0, 0i 0, 0, 0, 1, 2

h0, 0, 0, 1i h0, 0, 0, 0i

h0, 0, 0i

h0, 0i h0,1ih0,2ih0,3ih0,4i h0,5i

h0i

Figure 5.2.1: Visualization of a tree

T WT 0 1 5.2.2 Lemma The (0, 1)–ary relations and are Π1 and Π1, respectively.

5.2.3 Theorem (Bar induction) For well–founded trees we have the principle of bar induction, i.e. if B is a well-founded tree we have (BI) (∀s)[(∀x)(s_hxi∈B⇒ϕ(s_hxi)) ⇒ ϕ(s)] ⇒ ϕ(hi).

Proof: We prove (∀s)[(∀x)(s_hxi∈B⇒ϕ(s_hxi)) ⇒ ϕ(s)] ⇒ (∀s ∈ B)ϕ(s). (5.21) Towards an indirect proof assume (∀s)[(∀x)(s_hxi∈B⇒ϕ(s_hxi)) ⇒ ϕ(s)] (i) and s ∈ B ∧¬ϕ(s) for some s. We are going to construct an inﬁnite path s0,s1,...in B.Put

s0 := s and assume that s0,...,sn are already deﬁned such that

si ∈ B ∧¬ϕ(si)∧ “siimmediately succeeds si+1” holds for i =0,...,nor i =0,...,n−1, respectively. But then there is an x such that _h i∈ ∧¬ _h i sn x B ϕ(sn x) because otherwise we get ϕ(sn) by (i). Putting _h i sn+1 := sn x we obtain an inﬁnite path s0,s1,...in B which contradicts the well-foundedness of B.

47 5. The Theory of Countable Ordinals

There should be a connection between bar induction and transﬁnite induction along well–founded predicates. To make this explicit we introduce the predicate ≤∗ ⇔ ∈ ∧ ∈ ∧ ⊆ s B t : s B t B t s. (5.22) ≤∗ ∗ ≤∗ We denote the strict version of B by

≤∗ 5.2.4 Theorem AtreeBis well–founded iff the predicate B is well–founded. According to Theorem 5.2.4 we may regard bar induction as a special case of transfinite induction. For a tree B and a node s ∈ B we may regard the subtree of B above s which is defined by Bs := t ∈ Seq s_t ∈ B . (5.23) Then we have T(B) ∧ s ∈ B ⇒ T(Bs) and obviously also WT(B) ∧ s ∈ B ⇒ WT(Bs). We call a tree B finitely branching if every node in B has only finitely many predecessors, i.e. if _ (∀s ∈ B) | x s hxi∈B |<ℵ0 where |M| denotes the cardinality of a set M and ℵ0 the first infinite cardinal. An important property of finitely branching trees is KONIG¨ ’s lemma.

5.2.5 Lemma (KONIG¨ ’s Lemma) Any tree which is finitely branching but infinite possesses an infinite path.

Proof: We assume that B is finitely branching but infinite. We construct an infinite path s0,s1,... in B.Put

s0 =hi and assume that s0,...,sn are deﬁned such that

si ∈ B ∧|Bsi|≥ℵ0 holds for i =0,...,n.Since [ ℵ ≤| | {hi} ∪ _h i _h i∈ 0 B sn = B sn x sn x B x∈N and _h i∈ ℵ x sn x B < 0 there is an x such that _h i∈ ∧| _h i| ≥ ℵ sn x B B sn x 0. Deﬁning _h i sn+1 := sn x we obtain an inﬁnite path.

48 5.2. Trees

We call a tree boundedly branching if there is a k ∈ N such that (∀s ∈ B)(∀x)[s_hxi∈B⇒x≤k]. (5.24) We call k a branching bound.IfBis boundedly branching with branching bound k we obviously have

(∀s ∈ B)(∀i

Combining (5.25) with KONIG¨ ’s Lemma we get

5.2.6 Theorem (Finiteness Theorem) Let B be a boundedly branching tree. Then (∀β)(∃z) β(z) ∈/ B ⇔ (∃n)(∀s)[Seq(s) ∧ lh(s)=n⇒s/∈B]. (5.26)

The importance of the Finiteness Theorem is that it shows that for boundedly branching trees the 1 Π1–property of being well-founded can be expressed arithmetically. A binary tree is a boundedly branching tree with branching bound 1. The Finiteness Theorem for binary trees is also known as Weak KONIG¨ ’s Lemma. We also want to establish a connection between well–founded tress and ordinals. The key here is Theorem 5.2.4 and the deﬁnitions in (5.16) and (5.17), respectively.

5.2.7 Deﬁnition Let B be a well-founded tree. For s ∈ B we deﬁne

otyp (s):=otyp≤∗ (s) B B and

otyp(B)=otypB(hi). By (5.16) we have ∗ otypB (s)=sup otypB(t)+1 t

5.2.8 Deﬁnition (KLEENE–BROUWER–ordering) For a sequence number s and x

sx := h(s)0,...,(s)x−·1i. Let B be a tree. For s, t in B we deﬁne KB ⇔ ( ∨ ∃ ∧ s

49 5. The Theory of Countable Ordinals

KB The predicate

A visualization of the KLEENE–BROUWER–ordering is given in Figure 5.2.2.

s1 s2

s4 s6

Figure 5.2.2: Visualization of the KLEENE–BROUWER–ordering The nodes s1,...,s6 are in increasing order

≤KB KB 5.2.9 Lemma For any tree B the predicate B is an ordering on B,

≤KB Proof: It suffices to show that B is irreflexive, transitive and linear. Irreflexivity follows by definition. Transitivity is easy but a bit cumbersome because of the many cases one has to consider. A proof is sketched in Figure 5.2.3. To check linearity notice that for any s =6 t ∈ B there ⊆ ≤KB is a maximal x such that s x = t x.Iftx=sthen s t, hence t B s and if s x = t then ⊆ ≤KB KB t s, hence s B t. Otherwise we either have (s)x < (t)x and obtain s (t)x KB and obtain t

≤KB 5.2.10 Theorem AtreeBis well–founded iff its KLEENE–BROUWER–ordering B is well– founded.

Proof: We start with the easy direction. Assume that B is not well-founded. Then there exists an KB KB infinite path s0,s1,...in B. According to Definition 5.2.8 this implies s0 >B s1 >B ...and KB we obtain an infinite B s1 >B ...>B si >B si+1 ... KB ∈ N be an infinite

v v u u u w v w w

u w u w u v v w v

KB KB ⇒ KB Figure 5.2.3: How to prove u

B0 := t Seq(t) ∧ (∃s ∈ S)[t⊆s] . Then B0 ⊆ B is obviously an inﬁnite tree. We claim that B0 is ﬁnitely branching. Chose any t ∈ B0 and regard _ 0 Mt := x t hxi∈B . ∈ x ∈ _h i⊆ x x KB y For any x Mtthere is an s S such that t x s and for x, y in Mt we get s

5.3 Recursive Ordinals

This lecture is only concerned with countable ordinals. However, we don’t want to hide that there are also bigger – uncountable – ordinals. Usually one puts ω1 := sup σ σ is a countable ordinal . We have seen in Theorem 5.1.7 that the countable ordinals are unbounded in the countable ordinals. Therefore ω1 can’t be a countable ordinal itself. In this section we will introduce an even smaller class of ordinals.

5.3.1 Deﬁnition An ordinal is called recursive (in A) if it is represented by some (in A) decidable countable well-ordering. We deﬁne CK ∈ ω1 := sup σ On σ is recursive and CK ∈ ω1 [A]:=sup σ On σ is recursive in A .

51 5. The Theory of Countable Ordinals

CK ≤ CK ≤ ⊆ N It is obvious that we have ω1 ω1 and ω1 [A] ω1 for any set A . Observe that the (in A) recursive ordinals form a segment of the ordinals. To see that let σ be a (in A) recursive ordinal and a (in A) decidable well–ordering representing σ.Forτ<σthere is a z ∈ ﬁeld() such that z represents τ.Sincezis again decidable (in A) the ordinal τ is recursive (in A), too. Notice that we did not claim that the relation τ<σis decidable.

CK 5.3.2 Lemma The ordinal ω1 is a limit ordinal which is not recursive. Proof: Let σ be a recursive ordinal and a decidable well-ordering representing σ. We construct a well–ordering 0 as in the proof of Theorem 5.1.7. Obviously 0 is again decidable. Therefore CK ω1 cannot be recursive and also not a successor ordinal. As a consequence of Lemma 5.3.2 and the fact that the (in A) recursive ordinals form a segment of the countable ordinals we get CK ∈ ω1 = min ξ On ξ is not recursive (5.30) and CK ∈ ω1 [A]=min ξ On ξ is not recursive in A (5.31) which entails

CK 5.3.3 Lemma An ordinal σ is recursive iff σ<ω1 . An ordinal σ is recursive in A iff σ< CK ω1 [A].

CK The ordinal ω1 is therefore the least ordinal which cannot be represented by a decidable well– CK ordering. In that sense ω1 is the “effective” counterpart of the ordinal ω1 which is the least ordinal which cannot be represented by a countable well–ordering. We are going to introduce the light face versions of the relations CF, PO, LO, WF, T and WT. We put CF (e):⇔“eis index of a characteristic function”

PO(e):⇔“eis index of a partial ordering”

POA(e):⇔“eis A–index of a partial ordering”

LO(e):⇔“eis index of an ordering”

LOA(e):⇔“eis A–index of an ordering”

WF (e):⇔“eis index of a well–founded binary predicate”

WF A(e):⇔“eis A–index of a well–founded binary predicate”

WO(e):⇔“eis index of a well–ordering”

WO A(e):⇔“eis A–index of a well–ordering”

Tree (e):⇔“eis index of a tree”

Tree A (e):⇔“eis A–index of a tree”

52 5.3. Recursive Ordinals

WT (e):⇔“eis index of a well–founded tree”

WT A(e):⇔“eis A–index of a well–founded tree” All these predicates are arithmetical or analytical. To check their complexity recall that {e}n,0(~x) ' y ⇔ (∃u) Tn,0(e, ~x, u) ∧ U(u)=y . { }n,0 ' ∈ 0 Hence (e, ~x, y) e (~x) y Σ1. Therefore we get CF(e) ⇔ (∀x)(∀y)(∃z) {e}2,0(x, y) ' z ∧ z ≤ 1 and ⇔ PO(e) CF(e) ∧ (∀x)(∀y) {e}2,0(x, y)=0 →{e}2,0(x, x)=0∧{e}2,0(y,y)=0 ∧(∀x)(∀y) {e}2,0(x, y)=0∧{e}2,0(y,x)=0 → x=y ∧(∀x)(∀y)(∀z) {e}2,0(x, y)=0∧{e}2,0(y,z)=0 →{e}2,0(x, z)=0 . as well as LO(e) ⇔ PO(e) ∧ (∀x)(∀y)[({e}2,0(x, x)=0∧{e}2,0(y,y)=0) → ({e}2,0(x, y)=0∨{e}2,0(y,x)=0)]. similarly we get Tree (e) ⇔ (∀x)(∃z) {e}1,0(x) ' z ∧ z ≤ 1 ∧{ }1,0 hi e ( )=0 ∧(∀x) {e}1,0(x)=0→Seq(x) ∧ (∀x)(∀y) {e}1,0(x)=0∧y⊆x→{e}1,0(y)=0 . These predicates are arithmetical. As examples for analyitcal predicates we take ⇔ WF (e) CF(e) ∧ (∀α){(∃x)(α(x)=0)∧(∀x) α(x)=0→{e}2,0(x, x)=0 → (∃z) α(z)=0∧(∀u)({e}2,0(u, z)=0→u=z∨α(u)=1) }. Therefore we have WO(e) ⇔ LO(e) ∧ WF (e) and WT (e) ⇔ Tree (e) ∧ (∀α)(∃z) {e}1,0(α(z)) = 1 . Summing up we get the following theorem.

0 5.3.4 Theorem The predicates PO(e), LO(e) and Tree(e) are all Π2. The predicates WF(e), 1 WO(e) and WT(e) are Π1.

5.3.5 Deﬁnition If WO(e) we put otypWO(e):=otyp( (x, y) {e}2,0(x, y)=0 ). For WO(e)A let

A otypWO (e):=otyp( (x, y) {e}A,2,0(x, y)=0 ).

53 5. The Theory of Countable Ordinals

For WT (e) we put otypTree(e):=otyp( x {e}1,0(x)=0 ). And for WT A(e) we let

A otypTree (e):=otyp( x {e}A,1,0(x)=0 ).

5.4 KLEENE’s Ordinal Notations

Before we look closer at the connections between recursive ordinals and the ordinals which are given by well–founded trees we introduce another form of ordinals via effective abstract notations. This approach is due to S. C. KLEENE. The idea is to introduce simultaneously a set O of ordinal notations together with an evaluation function |·|O: O−→On and an order relation

5.4.1 Deﬁnition We deﬁne the set O of ordinal notations,theO–evaluation ||O and the order– predicate

1) 1 ∈O,|1|O := 0 and 1 ≤O a for all a ∈O.

a a a 2) If a ∈Othen 2 ∈O,|2 |O := |a|O +1and c

An ordinal σ is KLEENE–recursive iff there is an a ∈Osuch that σ = |a|O.

As a ﬁrst consequence of Deﬁnition 5.4.1 we obtain

5.4.2 Lemma The predicate

Proof: We show

5.4.3 Corollary The predicate

Proof: Any inﬁnite

54 5.4. KLEENE’s Ordinal Notations

5.4.4 Theorem The KLEENE–recursive ordinals form a segment of the countable ordinals, i.e. if a ∈Oand σ<|a|O then there is a b ∈Osuch that σ = |b|O.

a Proof: We induct on |a|O.Fora=1we have nothing to show. For a =2 0 and σ<|a|Owe get σ ≤|a0|O. Therefore we either have σ = |a0|O or obtain a b ∈Osuch that σ = |b|O by the induction hypothesis. e 1,0 For a =3·5 and σ<|a|O we get σ<|{e} (n)| for some n ∈ N. Then there is a b ∈Osuch that σ = |b|O by induction hypothesis.

5.4.5 Lemma There is a binary computable function +O such that for all a, b, c ∈ N the following hold

1) (a ∈O∧b∈O) ⇔ a+O b∈O

2) (a ∈O∧b∈O) ⇒|a+Ob|O=|a|O+|b|O

3) (a ∈O∧b∈O∧b=1)6 ⇒ a

4) (a ∈O∧c

5) (a ∈O∧b=c∈O) ⇔ a+Ob=a+O c Proof: Let h be a recursive function such that for all e, a, d, n ∈ N {h(e, a, d)}(n) '{e}(a, {d}(n)) holds. By using different indices for the same function we are able to make h one–one. Deﬁne  a b =1  if 2{e}(a,y) if b =2y =16 g(e, a, b)= · h(e,a,y) · y 3 5 if b =3 5 7 otherwise, and use the Recursion Theorem to obtain an index e such that {e}(a, b) ' g(e, a, b). (i) Putting

a +O b := {e}(a, b) we get a partial–computable function for which one easily sees by induction that a+O b is deﬁned for all a, b ∈O. Surprisingly, the new function is total. Suppose a +O b is not deﬁned. Then, as h is total, we have b =2y =16 for some y

We postpone the study of the complexities of the set O and the predicate

5.4.6 Lemma There is a computable function f such that for alle e ∈ N WT(e) ⇔ WO(f (e)) and WT(e) ⇒ WO(f (e)) ∧ otypTree(e) ≤ otypWO(f (e)).

55 5. The Theory of Countable Ordinals

The ordertype of a decidable tree is therefore a recursive ordinal.

≤KB Proof: If B is any decidable tree then the associated KLEENE–BROUWER–ordering B is also ≤KB decidable. Moreover, an index for B is effectively computable from an index of B.Since ∗ KB s< tentails s< t we get by induction on otypB(t) B B ∗ KB otypB (t)=sup otypB(s)+1 s< t ≤sup otyp≤KB (s)+1 s< t = otyp≤KB (t). B B B B Therefore we obtain together with Lemma 5.3.3 that the ordertypes of decidable trees are recursive ordinals. Unfortunately it is not sufﬁcient to take f as the function that takes each e ∈ N to an index of the KLEENE–BROUWER–ordering induced by {e}:Ifeis the characteristic function of a ﬁnite set of KB sequences that is not closed under initial segments then

5.4.7 Lemma (Recursion Lemma) Let R be an (m +2,n)-ary relation and ≺ be an irreflexive well–founded predicate. For any (m +2,n)–ary in A partial–computable functional H such that (∀a)(∀e)(∀x ∈ field(≺)) (∀y ≺ x)R(a,y,{e}A,m+1,n(a,y)) ⇒ R(a,x,H(a,x,e)) (5.32) there is an in A partial–computable functional F such that (∀a)(∀x ∈ field(≺)) [R(a,x,F(a,x))] . (5.33) If H is total, then so is F .

Proof: We use the Recursion Theorem to obtain an A–index f with {f}A(a,x)'H(a,x,f). (i) We show (∀a)(∀x ∈ field(≺))R(a,x,{f}A(a,x)) by transfinite induction along ≺.Wehave (∀a)(∀y ≺ x)R(a,y,{f}A(a,y)) (ii) by the induction hypothesis. Then by (i) and (5.32) we obtain from (ii) R(a,x,H(a,x,f)) which is R(a,x,{f}A(a,x)). Putting F := {f}A finishes this proof. The Recursion Lemma is the main tool in the proof of the following theorem which establishes the connections between recursive ordinals, order–types of decidable trees and KLEENE–recursive ordinals.

56 5.4. KLEENE’s Ordinal Notations

5.4.8 Theorem There are computable functions f and g such that a ∈ WT ⇒ f(a) ∈ WO ∧ otypTree(a) ≤ otypWO(f (a)) (5.34) and Tree a ∈O ⇒ g(a)∈WT ∧|a|O =otyp (g(a)). (5.35) For a ∈ WO let be the induced well–ordering. There exists a partial–computable function h with

(∀x ∈ ﬁeld()) [h(x) ∈O∧otyp(x) ≤|h(x)|O]. (5.36) Additionally we get WO a ∈ WO ⇒ (∃b ∈O) otyp (a) ≤|b|O . (5.37)

Proof: Equation (5.34) is Lemma 5.4.6. To show (5.35) we use the Recursion Lemma along the well–founded predicate

H(e, x) ∈O∧otyp(x) ≤|H(e, x)|O. (iii)

Here, however, we encounter the difficulty that we cannot in general decide whether otyp(x) ∈ Lim. As a remedy we use a trick. We introduce a new well–ordering 0 which is the reflexive hull of the predicate defined by a ≺0 b :⇔ Seq(a) ∧ Seq(b) ∧ lh(a)=lh(b)=2

∧ [(a)0 ≺ (b)0 ∨ ((a)0 =(b)0 ∧(a)0 (a)0 ∧(a)1 <(b)1)] . The Ordering 0 is again decidable and a well-ordering such that otyp() ≤ otyp(0). (It is otyp(0)=ω·otyp() for those who know ordinal arithmetic.) The ordering 0 has the advantage that we can decide whether x ∈ ﬁeld(0) is a limit point. We have

otyp0 (x) ∈ Lim ⇔ (x)0 =06 ∧(x)1 =0 where we assume without loss of generality that 0 is the –least element. Moreover we can also compute a fundamental sequence for otyp0 (hx, 0i). We put F (x, 0) := h0, 0i (iv)

57 5. The Theory of Countable Ordinals and hn, 0i if (F (x, n)) ≺ n ≺ x F (x, n +1):= 0 h(F (x, n))0, (F (x, n))1 +1i otherwise. Then F is a computable function. We have (∀n)[F(x, n) ≺0 F (x, n +1)] (v) and prove x =06 ⇒ (∀n)[F(x, n) ≺0 hx, 0i] . by induction on n.Forn=0this follows from x =06 and (iv). From the induction hypoth- 0 0 esis F (x, n) ≺ hx, 0i we get (F (x, n))0 ≺ x and obtain F (x, n +1)= hn, 0i≺ hx, 0i if 0 (F (x, n))0 ≺ n ≺ x or F (x, n +1)=h(F(x, n))0, (F (x, n))1 +1i≺ hx, 0i otherwise. Hence sup otyp0 (F (x, n)) n ∈ N ≤ otyp0 (hx, 0i). (vi) To obtain equalitity in (vi) we assume hy,ni≺0 hx, 0i and show that there is a k ∈ N such that 0 0 hy,ni≺ F(x, k).Fromhy,ni≺ hx, 0i we get y ≺ x.Ify(F(x, y))0 then

F (x, y +1)=h(F(x, y))0, (F (x, y))1 +1i 0 and we ﬁnd a k ∈ N such that hy,ni≺ F(x, k).If(F(x, y))0 ≺ y then F (x, y +1)=hy,0i and again we ﬁnd a k ∈ N such that hy,ni≺0 F(x, k). Hence sup otyp0 (F (x, n)) n ∈ N = otyp0 (hx, 0i).

Together with (v) this shows that (otyp0 (F (x, n)))n∈N is a fundamental sequence for otyp0 (hx, 0i). We use the Recursion Lemma to obtain (5.36) for 0 instead of and assume the recursion hypothesis (ii) for 0 instead of .Wedeﬁne  1 if x = h0, 0i {e}1,0(hu,vi) h i H(e, x):=2 if x = u, v +1 3·5z if x = hu, 0i and u =06 where z is such that {z}1,0(0) = 1 and

1,0 1,0 1,0 1 {z} (n +1)={z} (n)+O {e} (F(u, n)) +O 2 hold. Then, according to Lemma 5.4.5, H(e, x) satisfies (iii) with replaced by 0 and we have (5.36). If a ∈ WO we find a decidable well-ordering 0 and a z ∈ field(0) such that otypWO(a)= 0 otyp0 (z). Without loss of generality we may assume that is an ordering of the kind we just have considered. Hence

WO otyp (a)=otyp0 (z) ≤|h(z)|O by (5.36) which ﬁnishes the proof.

It follows from Theorem 5.4.8 that the different approaches to obtain representations for “effective” ordinals all lead to the same class. This proves the following theorem.

5.4.9 Theorem We have

58 5.4. KLEENE’s Ordinal Notations

 ωCK = sup σ ∈ On (∃a ∈ WO) σ = otypWO(a) 1 = σ ∈ On (∃a ∈ WT) σ = otypTree(a) sup = sup σ ∈ On (∃a ∈O)[σ =|a|O] .

59 5. The Theory of Countable Ordinals

60 6. Generalized Inductive Deﬁnitions

In the previous chapter we left the question for the complexity of KLEENE’s O unanswered. This chapter will show that the principles used in the deﬁnition of O can be systematically studied. This will lead to the fundamental notion of generalized inductive deﬁnitions.

6.1 Clauses and operators

Inductive definitions are ubiquitous in Mathematics and especially in Mathematical Logic. Usu- ally we use clauses in inductive definitions. The simplest example of an inductive definition is that of the set of natural numbers. We might say that the natural numbers are inductively defined by the following clauses: • 0 is a natural number • If 0 is a natural number, its successor S(n) is also a natural number. We develop an abstract notion for clauses. Let N be a nonempty set.

6.1.1 Definition A clause over N has the form (C) R → r where R ⊆ N n and r ∈ N n. We call R the set of premises and r the conclusion of the clause (C). AsetS⊆Nn satisfies clause (C) iff R ⊆ S implies r ∈ S. A system of clauses is a set Φ= Ri →ri i∈I of clauses Ri → ri. n AsetS⊆N is closed under Φ if S satisfies Ri → ri for all i ∈ I. The least subset of N n which is closed under a system Φ of clauses is the set which is inductively defined by Φ.

Examples for systems of clauses are: •∅→0 •{n}→n+1 which defines the natural numbers inductively on N. • ∅→s s∈S Xn • {x1,...,xn}→ αixi n ∈ N,α1,...,αn ∈K i=1 which defines the subspace of a vector space V over K spanned by some S ⊆ V . More examples are easy to find. The important feature of an inductively defined set S ⊆ N n is that we have a “principle of induction on the definition” of S, which is: n “If a set S ⊆ N is inductively defined by some system Φ= Ri →ri i∈I of clauses and ϕ is some ‘property’ which is preserved by all clauses in Φ,i.e.if

(∀x∈Ri)ϕ(x)⇒ϕ(ri)for all i ∈ I,

61 6. Generalized Inductive Definitions then ϕ(s) holds for all s ∈ S.” This principle is obvious from the definition of the set inductively defined by Φ as the least set which is closed under ϕ. Properly ϕ being preserved by all clauses in Φ means that x ϕ(x) is closed under Φ.SinceSis the least Φ–closed set, we have (∀s ∈ S)ϕ(s). Observe that the principle of induction on the definition of the natural numbers is exactly the familiar principle of Mathematical Induction. Most induction principles are instances of the principle of induction on some inductive definitions. We are going to study this on the example of transfinite induction along a well–founded predicate. Let ≺⊆N×Nbe a binary predicate. We introduce the system of clauses (A) y y ≺ x → x x ∈ N and call the set Acc(≺) ⊆ N which is inductively defined by (A) the accessible part of ≺.The principle of induction on the inductive definition of Acc(≺) takes the form (∀x)[(∀y)(y ≺ x → ϕ(y)) → ϕ(x)] ⇒ (∀x ∈ Acc(≺))ϕ(x). (6.1) If we assume that ≺ is well–founded we get Acc(≺)=N. (6.2) Acc(≺) ⊆ N holds by definition. Let x ∈ N.Ifx/∈field(≺) we have trivially (∀y)(y ≺ x → y ∈ Acc(≺)). This, however, implies x ∈ Acc(≺) by (A). If we assume that there is an x ∈ field(≺) which does not belong to Acc(≺) then we get a least such x by the well– foundedness of ≺.Buttheny∈Acc(≺) for all y ≺ x which again entails x ∈ Acc(≺) by (A). Hence field(≺) ⊆ Acc(≺). By (6.1) and (6.2) we obtain (∀x)[(∀y)(y ≺ x → ϕ(y)) → ϕ(x)] ⇒ (∀x)ϕ(x) and also (∀x ∈ field(≺)) [(∀y)(y ≺ x → ϕ(y)) → ϕ(x)] ⇒ (∀x ∈ field(≺))ϕ(x) which is the principle of transfinite induction. Towards a theory of inductively defined sets we generalize the notion of an inductive definition. A system of clauses C = Ri → ri i ∈ I on an infinite set N induces an operator n n ΓC: Pow(N ) −→ Pow(N ) which is defined by n ΓC(S)= r∈N (∃R)[R⊆S ∧R→r ∈C] .

If S ⊆ T we obviously have ΓC(S) ⊆ ΓC(T ). An operator Γ: Pow(N n ) −→ Pow(N n) having the property S ⊆ T → Γ(S) ⊆ Γ(T ) is called monotone. Generalizing the situation of systems of clauses we introduce the following deﬁnition.

6.1.2 Definition Let N be an infinite set. A monotone operator Γ: Pow(N n ) −→ Pow(N n) is a generalized inductive definition on N AsetS⊆Nn is Γ–closed if Γ(S) ⊆ S.AsetS⊆Nnis a fixed–point of Γ if

62 6.2. The stages of an inductive deﬁnition

Γ(S)=S.

We denote the – with respect to set inclusion – least fixed–point of an operator Γ by IΓ. We call IΓ the fixed–point of Γ. AsetS⊆Nn is inductively definable if there is an inductive definition Γ and a tuple ~k ∈ N m such that n S = ~x ∈ N (~x,~k) ∈ IΓ .

6.1.3 Lemma Let Γ be a generalized inductive deﬁnition on N. The ﬁxed–point of Γ is the least Γ–closed set, i.e. \ n IΓ = S ⊆ N Γ(S) ⊆ S .

Proof: Put D := S ⊆ N n Γ(S) ⊆ S and \ D = D.

For any S ∈Dwe have D ⊆ S and therefore also Γ(D) ⊆ Γ(S) ⊆ S by the monotonicity of Γ. Hence \ Γ(D) ⊆ D = D. (i)

From (i) we get again by the monotonicity of Γ Γ(Γ(D)) ⊆ Γ(D) (ii) which proves Γ(D) ∈D. Hence D ⊆ Γ(D). (iii) Thus D is a fixed–point by (ii) and (iii). Since D ⊆ F for any fixed–point F holds by definition of D, it is the least fixed–point.

6.2 The stages of an inductive deﬁnition

Describing inductively defined sets by fixed–points of monotone operators means to define them explicitly. This does not really meet the meaning we associate with the phrase “inductive”. An inductive definition should come step by step. Given a generalized inductive definition Γ: Pow(N n) −→ Pow(N n) we may try to construct the fixed–point stepwise by forming Γ(∅), Γ(Γ(∅)), Γ3(∅),... But in general we cannot expect to obtain the fixed–point after finitely many steps. Therefore we will have to iterate Γ transfinitely often.

6.2.1 Definition Let N be a countable infinte set and let Γ: Pow(N n ) −→ Pow(N n) be an inductive definition. We define by transfinite recursion [ σ τ IΓ := Γ( IΓ) τ<σ σ and call IΓ the σ–th stage of the fixed–point IΓ. 63 6. Generalized Inductive Definitions

The countability of N is needed since we have only introduced countable ordinals. It follows easily from Definition 6.2.1 that for finite ordinals n<ωwe have n 1+n ∅ IΓ =Γ ( ). To simplify notations we put [ <σ ξ IΓ := IΓ. (6.3) ξ<σ ⇒ <σ ⊆ <τ Then σ<τ IΓ IΓ and by the monotonicity of the operator Γ we obtain ⇒ σ <σ ⊆ <τ τ σ<τ IΓ =Γ(IΓ ) Γ(IΓ )=IΓ. (6.4) σ ⊆ n τ We have IΓ N by definition. Hence all IΓ are countable. By (6.4) it follows by a cardinality <σ σ argument that there is a countable ordinal σ<ω1, such that I = IΓ .Wedefine Γ || || <σ σ Γ := min σ IΓ = IΓ (6.5) and call ||Γ|| the closure ordinal of the inductive definition Γ.

σ 6.2.2 Theorem The ﬁxed–point IΓ of an inductive deﬁnition is the union of its stages IΓ .We have especially

||Γ|| IΓ = IΓ . Proof: First we show ξ ⊆ IΓ IΓ (i) <ξ ⊆ by induction on ξ. The induction hypothesis yields IΓ IΓ. By the monotonicity of Γ this ξ <ξ ⊆ || || <||Γ|| ||Γ|| <||Γ|| entails IΓ =Γ(IΓ ) Γ(IΓ)=IΓ. By deﬁnition of Γ we have Γ(IΓ )=IΓ = IΓ <||Γ|| which shows that IΓ is Γ–closed. Hence ⊆ <||Γ|| IΓ IΓ (ii) and the claim follows by (i) and (ii).

|| || σ ||Γ|| ≥|| || Observe that by (6.4) and the deﬁnition of Γ we have IΓ = IΓ for all σ Γ .

6.2.3 Deﬁnition Let Γ be an inductive deﬁnition on N.Forn∈Nwe put ∈ σ ∈ min σ n IΓ if n IΓ |n|Γ := ω1 otherwise and call |n|Γ the Γ–inductive norm of n.

6.2.4 Theorem Let Γ be an inductive definition. Then ||Γ|| = sup |x|Γ +1 x∈IΓ . Proof: We have |x|Γ < ||Γ|| for all x ∈ IΓ by definition. Hence σ := sup |x|Γ +1 x∈IΓ ≤ || || || || <σ $ σ ∈ ≤|| Γ . Assuming σ< Γ we get IΓ IΓ and find some x IΓ such that σ xΓ < |x|Γ+1≤σ. A contradiction. Determining the closure ordinal of an inductive definition is — as we will see — an interesting problem. In the general case, however, all we can say is that it is some countable ordinal. Yet, in special situations we may know more.

64 6.3. Arithmetically deﬁnable inductive deﬁnitions

6.2.5 Theorem Let Φ be a finite system of finite clauses, i.e. a finite set Φ such that for all R → r ∈ Φ the set R is finite. Let ΓΦ be the induced operator. Then ||ΓΦ|| ≤ ω.

Proof: Let IΦ be the fixed–point of ΓΦ.Weshow ∈ ⇒| | r IΦ rΓΦ <ω ∈ → | | by induction on the inductive definition of IΦ.Forr IΦand R r we get s ΓΦ <ωfor all s ∈ R.SinceRis finite and there are only finitely many R → r ∈ Φ we obtain | | ∃ ∈ ∧ → ∈ σ := sup s ΓΦ ( R)[s R R r Φ] <ω. | | ≤ || || ≤ Hence r ΓΦ σ +1<ω. By Theorem 6.2.4 we get ΓΦ ω.

6.3 Arithmetically deﬁnable inductive deﬁnitions

We will now concentrate on inductive deﬁnitions on the space Nm,n. To introduce deﬁnable operators we extend the language of arithmetic by n–ary predicate variables which we are going to denote by capital Roman letters in the end of the alphabet, e.g. X, Y , Z, X1, . . . We will moreover enrich the language by variables for functionals for which we are going to use F , G, F1, . . . as syntactical variables. Observe that we then obtain additional terms t(a) which may contain occurences of functional variables and new atomic formulas of the shape (~x ∈ X).

6.3.1 Definition An operator Γ: Pow(Nn) −→ Pow(Nn ) is definable if there is a formula ϕ(X, ~x) in the language of arithmetic whose only free variables are those shown such that Γ(S)= ~x∈Nn N|=ϕ[S, ~x] . We call Γ arithmetically or elementary definable if its defining formula ϕ(X, ~x) does not contain second order quantifiers, i.e. quantifiers ranging over functions. If there are additional function parameters in ϕ(X, ~x,α ~) we say that Γ is definable with parameters.

Observe that in the case that an operator is deﬁnable with parameters, say Γ= ~x N|=ϕ[~x,α ~] , we may denote the dependence on the parameters by Γ(α~), i.e. we obtain a relation

QΓ(~x,α ~):⇔~x∈Γ(α~). In this sense we will also talk about relations which are definable by operators. In order to obtain inductive definitions we need monotone operators. To ensure that definable operators are monotone we have to restrict the class of defining formulas.

6.3.2 Definition We inductively define the class of X–positive formulas by the following clauses: 1) If X does not occur in ϕ(X) then ϕ(X) is X–positive 2) The formula t ∈ X is X–positive 3) The X–positive formulas are closed under • the positive boolean operations ∨, ∧ • quantification over numbers and functions.

6.3.3 Lemma Let ϕ(X, x1,...,xm,α1,...,αn)be an X–positive formula without further free variables. The operator

65 6. Generalized Inductive Deﬁnitions

 m Γϕ(S):= (x1,...,xm)∈N N|=[S, x1,...,xm,f1,...,fn] , where f1,...,fn is a ﬁxed n–tuple of functions from N to N, is a monotone operator. Proof: Let S ⊆ T ⊆ N.Wehavetoshow

ϕ[S, x1,...,xm,f1,...,fn]⇒ϕ[T,x1,...,xm,f1,...,fn] (i) and prove (i) by induction on the definition of “ϕ(X, ~x) is an X–positive formula”. If X does not occur in ϕ(X, ~x,α ~) then both formulas in (i) are identical. If ϕ(X, ~x,α ~ ) ≡ (~t ∈ X) then (~t N ∈ S) ⇒ (~t N ∈ T ) holds by the hypothesis S ⊆ T . The remaining cases follow immediately from the induction hypothesis. A monotone operator which is definable by an X–positive formula is called positively definable. It is of course unlikely that all definable monotone operators are positively definable. However, it follows from the CRAIG–LYNDON interpolation theorem that at least those definable operators whose monotonicity is logically provable are positively definable. This is because if |=(∀x)(x ∈ X → x ∈ Y ) → (∀~y)[ϕ(X, ~y) → ϕ(Y,~y)] then there is an interpolation formula, say ψ(Y,~y),inwhichY occurs at most positively such that |=(∀x)(x ∈ X → x ∈ Y ) → (∀~y)[ϕ(X, ~y) → ψ(Y,~y)] (i) and |=(∀~y)[ψ(Y,~y) → ϕ(Y,~y)] . (ii) Choosing X = Y in (i) yields |=(∀~y)[ϕ(Y,~y) → ψ(Y,~y)] (iii) and (ii) and (iii) show that ϕ(Y,~y) is logically equivalent to a Y –positive formula. || || If Γ is an operator which is definable by some formula ϕ we write shortly Iϕ for IΓϕ , ϕ for || || | | | | Γϕ and n ϕ for n Γϕ .

6.3.4 Definition Let Γϕ be the operator which is definable by the X–positive formula ϕ(X, ~x,α ~ ) with parameters. For any choice of a tuple of functions α~ we obtain its fixed–point Iϕ(α~ ) which we denote by Iϕ(α~). This defines an (n, m)–ary relation. Observe that we may write

~x ∈ Iϕ(α~) ⇔ ϕ(Iϕ,~x,α ~) since α~ is not really an argument of Iϕ. A relation P ⊆ Nm,n is positively arithmetical inductive over N if there is an X–positive arithmetical formula ϕ(X, ~x, ~y,α ~) with no other free variables and a tuple m~ such that P = (~x,α ~) (~x,m ~ ) ∈ Iϕ(α~) .

6.3.5 Remark This is not the strongest way to obtain relations by fixed–points. Another way would be to augment the language by (m, n)–ary relation variables X, Y, ...and thendefine operators from a formula ϕ(X, a) by Γϕ(S):= a N|=ϕ[S,a] . Then one may regard fixed points of such operators. However, in this lecture we will only regard relations whose “function part” comes from the parameters in the defining formula. We usually omit the the phrase “positively arithmetical” and talk just about inductive relations or relations which are inductive with parameters. The rest of the section is devoted to the study of the closure properties of inductive relations.

66 6.3. Arithmetically deﬁnable inductive deﬁnitions

6.3.6 Lemma (Simultaneous inductive definitions) For any X, Y –positive formulas ϕ(X, Y, ~x,α ~) and ψ(X, Y, ~y,α ~) we define ξ ∈Nm <ξ <ξ Iϕ(α~):= ~x ϕ(Iϕ ,Iψ ,~x,α ~ ) and ξ ∈Nm <ξ <ξ Iψ(α~):= ~y ψ(Iϕ ,Iψ ,~y,α ~) . Then we find a Z–positive formula χ(Z, z, ~x, ~y,α ~) and tuples m,~ ~n of the adequate length such that

~x ∈ Iϕ(α~) ⇔ (0,~x,m ~ ) ∈ Iχ(α~ ) and

~y ∈ Iψ(α~) ⇔ (1,~y,~n) ∈ Iχ(α~) S S ξ ξ where Iϕ(α~):= ξ∈On Iϕ(α~) and Iψ(α~):= ξ∈On Iψ(α~ ). Proof: Choose m~ and ~n of the appropriate arity and put χ(Z, z, ~x, ~y,α ~):≡ z=0∧ϕ( ~u (0,~u,m ~ ) ∈ Z , ~v (1,~n, ~v) ∈ Z ,~x,α ~) ∨ z =1∧ψ( ~u (0,~u,m ~ ) ∈ Z , ~v (1,~n, ~v) ∈ Z ,~y,α ~) . Then χ(Z, z, ~x, ~y,α ~) is Z–positive and we show by transﬁnite induction on ξ ∈ ξ ⇔ ∈ ξ ~x Iϕ(α~) (0,~x,m ~ ) Iχ(α~ ) as well as ∈ ξ ⇔ ∈ ξ ~y Iψ(α~) (1,~n, ~y) Iχ(α~). From the induction hypothesis we get ~x ∈ Iξ (α~) ⇔ ϕ(I<ξ,I<ξ,~x,α ~) ϕ ϕ ϕ ⇔ ∈ <ξ ∈ <ξ ϕ( ~u (0,~u,m ~ ) Iχ , ~v (1,~n, ~v) Iχ ,~x,α ~) ⇔ <ξ χ(Iχ , 0,~x,m, ~ α ~) ⇔ ∈ ξ (0,~x,m ~ ) Iχ(α~ ). Completely analogously we get ∈ ξ ⇔ <ξ ~y Iψ(α~) χ(Iχ , 1,~n, ~y,α ~) ⇔ ∈ ξ (1,~n, ~y) Iχ(α~ ).

In a next step we want to show that the inductive predicates are closed under “positively inductive in”.

6.3.7 Lemma Let ϕ(X, ~x,α ~) be an X–positive arithmetical formula and let ψ(X, Y, ~y,α ~) be an X, Y –positive arithmetical formula. Put ψ˜(X, ~y,α ~):≡ψ(X, Iϕ(α~),~y,α ~). Then there is an X– positive arithmetical formula χ(X, z, ~x, ~y,α ~) without additional function parameters and a tuple m~ ∈ Nk such that ∈ ⇔ ∈ ~y Iψ˜(α~) (1,~m, ~y) Iχ(α~).

Proof: The difficulty is the fact, that ψ˜ is not longer an arithmetical formula. The idea of the proof is to construct Iϕ and Iψ˜ simultaneously instead of first finishing Iϕ and then start con- 67 6. Generalized Inductive Definitions

structing Iψ˜. To improve readability we suppress the parameters α~. We choose tuples m~ and ~n of adequate lengths and put χ(Z, z, ~x, ~y):≡ z=0∧ϕ( ~u (0,~u, ~n) ∈ Z ,~x) ∨ z =1∧ψ( ~v (1,~m, ~v) ∈ Z , ~u (0,~u, ~n) ∈ Z ,~y) . We introduce the abbreviations ξ ∈ ξ J0 := ~x (0,~x, ~n) Iχ and ξ ∈ ξ J1 := ~y (1,~m, ~y) Iχ . We ﬁrst prove

ξ ξ J0 = Iϕ (i)

<ξ <ξ by induction on ξ. From the induction hypothesis J0 = Iϕ we get ∈ ξ ⇔ ∈ ξ ~x J0 (0,~x, ~n) Iχ ⇔ <ξ χ(Iχ , 0,~x, ~n) ⇔ <ξ ϕ(J0 ,~x) ⇔ <ξ ϕ(Iϕ ,~x) ⇔ ∈ ξ ~x Iϕ. Obviously ||χ|| ≥ ||ϕ|| holds. Now we get ∈ ξ ⇔ ∈ ξ ~y J1 (1,~m, ~y) Iχ ⇔ <ξ χ(Iχ , 1,~m, ~y) (ii) ⇔ <ξ <ξ ψ(J1 ,Iϕ ,~y). <ξ It remains to show that this inductive deﬁnition which only uses the initial part Iϕ instead of Iϕ will eventually catch up with that of Iψ˜.Weﬁrstshow Jξ ⊆Iξ (iii) 1 ψ˜ <ξ ⊆ <ξ ⊆ by induction on ξ. This, however, is immediate from Iϕ Iϕ, the induction hypothesis J1 I<ξ, (ii) and the X, Y –positivity of ψ(X, Y, ~y). To obtain also the converse inclusion we show ψ˜ ~y ∈ Iξ ⇒ (1,~m, ~y) ∈ I (iv) ψ˜ χ by induction on ξ. Using the induction hypothesis and (i) we see ~y ∈ Iξ ⇔ ψ(I<ξ,I ,~y) ψ˜ ψ˜ ϕ <ξ ⇔ ψ(I ,J0,~y) ψ˜ ⇒ ψ( ~v (1,~m, ~v) ∈ Iχ , ~u (0,~u, ~n) ∈ Iχ ,~y)

⇔ χ(Iχ,1,~m, ~y)

⇔ (1,~m, ~y) ∈ Iχ. From (iii) and (iv) we ﬁnally get ∈ Iψ˜ = ~y (1,~m, ~y) Iχ .

68 6.3. Arithmetically deﬁnable inductive deﬁnitions

Lemma 6.3.7 generalizes of course to inductive relations. That means we have the following theorem.

6.3.8 Theorem (Substitution Theorem) Assume that S1,...,Sn are positively inductive relations and ϕ(X, Y1,...,Yn,~x,α ~ ) is an X, Y1,...,Yn–positive arithmetical formula. Then the ﬁxed–point of the operator deﬁned by ϕ(X, S1,...,Sn,~x,α ~ ) is positively inductive. As an easy observation we get

6.3.9 Theorem Every arithmetical deﬁnable relation is positively inductive. Proof: Let P = (~x,α ~) ∈ Nm,n ϕ(~x,α ~) for an arithmetical formula ϕ(~x,α ~).Thenϕ(~x,α ~) is X–positive. For its ﬁxed point we get

~x ∈ Iϕ(α~) ⇔ N |= ϕ [~x,α ~] , i.e. Iϕ = P .

6.3.10 Deﬁnition A relation S ⊆ Nm,n is coinductive if its complement Nm,n \ S is inductive. A relation is hyperelementary if it is both, inductive and coinductive. From Theorem 6.3.9 and the Substitution Theorem (Theorem 6.3.8) we already get the basic closure properties of inductive, coinductive and hyperelementary predicates.

6.3.11 Theorem The inductive and coinductive relations are closed under • positive boolean operations • quantification over numbers • substitution with hyperelementary relations and functions. The hyperelementary relations are closed under • all boolean operations • quantification over numbers • substitution with hyperelementary relations and functions. Proof: Assume that a relation Q is obtained from inductive relations by positive boolean operations and quantification over numbers from inductive relations S1,...,Sn. Then there is a Y1,...,Yn–positive formula ϕ(Y1,...,Yn,a)such that

Q(a) ⇔ ϕ(S1,...,Sn,a).

Regarding ϕ(Y1,...,Yn,a)as X–positive for a dummy variable X we obtain

Q(a) ⇔ ϕ(S1,...,Sn,a) ⇔ ∈ a Iϕ(S1,...,Sn) and Q is inductive by the Substitution Theorem 6.3.8. There are different possibilities to substitute functions or relations into inductive relations. The most simple one is to deﬁne a predicate Q := a (f(a), a) ∈ S (i) for an inductive set S and a hyperelementary function f. A function is hyperelementary iff its graph is hyperelementary. Recall that for total functions it sufﬁces to have an inductive (or coinductive) graph in order to be hyperelementary. We get from (i)

69 6. Generalized Inductive Deﬁnitions

a ∈ Q ⇔ (∃z)[f(a)=z∧(z,a) ∈ S]. By the already known closure properites of inductive sets and the fact that f(a)=zis inductive we conclude that the predicate Q is inductive. The other possibility to substitute hyperelelemen- tary predicates into relations is to form a relation Q := (~x,α ~) (~y,α, ~ x H(~y,α, ~ x) ) ∈ R for a hyperelementary relation H.Let ∈ ∈ H= (~y,α, ~ x) (~y,x,m ~ +) Iψ+ (α~) = (~y,α, ~ x) (~y,x,m ~ −) / Iψ− (α~) and ∗ ∗ R = (~y,α, ~ α ) (~y,m ~ 0) ∈ Iϕ(α,~ α ) . Then Q(~y,α ~) ⇔ (~y,m ~ ) ∈ I (α,~ x H(~y,α, ~ x) ) 0 ϕ ⇔ ϕ(I ,~y,α, ~ x H(~y,α, ~ x) ,~m ) ϕ 0 ⇔ ϕ(I ,~y,α, ~ x (~y,x,m ~ ) ∈ I (α~) ,~m ) ϕ + ψ+ 0 ⇔ ∈ ϕ(Iϕ,~y,α, ~ x (~y,x,m ~ −) Iψ− (α~) ,~m0).

Choosing the positive version Iψ+ or the negative version Iψ− according to the occurence of the predicate variable Y in ϕ(X, (~y,α ~),Y,~y)we obtain the claim from Lemma 6.3.7 By Lemma 6.1.3 we obtain an upper bound for the complexitiy of arithmetical positive inductive deﬁnitions in the analytical hierarchy. We get

1 1 6.3.12 Theorem Every inductive relation is Π1. The coinductive relations are therefore Σ1 and 1 the hyperelementary relations ∆1. Proof: Let S = (~x,α ~) (~x,~k) ∈ Iϕ(α~) (i) for some X–positive formula ϕ(X, ~x, ~y,α ~). By Lemma 6.1.3 Iϕ is the least Γϕ(α~)–closed set which implies (~x,α ~) ∈ S ⇔ (∀X)[(∀~u)(∀~v)(ϕ(X, ~u,~v,α ~) ⇒ (~u,~v) ∈ X) ⇒ (~x,~k) ∈ X] 1 This is a Π1–deﬁnition of S. The remaining claims follow immediately.

6.4 The stage comparison theorem

Deﬁning the inductive norm |n|ϕ for objects in Iϕ opens the possibility to use elements n ∈ Iϕ as ordinal notations. Ordinal notations, however, are of little use as long as we don’t know how to compare them. The aim of the present section is to show that the stage comparison predicate is also an inductive predicate.

6.4.1 Deﬁnition Let ϕ(X, ~x) and ψ(X, ~y) be X–positive elementary formulas. We introduce the stage comparison predicates ≤∗ ⇔ ∈ ∧ ∈ ⇒| | ≤| | ~x ϕ,ψ ~y : ~x Iϕ (~y Iψ ~xϕ ~yψ) (6.6) and ∗ ⇔ ∈ ∧ ∈ ⇒| | | | ~x<ϕ,ψ ~y : ~x Iϕ (~y Iψ ~xϕ < ~yψ). (6.7) 70 6.4. The stage comparison theorem

Recall that we deﬁned |~n|ϕ = ω1 for n/∈Iϕ. That means that we have

~n ∈ Iϕ ⇔|~n|ϕ<ω1. The deﬁnitions in (6.6) therefore simplify to ≤∗ ⇔ ∈ ∧| | ≤| | ~x ϕ,ψ ~y ~x Iϕ ~xϕ ~yψ (6.8) and ∗ ⇔ ∈ ∧| | | | ~x<ϕ,ψ ~y ~x Iϕ ~xϕ < ~yψ (6.9) respectively.

≤∗ ∗ 6.4.2 Theorem (Stage Comparison Theorem) The stage comparison predicates ϕ,ψ and <ϕ,ψ as deﬁned in (6.6) and (6.7) are positively inductive.

Proof: To find the defining formula for the stage comparison predicate we just rewrite its definition in modified form. We have | | ≤∗ ⇔ ∈ ~y ψ ~x ϕ,ψ ~y ~x Iϕ | | ⇔ < ~y ψ ϕ(Iϕ ,~x) (i) ⇔ ϕ( ~u |~u| <|~y| ,~x) ϕ ψ ⇔ ϕ( ~u ¬|~y|ψ ≤|~u|ϕ ,~x).

But for ~u ∈ Iϕ we get | | |~y| ≤|~u| ⇔ ~y∈I~uϕ ψ ϕ ψ ⇔ ψ( ~v |~v| <|~u| ,~y) (ii) ψ ϕ ⇔ ¬ ≤∗ ψ( ~v (~u ϕ,ψ ~v) ,~y). For the last equivalence observe that ¬ ≤∗ ⇔ ∈ ∨ ∈ ∧| | | | (~u ϕ,ψ ~v) ~u/Iϕ (~v Iψ ~vψ < ~uϕ).

Therefore assuming ~u ∈ Iϕ we have ~v |~v|ψ < |~u|ϕ = ~v ¬(~u ≤ϕ,ψ ~v) .

For ~u/∈Iϕ, however, we have n ~v ¬(~u ≤ϕ,ψ ~v) = N . Thus assuming ψ(Nn,~y) (iii) we can dispense with the premise ~u ∈ Iϕ. However, assuming (iii) means no loss of generality. If ¬ψ(Nn,~y)we modify the formula to ψ˜(X, ~y):≡ψ(X, ~y) ∨ (∀~z)(~z ∈ X) and observe that Iξ = Iξ ψ˜ ψ holds for all ξ ∈ On. Now, plugging (ii) into (i) we get ≤∗ ⇔ ¬ ¬ ≤∗ ~x ϕ,ψ ~y ϕ( ~u ψ( ~v (~u ϕ,ψ ~v) ,~y) ,~x). (iv) Deﬁning

71 6. Generalized Inductive Deﬁnitions

χ(Z, ~x, ~y):≡ϕ( ~u ¬ψ( ~v ¬(~u,~v) ∈ Z ,~y) ,~x) we obtain a Z–positive formula. By (iv) we have ≤∗ ⇔ ≤∗ ~x ϕ,ψ ~y χ( ϕ,ψ,~x, ~y). Hence ⊆≤∗ Iχ ϕ,ψ ≤∗ and it remains to show that ϕ,ψ is indeed the least ﬁxed–point. We prove ≤∗ ⇒ ∈ ~x ϕ,ψ ~y (~x, ~y) Iχ by induction on |~x|ϕ. Towards an indirect proof assume ≤∗ ∧ ∈ ~x ϕ,ψ ~y (~x, ~y) / Iχ. Then we have ¬ϕ( ~u ¬ψ( ~v ¬(~u,~v) ∈ Iχ ,~y) ,~x) and

<|x|ϕ ϕ(Iϕ ,~x) which implies | | < ~x ϕ * ¬ ¬ ∈ Iϕ ~u ψ( ~v (~u,~v) Iχ ,~y) . (v)

<|~x|ϕ By (v) there is a ~x0 ∈ Iϕ such that ψ( ~v ¬(~x0,~v) ∈ Iχ ,~y). (vi) By induction hypothesis, however, we have ~v ¬(~x0,~v) ∈ Iχ ⊆ ~v ¬(~x0 ≤ϕ,ψ ~v) . (vii) From (vi) and (vii) we obtain ψ( ~v ¬(~x0 ≤ϕ,ψ ~v) ,~y) which is

<|~x0|ϕ ψ(Iψ ,~y). | | ∈ ~x0 ϕ Hence ~y Iψ which means

|~y|ψ ≤|~x0|ϕ <|~x|ϕ ≤∗ in contradiction to ~x ϕ,ψ ~y. ∗ The proof of the fact that <ϕ,ψ is a fixed–point is completely dual and left as an exercise. ≤∗ ∗ ≤∗ ∗ If there is no danger of confusion we write ~x ~y and ~x< ~yinstead of ~x ϕ,ψ ~y and ~x<ϕ,ψ ~y. We want to extend the norm definition |x|ϕ to elements of inductive sets. If S = ~x (~x,~k) ∈ Iϕ it makes no sense to define |~x|S = |(~x,~k)|ϕ since this would leave gaps. However, if we define a predicate

~~72 6.4. The stage comparison theorem~~

~~ |~x|S := sup |~y|S +1 ~y~~

6.4.3 Deﬁnition Let S ⊆ Nm,n and µ: S −→onto λ ∈ On be a mapping. We call µ an inductive norm if there are an inductive relation J and a coinductive relation J˘ such that for all b ∈ S we have a ∈ S ∧ µ(a) ≤ µ(b) ⇔ J(a, b) (6.12) ⇔ J˘(a, b).

~~There is a uniform way of expressing J and J˘. We prove~~

onto 6.4.4 Lemma Let λ be an ordinal and µ: S −→ λ be a mapping onto λ. The norm given by µ is inductive iff the relations ∗ ⇔ ∈ ∧ ∈ ⇒ ≤ a S b : a S [b S µ(a) µ(b)] (6.13) and ≺∗ ⇔ ∈ ∧ ∈ ⇒ a S b : a S [b S µ(a) <µ(b)] (6.14) are inductive. ∗ ≺∗ Proof: If S and S are both inductive we put ⇔ ∗ J(a, b): a Sb and ˘ ⇔¬ ≺∗ J(a, b): (b Sa) and check easily that J and J˘ satisfy (6.12). Thus assume that µ is an inductive norm whose accompanying predicates are J and J˘.Thenwe obtain h i ∗ ⇔ ∈ ∧ ∨¬˘ a S b a S J(a, b) J(b,a) and ≺∗ ⇔ ∈ ∧ ˘ a S b a S J(b, a). It will follow that the norm deﬁned in (6.10) is inductive. We prove 6.4.5 Theorem Let S be an inductive relation. Say S = (α,~ ~x) (~x,~k) ∈ Iϕ(α~ ) for some X–positive arithmetical formula ϕ(X, ~x, ~y,α ~). Then the norm deﬁned in (6.10)

~~73 6. Generalized Inductive Deﬁnitions~~

~~ || :S −→ | | S || := | | +1 ∈S S sup a S a a 7−→ |a|S := sup |b|S +1 b~~

Proof: By deﬁnition ||S is a map from S onto ||S||. Because of (α,~ ~x) ∗ (β,~~ y) ⇔ (~x,~k) ≤∗ (~y,~k) S ϕ(α~),ϕ(β~) and (α,~ ~x) ≺∗ (β,~~ y) ⇔ (~x,~k) <∗ (~y,~k) S ϕ(α~),ϕ(β~) ∗ ≺∗ || we obtain S and S as inductive. Hence S is inductive by Lemma 6.4.4.

~~74 1 7. Inductive Deﬁnitions, Π1–sets and the CK ordinal ω1~~

~~1 7.1 Π1–sets vs. inductive sets~~

1 In Theorem 6.3.12 we have shown that all elementary positive inductive sets are Π1–definable. 1 Our next aim is to show that conversely every Π1–set is inductively definable. The first step is to 1 1 define a normal–form for Π1–relations. Let P be some (m, n)–ary Π1–relation. Then P (a) ⇔ (∀α)(∃y)R(α, y, a) and the relation (∃y)R(α, y, a) is semi–decidable. But then there is some decidable predicate R0 such that (∃y)R(α, y, a) ⇔ (∃y)R0(α(y),y,a(y)) and we define 0 RP (s, a):⇔R(s, lh(s), a(lh(s))). Then we get

~~1 1 7.1.1 Lemma (Π1–normal form) For every Π1–relation P there is a decidable relation RP such that~~

~~P (a) ⇔ (∀α)(∃y)RP (α(y), a).~~

~~We use Lemma 7.1.1 in the following deﬁnition~~

~~7.1.2 Deﬁnition Let~~

~~P (a) ⇔ (∀α)(∃y)RP (α(y), a) 1 be a Π1–relation in normal form. We deﬁne TP (a):= s∈Seq (∀s0)(s0 ( s ⇒¬RP(s0,a)) (7.1) and call TP the tree of unsecured sequences for P .~~

~~It is an immediate consequence of (7.1) that TP (a) is a tree. We have~~

~~P (a) ⇔ (∀α)(∃y)RP (α(y), a)~~

~~⇔ (∀α)(∃y)[α(y)∈/ TP(a)]~~

~~⇔ TP (a) is well-founded.~~

Observe that the quantiﬁer in (7.1) is bounded. Hence TP (a) is decidable in a which means that its characteristic function has the form λx. FP (a,x) for some computable functional FP and we have shown

~~1 7.1.3 Theorem For every Π1–relation P there is a computable functional FP such that~~

~~P (a) ⇔ λx. FP (a,x) ∈WT.~~

~~75 1 CK 7. Inductive Deﬁnitions, Π1–sets and the ordinal ω1~~

~~If P is an n–ary predicate then TP (~x) is decidable and an index for TP (~x) can be computed from ~x. Thus Theorem 7.1.3 modiﬁes to~~

~~1 7.1.4 Theorem For every Π1–predicate P there is a computable function TP such that~~

P (~x) ⇔ TP (~x) ∈ WT. ≤ 1 1 By Theorem 7.1.4 we have P m WT for every Π1–predicate P . We say that WT is Π1– complete. 1 To establish the connection between Π1–predicates and inductive sets we study well–founded trees in terms of ﬁxed–points. Let T be a tree and put _ _ ϕT (X, x):≡(∀y)[x hyi∈T ⇒ x hyi∈X]. (7.2)

~~Then ϕT (X, x) is an X–positive formula. Denote its ﬁxed–point by IT . We prove~~

∈ WT ∧ ∈ ⇒ ∈ otypT (s) T s T s IT (7.3) by induction on otypT (s).If ∗ otypT (s)=sup otypT (t)+1 t0then otypT (s y ) <σfor all y such that s y T.Bythe induction hypothesis we get ∀ _h i∈ ⇒ _h i∈ <σ ( y) s y T s y IT <σ ∈ σ which is ϕT (IT ,s). Hence s IT . From (7.3) we get

s ∈ T ∧ s ∈ IT ⇒ T s ∈ WT ∧ otyp(T s) ≤|s|T (7.5) _ by induction on |s|T .If|s|T =0we have (∀y)[s hyi∈/ T] which shows T s = hi and σ <σ otyp(T s)=0. So assume |s|T =: σ>0.Sinces∈I we get ϕT (I ,s)which is T T ∀ _h i∈ ⇒ _h i∈ <σ ( y) s y T s y IT . (i) By induction hypothesis we get (∀y)[s_hyi∈T ⇒Ts_hyi∈WT ∧ otyp(T s_hyi) <σ]. (ii) An inﬁnite path in T s would induce an inﬁnite path in one of the trees T s_hyi which is impossible by (ii). So T s is well–founded and by (5.28) we get _ _ otyp(T s)=otypT s(hi)=sup otyp(T s hyi)+1 s hyi∈T ≤σ.

~~It follows from (7.5) that a tree T is well–founded if hi ∈ IT . Conversely we have hi ∈ IT for well–founded trees T by (7.4). Therefore we have shown~~

~~7.1.5 Theorem AtreeT is well–founded iff hi ∈ IT . For well–founded trees we get~~

~~s ∈ T ⇒ otypT (s)=|s|T (7.6) and~~

~~otyp(T )+1=||ϕT ||. (7.7)~~

~~76 1 7.1. Π1–sets vs. inductive sets~~

~~Proof: We proved everything but (7.7). But this is simple because otyp(T )=otypT (hi)= |hi|T > |s|T for all s ∈ T such that s =6 hi. By Theorem 6.2.4, however, it then follows~~

~~||ϕT || = |hi|T +1=otyp(T )+1. 1 The link between Π1–relations and inductive relations is given by Theorems 7.1.3 and 7.1.5. We get~~

~~1 N 7.1.6 Theorem The Π1–relations are exactly the positively inductive relations on .~~

~~1 Proof: We have by Theorem 6.3.12 that positively inductive relations are Π1. Conversely if P is 1 a Π1–relation then we get by Theorem 7.1.3~~

P (a) ⇔ λx. FP (a,x) ∈WT. _ _ Putting ϕP (X, s, a):⇔(∀y)[FP(a,s hyi)=0⇒s hyi∈X]we get by Theorem 7.1.5 ⇔ hi ∈ P (a) ( , a) IϕP . Hence P is inductive. Dealing with predicates we can sharpen Theorem 7.1.6 as follows

~~1 7.1.7 Theorem There is an X–positive elementary formula ϕP (X, s, ~x) such that for any Π1– predicate P we have ⇔ hi ∈ P (~x) ( ,~x) IϕP .~~

~~Proof: By Theorem 7.1.4 we have~~

~~P (~x) ⇔ TP (~x) ∈ WT for a computable function TP .Wedeﬁne 1,0 _ _ ϕP(X, s, ~x) ⇔ (∀y) {TP (~x)} (s hyi)=0 ⇒ (s hyi,~x)∈X . (7.8)~~

~~Then ϕP (X, s, ~x) is X–positive and elementary and by Theorem 7.1.5 we get ∈ ⇔ hi ∈ TP (~x) WT ( ,~x) IϕP .~~

7.1.8 Remark Although we proved in Theorem 6.3.12 that fixed–points of arithmetically defin- 1 able monotone operators are Π1–definable we did not prove the converse proposition in Theo- 1 rem 7.1.6 (or 7.1.7). All we showed is that Π1–relations are inductive but not necessarily fixed– points. The additional parameter — which is hi in our setting — is indispensable, even for certain 1 ∆1–relations. A proof of this fact, however, is outside the scope of this lecture. It can be found in [1].

~~As a consequence of Theorem 7.1.6 and 7.1.7 we get the following corollaries.~~

1 N 1 7.1.9 Corollary The Π1–relations are exactly the positively inductive relations on .TheΣ1– N 1 relations are exactly the positively coinductive relations on .The∆1–relations are exactly the hyperelementary relations on N.

1 7.1.10 Corollary The Π1–predicates are exactly the positively elementary inductive predicates N 1 N on .TheΣ1–predicates are exactly the positively elementary coinductive predicates on and 1 the ∆1–predicates exactly the hyperelementary predicates.

~~77 1 CK 7. Inductive Deﬁnitions, Π1–sets and the ordinal ω1~~

~~7.2 The inductive closure ordinal of N~~

Let us return to the general situation. Developing the theory of inductive definitions in Chapter 6 we did not make use of special features of the structure N of natural numbers. The fact that we restricted ourselves to unary predicate variables was sheer lazyness. Without the possibility of contracting n–ary predicate variables to unary ones we could have developed the same theory using n–ary predicate variables. [But observe that we did make use of special features of N in Section 7.1.] Let A by any structure and call a first order formula in the language of ALA– elementary if it contains no function or set parameters. We define A κ := sup ||ϕ|| ϕ(X, ~x) is an X–positive LA–elementary formula . and call κA the (inductive) closure ordinal of the structure A. Our aim is to characterize κN.But before doing that we give some abstract consequences of the Stage Comparison Theorem.

~~ξ 7.2.1 Lemma Let ϕ(X, ~x) be an elementary X–positive formula. Then Iϕ is hyperelementary N N for any ξ<κ . Especially if ||ϕ|| <κ then Iϕ is hyperelementary.~~

Proof: The proof depends heavily on the Stage Comparison Theorem. The Lemma is true for arbitrary structures A replacing N. But, since we want to concentrate on N, we only stated it as N above. For ξ<κ we ﬁnd an elementary inductive deﬁnition ψ(Y,y) and an n ∈ Iψ such that |n|ψ = ξ. Using stage comparison we get ∈ ξ ⇔ ≤∗ ~x Iϕ ~x ϕ,ψ n ⇔¬ ∗ (n<ψ,ϕ ~x). ξ Hence Iϕ is hyperelementary.

~~7.2.2 Theorem (Closure Theorem) The ﬁxed–point of an elementary inductive deﬁnition ϕ(X, ~x) is hyperelementary iff ||ϕ|| <κN.~~

~~Proof: One direction is Lemma 7.2.1. For the other direction let Iϕ be hyperelementary and deﬁne χ(Z, z, ~x):≡ z=0∧ϕ( ~u (0,~u)∈Z ,~x)~~

∨ [z =1∧(∀~y)(~y ∈ Iϕ → (0,~y)∈Z)] . A close look at the proof of Lemma 6.3.7 shows that there is a positively elementary formula θ N with κ ≥||θ|| ≥ ||χ||,furthermoreIχ is trivially contained in the elementary inductive set Iθ, thus it is elementary inductive, too. First we show ξ ∈ ξ Iϕ = ~x (0,~x) Iχ (i) by induction on ξ. From the induction hypothesis we get ~x ∈ Iξ ⇔ ϕ(I<ξ,~x) ϕ ϕ ⇔ ∈ <ξ ϕ( ~x (0,~x) Iχ ,~x) ⇔ <ξ χ(Iχ , 0,~x) ⇔ ∈ ξ (0,~x) Iχ. As a consequence of (i) we get ∈ <||ϕ|| Iϕ = ~x (0,~x) Iχ . (ii) || || ∈ ∈ ξ ∈ ξ For any ξ< ϕ there is a ~y Iϕ such that ~y/Iϕ,i.e.(0,~y) / Iχ by (i). Therefore we have

~~78 7.2. The inductive closure ordinal of N~~

¬ ξ χ(Iχ, 1,~x) (iii) for any ~x and ξ<||ϕ||. By (ii), however, we have <||ϕ|| χ(Iχ , 1,~x) (iv) for all ~x. Hence by (iii) and (iv) ∈ ||ϕ|| \ <||ϕ|| (1,~x) Iχ Iχ . (v) N From (v) we ﬁnally obtain ||ϕ|| = |(1,~x)|χ <κ . As a consequence of the Closure Theorem (Theorem 7.2.2) we obtain a characterization of the closure ordinal κN.

CK 7.2.3 Theorem The inductive closure ordinal of the structure of natural numbers is ω1 . Proof: By Theorem 5.4.9 we have CK Tree ∈ ω1 = sup otyp (e) e WT . However, if T is a decidable well–founded tree, we get by (7.7) N otyp(T )+1=||ϕT || ≤ κ since ϕT is an elementary formula. Hence CK ≤ N ω1 κ . CK N ∈ 1 \ 1 Assume ω1 <κ. Choose some predicate P Π1 ∆1.SuchPexists by the Analytical Hierarchy Theorem. Now we apply Theorem 7.1.7 to obtain

~~P (~x) ⇔ TP (~x) ∈ WT ⇔ hi ∈ ( ,~x) IϕP . By (7.7) we have ||ϕ || ≤ sup otypTree(T (~x)) + 1 ~x ∈ P P P ≤ sup otypTree(e)+1 e∈WT CK N = ω1 <κ .~~

It follows from the Closure Theorem (Theorem 7.2.3) that Iϕ is hyperelementary which by Corol- 1 lary 7.1.10 entails that Iϕ is ∆1. But this contradicts the choice of P . N CK To obtain further characterizations of κ — and thus also of ω1 — we introduce some notations.

~~7.2.4 Deﬁnition A binary well–founded predicate ≺ is a pre–well–ordering iff~~

~~x ≺ y ⇔ x ∈ ﬁeld(≺) ∧ y ∈ ﬁeld(≺) ∧ otyp≺(x) < otyp≺(y).~~

~~Pre–well–orderings are closely connected to norms.~~

~~onto 7.2.5 Lemma Let µ: S −→ λ be a norm. The predicate ≺µ deﬁned by~~

~~~x ≺µ ~y :⇔ ~x ∈ S ∧ ~y ∈ S ∧ µ(~x) <µ(~y) is a pre–well–ordering such that~~

~~otyp≺µ (~x)=µ(~x) holds for all ~x ∈ S.~~

~~79 1 CK 7. Inductive Deﬁnitions, Π1–sets and the ordinal ω1~~

~~Proof: The predicate ≺µ is obviously well–founded. So we only have to prove ∈ ⇒ ~x S otyp≺µ (~x)=µ(~x). (i)~~

This is done by induction on ≺µ. Using the induction hypothesis we compute otyp≺ (~x)=sup otyp≺ (~y)+1 ~y≺ ~x µ µ µ =sup otyp≺ (~y)+1 µ(~y)<µ(~x) µ =sup µ(~y)+1 µ(~y)<µ(~x) =µ(~x).

7.2.6 Theorem We have κN = otyp(≺) ≺ sup is a hyperelementary pre–well–ordering = otyp(≺) ≺ sup is a hyperelementary well–founded binary predicate = sup otyp(≺) ≺ is a coinductive well–founded binary predicate . However, none of these suprema is attained.

Proof: Before we start proving the theorem we want to mention that it is true for arbitrary structures. Put σhp := sup otyp(≺) ≺ is a hyperelementary pre–well–ordering σhf := sup otyp(≺) ≺ is a hyperelementary well–founded binary predicate and σcf := sup otyp(≺) ≺ is a coinductive well–founded binary predicate .

Starting with an elementary X–positive formula ϕ(X, ~x) we construct for every ~x0 ∈ Iϕ a hy- ≺ perelementary pre–well–ordering ~x0 such that | | ≤ ≺ ~x0 ϕ +1 otyp( ~x0 ). (i) N Then (i) proves κ ≤ σhp.Since

σhp ≤ σhf ≤ σcf holds trivially it then remains to show N σcf ≤ κ (ii) to ﬁnish the proof. Let’s prove (i). Choose ~x0 ∈ Iϕ and deﬁne ≺ ⇔|| ||≤| | ~x ~x0 ~y : ~xϕ<~yϕ ~x0 ϕ ⇔ ∗ ≤∗ x<ϕ,ϕ ~y ϕ,ϕ ~x0 (iii) ⇔¬ ≤∗ ∧¬ ∗ (~y ϕ,ϕ ~x) (~x0 <ϕ,ϕ ~y). ≺ Then it is clear from (iii) that ~x0 is hyperelementary and well–founded. By Lemma 7.2.5 it is also a pre–well–ordering such that

~~otyp≺ (~x)=|~x| . ~x0 ϕ Therefore we obtain~~

~~80 7.2. The inductive closure ordinal of N~~

~~ otyp(≺~x )=sup otyp≺ (~x)+1 |~x|ϕ ≤|~x0|ϕ 0 ~x0 =sup |~x|ϕ +1 |~x|ϕ ≤|~x0|ϕ~~

~~=|~x0|ϕ+1. To prove (ii) let ≺ be a coinductive well–founded binary predicate. Recall the deﬁnition of the accessible part Acc(≺) of ≺ which is the ﬁxed–point of the formula~~

ϕ≺(X, x):≡(∀y)(y ≺ x → y ∈ X). Denote by Accξ(≺) the ξ–th stage of this ﬁxed–point. We prove ξ x ∈ Acc (≺) ⇒ otyp≺(x) ≤ ξ (iv) by transﬁnite induction on ξ.Forx∈Accξ(≺) we get (∀y) y ≺ x → y ∈ Acc<ξ(≺) which by induction hypothesis gives otyp≺(x)=sup otyp≺(y)+1 y≺x ≤ξ. Now we prove x ∈ Accotyp≺(x)(≺) (v) by induction on ≺. From the induction hypothesis we get (∀y)(y ≺ x ⇒ y ∈ Acc

otyp≺(x)=|x|Acc(≺) (7.9) which holds for arbitrary well–founded predicates. From (7.9) and (6.2) we get otyp(≺)= otyp≺(x)+1 x∈ﬁeld(≺) sup ≤ sup |x|Acc(≺) +1 x∈Acc(≺) (vi) = ||ϕ≺||. ≺ ≺ Since is coinductive we get by Lemma 6.3.7 that Acc( )=Iϕ≺ is inductive. Hence N ||ϕ≺|| ≤ κ (vii) and we get from (vi) and (vii) N σcf ≤ κ .

It remains to show that none of the suprema is attained. For that it sufﬁces to show that σcf is not attained. This, however, is obvious since for a given coinductive well–founded predicate ≺ we deﬁne x ≺0 y :⇔ Seq(x) ∧ Seq(y) ∧ lh(x)=lh(y)=2

∧ [((x)0 =(y)0 =0∧(x)1 ≺(y)1)∨(y)0 =(y)1 =1]. Then ≺0 is a coinductive well–founded predicate, too, and otyp(≺0) ≥ otyp(≺)+1. Recalling Theorems 6.3.12, 7.2.6 and 7.2.3 we have shown

CK 1 7.2.7 Theorem The ordinal ω1 is the supremum of the order–types of Σ1–definable well–orderings. 1 This supremum is not attained, i.e. the order–type of any well–founded Σ1–definable predicate is CK less than ω1 . 81 1 CK 7. Inductive Definitions, Π1–sets and the ordinal ω1

~~1 There is an extension of Theorem 7.2.7 to Σ1–deﬁnable collections of well–orderings.~~

1 7.2.8 Theorem (Boundedness Principle) Let P be a Σ1–definable subset of WO (or WT). Then WO ∈ CK sup otyp (e) e P <ω1 Tree ∈ CK (or sup otyp (e) e P <ω1 ). 1 WO WT If P is a Σ1–definable subset of (or )then ∈ CK sup otyp(α) α P <ω1 . Proof: Similarly to Theorem 7.2.3, the key to the proof will be the Analytical Hierarchy Theorem. ⊆ 1 Let P WO be Σ1–definable and put Q(a, b):⇔a∈LO ∧ P (b) ∧ (∃α)(∀x)(∀y)[{a}2,0(x, y)=0⇒{b}2,0(α(x),α(y)) = 0]. Then Q(a, b) says that a is the index of an ordering which is order preserving embeddable into an ordering in P . This implies that a is a well–ordering. Hence (∃b)Q(a, b) ⇒ a ∈ WO. (i) WO ∈ CK ∈ ∈ Now assume sup otyp (e) e P = ω1 .Thenwegetforanya WO a b P such that otypWO(a) ≤ otypWO(b) and therefore also an order–preserving embedding from field({a}2,0) into field({b}2,0),i.e.weget a∈WO ⇒ (∃b)Q(a, b). (ii) From (i) and (ii) we obtain a ∈ WO ⇔ (∃b)Q(a, b). 1 ≤ ≤ For any Π1–predicate R, however, we have R m WT m WO by Theorem 7.1.4 and Lemma 5.4.6. ∃ 1 1 1 Since ( b)Q(a, b) is a Σ1–predicate every Π1–predicate would already be Σ1. This contradicts the Analytical Hierarchy Theorem. The same proof works for WO replaced by WT . If P ⊆ WO then we define Q(α, β) ⇔ α ∈ LO ∧ β ∈ P ∧ (∃η)(∀x)(∀y)[α(hx, yi)=0⇒β(hη(x),η(y)i)=0] 1 which again is Σ1 and copy the above argument. In the Closure Theorem we have seen that the complexity of the obtained fixed–point depends on the number of steps which are needed to construct the fixed–point, i.e. on ||ϕ||. An interesting question to ask is whether ||ϕ|| depends on the complexity of the defining formula ϕ or not. Let us regard the formula 1,0 _ _ ϕC (X, x, e):≡(∀y) {e} (x hyi)'0⇒(x hyi,e)∈X . (7.10) 0 Then ϕC is Π1. We know from Theorem 7.1.5 ∈ ⇒ ∈ ⇔ hi ∈ e Tree (e WT ( ,e) IϕC). and for e ∈ WT Tree |(hi,e)|ϕ =otyp (e). C CK Tree ∈ CK ≤ Since ω1 = sup otyp (e)+1 e WT we obtain for every ξ<ω1 an e such that ξ |(hi,e)|ϕ which shows C || || 0 CK sup ϕ ϕ is an X–positive Π1–formula = ω1 . (7.11) 82 0 7.3. Σ1–inductive definitions and semi–decidable sets

0 It follows from (7.11) that restricting the inductive definition to Π1–definable ones does not de- 0 crease the inductive closure ordinal. In the next section we are going to study the case of Σ1– definable operators.

~~0 7.3 Σ1–inductive deﬁnitions and semi–decidable sets~~

0 0 7.3.1 Lemma (Σ1–Reflection) Let ϕ(X, ~x) be an X–positive Σ1–formula and Iψ any fixed– <ω n point such that ϕ(Iψ ,~x). Then there is some n<ωsuch that ϕ(Iψ ,~x). Proof: We induct on the definition of “ϕ(X, ~x) is an X–positive formula”. The claim is obvious ≡ ∈ ∈ <ω if X does not occur in ϕ(X, ~x).Ifϕ(X, ~x) t(~x) X and t(~x) Iψ then there is some ∈ n ≡ ∧ n<ωsuch that t(~x) Iψ .Ifϕ(X, ~x) ϕ1(X, ~x) ∨ ϕ2(X, ~x) we find n1,n2 <ωsuch n1 ∧ n2 { } n that ϕ1(Iψ ,~x)∨ ϕ2(Iψ ,~x). Putting n := max n1,n2 we get ϕ(Iψ ,~x)by the X–positivity of ≡ ∃ <ω ϕi(X, ~x). The last possibility is that ϕ(X, ~x) ( y)ϕ0(X, ~x, y).Ifϕ(Iψ ,~x)then we find some <ω n y<ωsuch that ϕ0(Iψ ,~x, y) and by induction hypothesis an n<ωsuch that ϕ0(Iψ ,~x, y).But n this implies ϕ(Iψ ,~x). 0 Observe that the above proof depended heavily on the fact that ϕ(X, ~x) was Σ1. The above argument would break down for ϕ(X, ~x) ≡ (∀y)ϕ0(X, ~x, y). Observe further that the opposite direction in Lemma 7.3.1 holds by monotonicity. Hence N | <ω ⇔ ∃ N| n = ϕ(Iψ ,~x) ( n<ω) =ϕ(Iψ,~x) . (7.12) As a consequence of Lemma 7.3.1 we obtain

0 || || ≤ 7.3.2 Theorem Let ϕ(X, ~x) be an X–positive Σ1–formula. Then ϕ ω. Proof: By (7.12) we have ~x ∈ Iω ⇔ N |= ϕ(I<ω,~x) ϕ ϕ ⇔ ∃ ∈ n+1 ( n<ω) ~x Iϕ ⇔ ∈ <ω ~x Iϕ .

0 It follows from Theorem 7.3.2 and the Closure Theorem 7.2.2 that every X–positive Σ1–formula 1 has ∆1 ﬁxed–point. This estimate, however, is much too crude. It follows from Theorem 7.3.2 that ∈ ⇔ ∃ ∈ n ~x Iϕ ( n)(~x Iϕ ). ∈ n 0 Thus, if we succeed to show that (~x, n) ~x Iϕ is arithmetical or even Σ1,wegetamuch lower complexity of the ﬁxed–point. The key here is a restatement of the Recursion Theorem.

~~7.3.3 Theorem (Recursion Theorem for semi–decidable predicates) Let ϕ(X, ~x) be an X–positive 0 Σ1–formula. There is an index e such that ∈ n,0 ⇔ n,0 ~x We ϕ(We ,~x)~~

~~0 Proof: Observe ﬁrst that substituting a semi–decidable set R into an X–positive Σ1–formula ϕ(X, ~x) yields a semi–decidable predicate (~x, ~y) ϕ( ~z (~z,~y) ∈ R ,~x) .~~

~~0 The proof is by induction on the deﬁnition of “ϕ(X, ~x) is an X–positive Σ1–formula” and is straight forward using the closure properties of semi–decidable predicates. Now we regard~~

~~83 1 CK 7. Inductive Deﬁnitions, Π1–sets and the ordinal ω1~~

n,0 Q = (~x, y) ϕ(WS(y,y),~x) which is semi–decidable and therefore has an index e0. Putting e := S(e0,e0)we obtain ∈ n,0 ⇔ ∈ n,0 ~x We (~x, e0) We0 ⇔ ϕ(Wn,0 ,~x) S(e0,e0) ⇔ ϕ(Wn,0,~x) e In consequence of the Recursion Theorem for semi–decidable predicates we get that the semi– decidable predicates are closed under inductive deﬁnitions.

0 0 7.3.4 Theorem The ﬁxed–point of an X–positive Σ1–formula ϕ is a Σ1–predicate. Proof: We use the Recursion Theorem to obtain an index e such that ∈ ⇔ ∧ ∅ ∨ ∧ ∈ (~x, m) We [m =0 ϕ( ,~x)] m = k +1 ϕ( ~u (~u, k) We ,~x) . We prove m ∈ Iϕ = ~x (~x, m) We by induction on m and obtain the claim since ~x ∈ I ⇔ ~x ∈ I<ω ϕ ϕ ⇔ ∃ ∈ n ( n) ~x Iϕ ⇔ (∃n)[(~x, n) ∈ W ] . e

~~1 7.4 Some properties of Π1– and related predicates~~

~~1 We will apply the theory of inductive sets to pursue the study of Π1–predicates. Recalling (7.10) we put 1,0 _ _ ϕTree(X, x, e):≡e∈Tree ∧ (∀y )({e} (x hyi)=0 ⇒ (x hyi,e)∈X).~~

Let ITree := Iϕ and put Tr ee hi ∈ σ WT σ := e ( ,e) ITree . (7.13) CK 1 For σ<ω the set WT σ is ∆ by Theorem 7.2.2 and Corollary 7.1.9. We prove 1 1 Tree WT σ = e ∈ WT otyp (e) ≤ σ . (7.14) 1,0 Assume e ∈ WT σ and put Te := s {e} (s)=0 . By (7.5) we get hi ∈ σ ⇒ hi ∈ ∧ hi ≤ ( ,e) ITree Te WT otyp(Te ) σ. Hence e ∈ WT and otypTree(e) ≤ σ. ∈ Tree ≤ hi ∈ σ For the converse inclusion assume e WT and otyp (e) σ. Then by (7.3) ( ,e) ITree. As a consequence of the Boundedness Principle (Theorem 7.2.8) we get

⊆ 1 CK ⊆ 7.4.1 Lemma Let S WT be a Σ1–set. Then there is an ordinal σ<ω1 such that S WTσ. CK Tree ∈ ≤ Proof: By the Boundedness Principle there exists a σ<ω1 such that sup otyp (e) e S σ. This implies S ⊆ WT σ. 1 From Lemma 7.4.1 we get a characterization of the ∆1–sets. 84 1 7.4. Some properties of Π1– and related predicates

~~1 CK ≤ 7.4.2 Theorem Let H be a ∆1–set. Then there is a σ<ω1 such that H m WTσ. ∈ 1 1 Proof: Since H Π1 and WT is Π1–complete we have~~

H ≤m WT (i) say via f. Because H is also Σ1 we get 1 M := f[H]= f(x) x∈H ⊆WT (ii) 1 ⊆ CK as a Σ1–subset of WT . Hence M WT σ for some σ<ω1 by Lemma 7.4.1. By (i) and (ii), however, we get

~~H ≤m WT σ via f.~~

~~1 1 7.4.3 Theorem (Reduction Theorem) Let P and Q be Π1–predicates. Then there are Π1– predicates P1 ⊆ P and Q1 ⊆ Q such that~~

~~P1 ∩ Q1 = ∅ and~~

~~P1 ∪ Q1 = P ∪ Q. Cf. Figure 7.4.1.~~

~~PP1Q1Q~~

~~Figure 7.4.1: Reducing sets P1 and Q1 for P and Q~~

~~Proof: The theorem is a consequence of the Stage Comparison Theorem. Put R(z,~x):⇔[z=0∧P(~x)] ∨ [z =1∧Q(~x)] .~~

1 Thus R is Π and hence inductive. Thus R admits an inductive norm ||R by Theorem 6.4.5. Put 1 ∗ P1 := ~x (0,~x) R (1,~x) and ≺∗ Q1 := ~x (1,~x) R (0,~x) ∗ ≺∗ ∗ where R and R are the predicates deﬁned in (6.13) and (6.14) on page 73. Then R as well ≺∗ 1 as R are inductive, i.e. Π1–relations such that 85 1 CK 7. Inductive Deﬁnitions, Π1–sets and the ordinal ω1

P1 ∩ Q1 = ∅. Moreover we have P1 ∪ Q1 ⊆ ~x (∃z)[(z,~x) ∈ R] ⊆ P ∪ Q (i) ∈ ∪ ∈ ∈ ∗ ≺∗ and for ~x P Q we either get (0,~x) Ror (1,~x) R. Hence (0,~x) R (1,~x)or (1,~x) R (0,~x) which implies ~x ∈ P1 or ~x ∈ Q1. This gives also the converse inclusion of (i) and the proof is ﬁnished. As a consequence of the Reduction Theorem we get

1 7.4.4 Theorem (Separation Theorem) Let P and Q be two disjoint Σ1–predicates. Then there 1 is a ∆1–predicate H which separates P and Q,i.e.whichsatisﬁes P ⊆H and H ∩ Q = ∅. Cf. Figure 7.4.2.

~~1 Figure 7.4.2: Separating P and Q by a ∆1–setH~~

~~Proof: We regard the complements ¬P and ¬Q and reduce them to P1 ⊆¬Pand Q1 ⊆¬Qby the Reduction Theorem. Because of n P1 ∪ Q1 = ¬P ∪¬Q=¬(P ∩Q)=N and~~

~~P1 ∩ Q1 = ∅ we get~~

~~P1 = ¬Q1. ¬ 1 Putting H := P1 we get H as a ∆1–predicate such that 86 1 7.4. Some properties of Π1– and related predicates~~

~~P ⊆ H and~~

~~Q ∩ H = Q ∩ Q1 = ∅ because Q1 ⊆¬Q.~~

~~1 1 7.4.5 Theorem (Weak Π1–uniformization) Let P be an (m+1,n)–ary Π1–relation. Then there is a partial functional FP such that dom(FP )= a (∃x)P(a,x)~~

~~(∀a∈dom(FP )) [P (a,FP(a)] 1 The graph of FP is Π1–deﬁnable. Cf. Figure 7.4.3.~~

~~P F~~

~~Figure 7.4.3: Uniformizing P by F~~

~~Proof: Thenaivetrytoput~~

~~FP(a):'µx. P (a,x) fails, because expressing that x is the least element such that P (a,x)requires to say (∀y~~

~~FP (a) ' y :⇔ P (a,y) ∧ ∀ ∗ ( z)[(a,y) P (a,z)] ∧ ∀ ≺∗ ( z~~

1 1 7.4.6 Theorem (Strong Π1–Uniformization) Let P be an (m, n +1)–ary Π1–relation. Then 1 there is an (m, n +1)–ary Π1–relation Q such that (∀a)(∀α)[Q(a,α)⇒P(a,α)] (i) (∀a)(∀α)(∀β)[Q(a,α)∧Q(a,β) ⇒ α=β] (ii) (∀a)[(∃α)P(a,α)⇒(∃α)Q(a,α)] . (iii)

Proof: Fix a.If¬(∃α)P(a,α) we trivially put Q := ∅. Thus assume (∃α)P (a,α). By Theo- rem 7.1.3 we have a computable functional F such that P (a,α) ⇔ λx. F (a,α,x) ∈WT. (iv) Let Tα := s ∈ Seq F (a,α,s)=0 be the associated tree. Put σ := min otyp(Tα) P (a,α) and let Q0 := α P (a,α) ∧otyp(Tα)=σ . (v)

We are going to define relations Qn by induction on n and assume that Qn is already defined. We put sn := min α(n) P (a,α) , σn := min otyp(Tαn) P (a,α)∧α(n)=sn and define Qn+1 := α ∈ Qn α(n)=sn ∧otyp(Tαn)=σn . (vi) Let \ Q := Qn. n∈ω From (v) and (vi) we get

(∀n<ω)[α ∈ Qn ⇒ P (a,α)] by induction on n.ByQ⊆Q0and (v) we have α ∈ Q ⇒ P (a,α). (vii) Another immediate consequence is Q(α) ∧ Q(β) ⇒ (∀n ∈ ω)[α(n)=s =β(n)] n (viii) ⇒ α = β. By (vii) and (viii) we obtain claims (i) and (ii) of the theorem. The real work is to prove (iii) and 1 ∃ the fact that Q is Π1–deﬁnable. Since we assumed ( α)P (a,α)it sufﬁces to prove (∃α)Q(α) to show (iii). Since Qn+1 ⊆ Qn we have sn ⊆ sn+1. Hence

~~m ≤ n ⇒ sm ⊆ sn. Therefore there is a unique function, say γ, such that~~

~~88 1 7.4. Some properties of Π1– and related predicates~~

~~(∀n ∈ ω)[γ(n)=sn]. (ix) We claim Q(γ). (x) In a ﬁrst step we prove ∗ ⇒ m~~

~~(∀α)[α(k)=γ(k) ⇒ ({n, m}⊆Tγ ⇔{n, m}⊆Tα)]. (xii)~~

We may choose k bigger than m and n.Pickα∈Qk+1.Thenα(n)=sn =γ(n)as well as α(m)=s =γ(m)and by (xii) we get m, n ∈ T .Butthenm<∗ nimplies m<∗ nand m α Tγ Tα we obtain otyp(Tαm) < otyp(Tαn). But since α ∈ Qk+1 ⊇ Qi+1 for i = m, n we ﬁnally obtain σm = otyp(Tαm) < otyp(Tαn)=σn. This terminates the proof of (xi). By a similar argument we also obtain

~~m ∈ Tγ ⇒ σm <σ. (xiii)~~

We choose k>msuch that (xii) and pick α ∈ Qk+1.Butthenσm=otyp(Tαm) < otyp(Tα)=σsince α ∈ Qk+1 ⊆ Qm+1 ⊆ Q0. It follows from (xi) that Tγ is well–founded. Hence P (a,γ). (xiv) Next we prove

n ∈ Tγ ⇒ otyp(Tγn) ≤ σn (xv) by induction on <∗ .Wehave Tγ otyp(T n)=sup otyp (m)+1 m∈T n γ Tγ n γ =sup otyp (n_m)+1 n_m∈T Tγ γ ∗ =sup otypT (m)+1 m< n γ Tγ ∗ =sup otyp(Tγm)+1 m< n Tγ ≤ ∗ ≤ sup σm +1 m

(∀n)[γ ∈ Qn] (xvi) by induction on n. From (xiii) we get otyp(Tγ) ≤ σ which together with (xiv) shows γ ∈ Q0. If γ ∈ Qn then we obtain from (ix) and (xv) γ ∈ Qn+1. 1 Now (x) follows from (xvi) and it remains to show that Q is Π1–definable. First observe that for T ∈ WT the relation otyp(S) ≤ otyp(T ) (xvii) as well as otyp(S) < otyp(T ) 1 ∈ WT are both Σ1–definable. To see this recall the formula ϕT in (7.2) and assume T .Then 89 1 CK 7. Inductive Definitions, Π1–sets and the ordinal ω1

≤ ⇔ |hi| ≤ |hi| otyp(S) otyp(T ) ϕS ϕT ⇔hi∈ ∨hi ∈ ∧ ¬|hi| |hi| /IϕT ( ϕS ϕT < ϕS ) ⇔¬hi ∗ hi ( <ϕS ,ϕT ) 1 and the last line is Σ1 by Stage Comparison. Analogously we also obtain ∈ WT ⇒ ⇔¬hi ≤∗ hi T (otyp(S) < otyp(T ) ( ϕT ,ϕS )).

~~Regard that according to the deﬁnition (vi) of Qn we have~~

~~β ∈ Qn ⇔ otyp(Tβ) ≤ σ ∧ (∀m~~

~~Thus, if we assume α ∈ Qn,~~

~~β ∈ Qn ⇔ otyp(Tβ) ≤ otyp(Tα) (xviii) ∧ (∀m~~

~~1 According to (xvii) the right hand side in (xviii) is a Σ1–relation, say R0(α, β, n) (where we suppress the parameters a which are hidden in Qn). Still assuming α ∈ Qn we thus get~~

~~α/∈Qn+1 ⇔ (∃β){β ∈ Qn ∧ [β(n) < α(n) ∨ (β(n)=α(n)~~

~~∧otyp(Tβn) < otyp(Tαn))]}~~

~~⇔ (∃β){R0(α, β, n) ∧ [β(n) < α(n) ∨ (β(n)=α(n) (xix)~~

~~∧otyp(Tβn) < otyp(Tαn))]}~~

~~⇔: R1(α, n). 1 By (xix) we see that R1(α, n) is a Σ1–relation. Using (xix) we ﬁnally get~~

α ∈ Q ⇔ α ∈ Q0 ∧ (∀n)¬R1(α, n) ⇔ ∧ ∀ ∗ ∧ ∀ ¬ P (a,α) ( β)[(a,α) P (a,β)] ( n) R1(α, n) ∗ 1 where P is the relation deﬁned in (6.13). Since P is Π1 and thus inductive we get by Theo- ∗ 1 rem 6.4.5 that P is inductive and thus Π1–deﬁnable.

~~7.5 Basis Theorems~~

Let P be an (0, 1)–ary relation, i.e. P is a collection of functions. Even if P can be classified in the arithmetical or analytical hierarchy we cannot hope to get some information about the members of P . Regard for example the collection of all functions which is decidable but contains functions of arbitrary complexity. All we can say is that there are computable functions among all functions. We are going to prove that in many cases we have a similar situation. If P is a collection having a simple classification then some functions in P can be classified in a simple way. This is made precise in the following definition.

~~7.5.1 Deﬁnition Let C be a collection of (0, 1)–ary relations. A class B of functions is called a basis for C if for every P in C we have (∃α)P (α) ⇒ (∃α ∈ B)P (α).~~

C 0 ∈C 6 ∅ As an example we regard the collection of all Σ1–classes of functions. Let P and P = . Then α ∈ P ⇔ (∃x)R(α(x)) for some decidable predicate R.SinceP=6 ∅there is some s ∈ Seq such that R(s). Deﬁning (s) if x

90 7.5. Basis Theorems we get β ∈ P and see that the class of functions which have value 0 almost everywhere form a 0 basis for the collection of Σ1–classes of functions. 0 ∈ 7.5.2 Lemma Let B be a basis for the collection of Π1–classes of functions. Then (γ)0 γ B 1 is a basis for the collection of Σ1–classes of functions.

1 Proof: Let P be in Σ1.Then P(α) ⇔ (∃β)Q(α, β) (i) 0 ∃ ∃ for some Π1–relation Q(α, β).From(α)P(α)it follows ( γ)Q((γ)0, (γ)1) and, since B is a 0 ∈ basis for the collection of Π1–classes of functions, we obtain a γ B such that Q((γ)0, (γ)1). But then P ((γ)0). By literally the same proof we obtain also 1 ∈ 7.5.3 Lemma Let B be a basis for the collection of Πn–classes of functions. Then (γ)0 γ B 1 is a basis for the collection of Σn+1–classes of functions. Let P be a class of functions. We deﬁne In(P )= s∈Seq (∃α)[P(α) ∧α(lh(s)) = s] , (7.15) i.e. In(P ) is the set of initial segments of functions in P . Generalizing our above example we obtain

0 ≤ 7.5.4 Lemma If P is a nonempty Π1–class of functions then P (β) for some β T In(P ). Proof: We deﬁne F (n):'µx. (F (n)_hxi∈In(P )). Then F is computable from In(P ).Weshow (∀n) F(n)∈In(P ) by induction on n. F (0) = hi ∈ In(P ) follows from the hypothesis (∃α)P (α). Now assume F (n) ∈ In(P ).Butthen F(n)=min α(n) P (α) ∧ α(n)=F(n) _h i∈ 0 is deﬁned and F (n +1)=F(n) F(n) In(P ).SincePis Π1 we get P (α) ⇔ (∀x)R(α(x)) for some decidable predicate R. Hence In(P ) ⊆ R and we get (∀x)R(F (n)). This proves P (F ).

~~As a consequence we obtain the ﬁrst half of KLEENE’s Basis Theorem.~~

1 7.5.5 Theorem The functions which are computable in the class of Σ1–predicates are a basis 0 1 for the collection of Π1–classes of functions and hence also for the collection of Σ1–classes of functions.

0 1 Proof: For a Π1–class P of functions we see from (7.15) that In(P ) is Σ1. By Lemma 7.5.4 1 it follows that the class of functions computable in the class of Σ1–predicates is a basis for the 91 1 CK 7. Inductive Deﬁnitions, Π1–sets and the ordinal ω1

~~0 1 collection of Π1–classes of functions and by Lemma 7.5.2 also for the collection of Σ1–classes of functions.~~

As already remarked Theorem 7.5.5 is only one half of KLEENE’s Basis Theorem which will be 1 our Theorem 8.2.3. The second half says that the class of ∆1–deﬁnable functions is not a basis 0 for the collection of Π1–classes of functions. We have to postpone this part until we have a better 1 characterization of the ∆1–deﬁnable functions. Recall that we identify sets with their characteristic functions. Therefore we may talk about bases for collections of classes of sets. A remarkable result is

0 7.5.6 Theorem (KREISEL’s Basis Theorem) The class of ∆2–functions is a basis for the collec- 0 tion of Π1–classes of sets. To prepare the proof we formulate a lemma which on its turn is an easy consequence of the Finiteness Theorem (Theorem 5.2.6).

~~0 7.5.7 Lemma Let P be a Π1–relation and deﬁne Q(a):⇔(∃α∗)P(a,α∗).~~

~~0 Then Q is also Π1. ∈ 0 Proof: Since P Π1 we have P (a,α) ⇔ (∀x)R(a(x),α(x)) for some decidable relation R.Thetree s (∀i~~

~~∧(∀i~~

0 0 holds. Thus In(P ) is Π1 by Lemma 7.5.7. The functions which are computable in the Π1–classes 0 of functions are therefore by Lemma 7.5.4 a basis for the collection of Π1–classes of sets. By 0 0 0 POST’s Theorem (Theorem 3.2.6) these are the functions which are ∆1[Π1],i.e.∆2. To obtain even further reaching basis theorems we introduce some notations.

~~7.5.8 Deﬁnition A (0, 1)–ary relation P deﬁnes a function γ implicitly if (∀α)(∀β)[P(α) ∧ P(β)⇒α=β] and P (γ). The function γ is called a singleton.~~

~~92 7.6. The complexity of KLEENE’s O~~

7.5.9 Lemma Let P deﬁne a function γ implicitly. Then ∈ 1 ⇔ ∈ 1 P ∆n γ ∆n for all n. ∈ 1 Proof: Assume ﬁrst P ∆n.Then γ(x)'y ⇔ (∃α)[P(α) ∧α(x)=y] ⇔ (∀α)[P(α)⇒α(x)=y]. If γ ∈ ∆1 we get n P = α (∀x)(∀y)[α(x)=y⇔γ(x)=y] . As an immediate consequence we get

7.5.10 Corollary Let C be a collection of classes of functions such that every nonempty class in C 1 1 has a ∆n–subclass which contains exactly one function. Then the class of ∆n–functions is a basis for the collection C.

~~1 From the strong Π1–uniformization and Corollary 7.5.10 we get the following theorem.~~

~~1 1 7.5.11 Theorem The class of ∆2–functions is a basis for the collection of Π1–classes of functions.~~

~~By Lemma 7.5.3 we get~~

~~1 1 7.5.12 Theorem The class of ∆2–functions is a basis for the collection of Σ2–classes of functions.~~

~~7.6 The complexity of KLEENE’s O~~

We will now settle the still open question for the complexity of KLEENE’s O within the analytical hierarchy. We defined O in Definition 5.4.1 by a rather complicated simultaneous inductive definition. Now we are going to unravel this definition into single steps.

~~0 7.6.1 Deﬁnition We deﬁne inductively the binary predicate~~

~~0 0 7.6.2 Lemma The predicate~~

~~In the next step we show~~

~~7.6.3 Lemma 0 1)~~

~~0 0 0 0 Proof: We prove a~~

~~7.6.4 Deﬁnition We deﬁne inductively the set O0 by the following clauses. 1) 1 ∈O0 2) a ∈O0 ⇒2a ∈O0 1,0 0 1,0 0 1,0 e 0 3) (∀n) {e} (n) ∈O ∧(∀n) {e} (n)~~

~~0 0 7.6.5 Lemma We have O = O and~~

Proof: We show x ∈O ⇔ x∈O0 (i) simultaneously by induction on the deﬁnition of x ∈Oand x ∈O0, respectively. Claim (i) is obvious for x =1and immediate from the induction hypothesis in case that x =2y =16 . Thus 0 let x =3·5y.Ifx∈Owe get {y}1,0(n) ∈Ofor all n ∈ N and therefore {y}1,0(n) ∈O for all 1,0 1,0 n ∈ N. We moreover have (∀n) {y} (n)

~~94 7.6. The complexity of KLEENE’s O~~

~~1,0 1,0 0 1,0 (∀n) {y} (n) ∈O ∧(∀n) {y} (n)~~

~~O 1 7.6.6 Theorem The set is Π1–complete.~~

Proof: By Theorem 7.1.7 there is a formula ϕP (the formula in (7.8)) such that ≤ P m IϕP . Thus it suffices to show ≤ O IϕP m . We want to get a computable function G such that ∈ ⇔ ∈O (s, x) IϕP G(s, x) . (i) First we define a function 1,0 1 if {TP (x)} (s) ' 1 G0(e, s, x)= z 1,0 3 · 5 if {TP (x)} (s) ' 0 where z is an index of the function F defined by F (0) = 1 2,0 _ F (n +1)=F(n)+O {e} (s hni,x)+O 2.

Note that the case distinction in the defintion of G0 is decidable because TP (x) ∈ Tree .Using the Recursion Theorem we get an index e0 such that 2,0 {e0} (s, x) ' G0(e0,s,x) 2,0 and we put G := {e0} . By definition G is computable. We show that G satisfies (i) and start to prove ∈ ⇒ ∈O (s, x) IϕP G(s, x) by induction on |(s, x)|ϕ . With Bx we denote the tree given by TP (x), P 1,0 Bx := s {TP (x)} (s) ' 0 . _ If s/∈Bxthen G(s, x)=1∈O.Nowlets∈Bxand n ∈ N.Ifwehaves hni∈Bxthen | _h i | | | _h i ∈O (s n ,x) ϕP < (s, x) ϕP and we obtain G(s n ,x) by the induction hypothesis. If on _ _ the other hand s hni ∈/ Bx then G(s hni,x)=1∈O. Hence (∀n)[G(s_hni,x)∈O]. (ii)

~~_ Since F (n +1)=F(n)+O G(s hni,x)+O 2and F (0) = 1 we get from Lemma 5.4.5 and (ii) (∀n)[F(n)∈O] as well as~~

~~(∀n)[F(n)~~

~~95 1 CK 7. Inductive Deﬁnitions, Π1–sets and the ordinal ω1~~

G(s, x)=3·5z ∈O. For the opposite direction we have to prove ∈O⇒ ∈ G(s, x) (s, x) IϕP 0 | |O ∈ ∅ ∈ ⊆ ∈ by induction on G(s, x) .Fors/Bxwe have ϕP ( ,s,x), thus (s, x) IϕP IϕP .Ifs Bx then G(s, x)=3·5z and 1,0 1,0 _ (∀n) {z} (n +1)={z} (n)+O G(s hni,x)+O 2 .

_ _ From Lemma 5.4.5 we can infer (∀n)[G(s hni,x)∈O], hence |G(s hni,x)|O < |G(s, x)|O _ for all s hni∈Bx. By induction hypothesis this implies ∀ _h i∈ ⇒ _h i ∈ ( n)[s n Bx (s n,x) IϕP ] which is

~~ϕP (IϕP ,s,x). ∈ Hence (s, x) IϕP . As a consequence of Theorem 7.6.6 and the Analytical Hierarchy Theorem we get the following corollary.~~

~~1 O 7.6.7 Corollary There is no Σ1–deﬁnition of . We can even strengthen the statement of Corollary 7.6.7 to get the Boundedness Principle for O.~~

7.6.8 Lemma Let P be a Σ1–deﬁnable subset of O.Then 1 | | ∈ CK sup a O a P <ω1 . Proof: By Theorem 5.4.8 there is a computable function g such that Tree a ∈O ⇒ g(a)∈WT ∧|a|O =otyp (g(a)). (i)

1 Thus g[P ] is a Σ1–deﬁnable subset of WT . Hence the Boundedness Principle (Theorem 7.2.8) and (i) yield | | ∈ Tree ∈ CK sup a O a P = sup otyp (g(a)) a P <ω1 . The tree–like structure of O leads to the following deﬁnition.

~~7.6.9 Deﬁnition Aset P ⊆Owhich is linearly ordered by~~

~~1 O 7.6.10 Corollary There are no Σ1–deﬁnable paths through . 1 O However, as we will see in section 9.1, there are Π1–deﬁnable paths through .~~

~~96 8. The Hyperarithmetical Hierarchy~~

~~8.1 Hyperarithmetical sets~~

~~We are now prepared for the study of inﬁnite iterations of the jump operator.~~

~~8.1.1 Deﬁnition For a ∈Owe put  ∅ if a =1,i.e.|a|O =0 b Ha := j(Hb) if a =2 =16 ,i.e.|a|O =|b|O +1  e hx, yi y~~

~~We say that a set S ⊆ N is hyperarithmetical if there is an a ∈Osuch that S ≤T Ha. The class Hyp := Ha a ∈O is the hyperarithmetical hierarchy.~~

The deﬁnition of the set Ha depends heavily on the ordinal notation a ∈O. It will take some effort to obtain the independence of the hyperarithmetical hierarchy from the ordinal notation. This will be achieved as soon as we are able to prove

~~8.1.2 Theorem For a, b ∈Osuch that |a|O = |b|O we have Ha ≡ Hb.~~

~~The proof needs some effort and is done in several steps. We ﬁrst prove~~

~~8.1.3 Lemma Let a ≤O b.ThenHa ≤mHb. This holds uniformly in a and b, i.e. an index for the reducing function can be computed from a and b.~~

Proof: Each b consists of an m-fold (m ≥ 0) iteration of exponentiations by 2 starting at a b0 ∈ N which is not of the form 2z. We descend this tower of exponentiations until we reach a or until we cannot descend any further. Let c be the element of O we reached and let n be the number (n) of steps we took. By Hb = Hc and (3.2) of Lemma 3.1.3 there exists a computable function f with

~~1,0 Hc ≤m Hb via {f(0,n)} . (i)~~

~~If a = c we are done. Otherwise we have a~~

~~x∈Ha ⇔hx, ai∈Hc 1,0 ⇔{f(0,n)} (hx, ai) ∈ Hb holds. Observe that the algorithm described above terminates even if a/∈O.~~

~~8.1.4 Lemma For a ∈Owe put Oa := x ∈O |x|O <|a|O .~~

~~O a a Then a is computable in H2 uniformly in a,i.e.aH2 –index for χOa is computable from a.~~

~~97 8. The Hyperarithmetical Hierarchy~~

~~Proof: We use the Recursion Lemma along~~

{ }H2a χOa = G(a, e) . We distinguish the following cases. a =1.ThenOa =∅and we choose G(a, e) to be an H2a –index of the empty set. a =2.ThenOa ={1}and we choose G(a, e) to be an H2a –index of {1}. a =2b and b =2c =16 .Then x Oa =Ob∪ 2 x∈Ob , 1,0 so Oa is decidable in Ob and {e} (b) is an H2b –index for Ob. By Lemma 8.1.3 and b2we obtain a as Π1 in H2b , hence decidable in H2a and an H2a –index for Oa is computable from e and a. a =3·5b.Then Oa= x (∃n)[x ∈O{b}(n)] . { } { } O By recursion hypothesis we obtain e ( b (n)) as an H2({b}(n)) –index for {b}(n). Using Lemma 8.1.3 we obtain Oa as semi–decidable in Ha and hence decidable in H2a .AnH2a–index for Oa depends computably on e and a. If a is of any other shape then we put G(a, e):=0. A close look at our construction shows that G(a, e) is deﬁned even if a/∈O. Thus the function g given by the Recursion Lemma is computable. As an easy consequence of Lemma 8.1.4 we obtain the next lemma.

~~8.1.5 Lemma For a ∈O x∈O |x|O =|a|O a a ∈O | | | | is decidable in H22 uniformly in a,i.e.anH22 –index for x xO = aO is computable from a.~~

~~Proof: We get~~

~~a x ∈O∧|x|O =|a|O ⇔ (x∈O∧|x|O <|2 |O)∧¬(x∈O∧|x|O <|a|O).~~

~~By Lemma 8.1.4 the ﬁrst conjunct is decidable in H22a and the second in H2a . Both statements a hold uniformly in a. Thus their conjunction is decidable in H22 uniformly in a.~~

~~8.1.6 Lemma If a ∈Oand b ∈Osuch that |a|O = |b|O then Ha ≤T Hb.~~

~~98 8.1. Hyperarithmetical sets~~

~~Proof: We deﬁne a well–founded predicate~~

~~hc, di~~

~~{ }Hb χHa = g(a, b) (i) holds. Let ha, bi∈ﬁeld(~~

~~h i h i⇒ {{ } }Hd c, d~~

~~{ }Hb χHa = G(e, a, b) . We distinguish the following cases: X a =1.LetG(e, a, b) be an e0 with {e0} = χ∅ for all X ⊆ N. c 6 d h i h i a =2 =1and b =2.Then c, d~~

|a|O < |{u}(n)|O. (ii) | | |{ } | By Lemma 8.1.4 “ a O < u (n) O” is uniformly decidable in H2{u}(n) , which in turn is uniformly decidable in Hb by Lemma 8.1.3. Thus an n satisfying (ii) is uniformly computable in Hb. Because of ha, {u}(n)i

~~{{ } { } }H{u}(n) χHa = e (a, u (n)) .~~

~~By Lemma 8.1.3 H{u}(n) is uniformly decidable in Hb, and so, with some considerable effort, G(e, a, b) cen be deﬁned appropriately. u a =3·5 .Ashc, bi~~

~~y ∈ Ha ⇔ y = hx, ci∧c~~

~~8.1.7 Lemma There is a computable function h such that for every a ∈Othe value h(a) is a 1 ∆1–index for the set Ha.~~

~~Proof: We use the Recursion Lemma (Lemma 5.4.7) along~~

~~∆1 1 where U 1 is the universal predicate for ∆1–sets as deﬁned in Theorem 4.2.6. By this theorem we obtain~~

~~99 8. The Hyperarithmetical Hierarchy~~

~~ 1 1 Σ1 Π1 Hb = x U 1,0 (x) = x U 1,0 (x) . ({e} (b))0 ({e} (b))1 The recursion step consists in deﬁning a partial computable function G such that~~

~~ 1 ∆1 Ha = x UG(a,e)(x) . We distinguish the following cases: ∅ 1 a =1.ThenHa= and we deﬁne G(a, e) to be a ∆1–index of the empty set. c a =2 =16 .Thenc~~

~~x ∈ Ha ⇔ (∃z)(∃s)[Seq(s) ∧ lh(s)=z∧(∀i~~

~~∧ (∀i~~

~~⇔ (∃z)(∃s)[Seq(s) ∧ lh(s)=z∧(∀i~~

~~x ∈ Ha ⇔ (∃z)(∃s)[Seq(s) ∧ lh(s)=z∧(∀i~~

~~8.1.8 Lemma There is a computable function d such that~~

~~a { }H22 ,1,0 WT|a|O = d(a)~~

~~100 8.1. Hyperarithmetical sets for all a ∈O.~~

~~Proof: By (7.13) we have ∈ ⇔ hi ∈ σ x WT σ ( ,x) ITree, hence it sufﬁces to show that there is a computable function g such that for all a ∈Owe have~~

~~a H 22 ,2,0 χ |a|O = { g (a)} . (i) ITr ee~~

~~We are going to prove (i) by the recursion lemma along |a|O. Therefore we have the recursion hypothesis h i H b 22 ,2,0 (∀b~~

~~H 2a ,2,0 χ |a|O = { G(e, a)} 2 . ITr ee We distinguish the following cases: a =1.Wehave 0 ∈ ∧{ }1,0 ∧ ∀ { }1,0 _h i ITree = (s, x) x Tree x (s)=0 ( y) x (s y )=1 .~~

0 0 a a This shows that ITree is Π2 and hence decidable in H22 .WedeﬁneG(e, a) to be an H22 –index 0 of ITree. c 6 |a|O |c|O+1 a =2 =1. Then, using ITree = ITree , we obtain h i | | | | (s, x) ∈ I a O ⇔ x ∈ Tree ∧ (∀y ) {x} 1,0 (s _ hy i)=0⇒(s_hyi,x)∈I c O Tree Tree (ii) 1,0 _ H c,2,0 _ ⇔ x ∈ Tree ∧ (∀y ) {x} (s hy i)=0⇒{{e}(c)} 22 (s hyi,x)=0 .

|a|O ∈ 0 0 c ¬ 0 The formula “x Tree ”isΠ2, the second conjunct in (ii) is Π1 in H22 . Thus ITree is Σ1 in |a|O |a|O c ¬ c ≤ H22 and by Theorem 3.1.1 it follows that ITree is m–reducible to j(H22 ). Hence ITree T |a|O c ≤ a a c { } j(H22 ) T H22 and an H22 –index for ITree is computable from the H22 –index e (c).Since cis computable from a we get a computable function G such that

c a |2 |O { c }H22 ,2,0 G(e, 2 ) = ITree . c a =3·5.Then|a|O ∈Lim and we get h i | | | | (s, x) ∈ I a O ⇔ x ∈ Tree ∧ (∀y ) {x} 1,0 (s _ hy i)=0⇒(s_hyi,x)∈I< a O Tree h Tree i 0 1,0 _ _ |v|O ⇔ x ∈ Tree ∧ (∀y )(∃v) v

~~0 v v ≤ ∈ v ⇔h 2 i∈ But observe that for v~~

H 2a ,2,0 χ |a|O = { G(e, a)} 2 . ITr ee By the Recursion Lemma we get a partial–computable function g such that, for all a ∈O,g(a) is an H22a –index for χ |a|O and we deﬁne d(a) as an index for the set ITr ee 101 8. The Hyperarithmetical Hierarchy

~~ H a ,2,0 x {g(a)} 22 (hi,x)=0 . Observe that d is computable.~~

1 1 8.1.9 Theorem (Characterization Theorem for ∆1–sets) The hyperarithmetical sets are the ∆1– sets. 1 ≤ Proof: It is an easy exercise to show that the ∆1–sets are closed under T . From this and 1 1 Lemma 8.1.7 it follows that every hyperarithmetical set is ∆1. Conversely, if a set S is ∆1 then, ≤ CK according to Theorem 7.4.2, S m WT σ for some σ<ω1 . By Lemma 8.1.8 there is some ∈O ≤ ≤ a a such that S m WT |a|O T H22 . Hence S is hyperarithmetical.

~~8.2 Hyperarithmetical functions~~

8.2.1 Definition A function α: N −→ N is hyperarithmetical if its graph Gα is a hyperarithmetical predicate. Since we are talking about total functions we have α(x) =6 y ⇔ (∃z)[α(x)=z∧z=6 y] which implies ∈ 1 ⇔ ∈ 1 ⇔ ∈ 1 Gα ∆1 Gα Π1 Gα Σ1. 1 Therefore a function is hyperarithmetical if it possesses a Π1–graph. This opens the possibility to 1 define indices for hyperarithmetical functions via the weak Π1–uniformization Theorem (Theo- rem 7.4.5). Though we did not emphasize it in the proof of Theorem 7.4.5 it should be clear that 1 1 a Π1–index of the uniformizing function is computable from a Π1–index of the original predicate via a computable function, say k. Then we define

~~Π1 { }I ' ⇔ 1 e (x) y Uk(e)(x, y) (8.1) and call {e}I a hyperarithmetical index. Note that {e}I is not necessarily total. We denote by H the class of hyperarithmetical functions. Then we obtain~~

~~Π1 { }I ∈ ⇔ ∀ ∃ 1 e H ( x)( y)Uk(e)(x, y) (8.2) 1 which is a Π1–statement. 1 We are going to prove that H is a genuine Π1–relation.~~

~~1 1 8.2.2 Lemma The relation H is Π1 but not Σ1. Proof: Because of α ∈ H ⇔ (∃e)[{e}I ∈ H ∧ (∀x)(α(x)={e}I(x))]~~

1 and (8.2) we obtain H as a Π1–relation. ∈ 1 Now assume H Σ1.Deﬁne ⇔ ∈ ∧ ∈O∧ ≤ ∨ ∧ ∈ P(α, a): (α H a α m WT |a|O ) (a =1 α/H). (i) By Lemma 8.1.8 and Lemma 8.1.7 the predicate Q deﬁned by ⇔ ∈O∧ ∈ Q(x, a): a x WT |a|O 1 is Π1.Since 102 8.2. Hyperarithmetical functions

≤ ⇔ ∃ ∀ ∀ ⇔ ∃ 1,0 h i ∧ ∈ α m WT |a|O ( e)( x)( y) α(x)=y ( z)[T (e, x, y ,z) U(z) WT |a|O ] ∈O 1 1 for a the relation P (α, a) is Π1. Using weak Π1 uniformization we obtain a functional FP 1 whose graph is Π1–deﬁnable. By Theorem 7.4.2 we get (∀α)(∃a)P (α, a)

1 which shows that FP is a total functional, hence the graph of FP is ∆1–deﬁnable. ∈O ∈ CK On the other hand, for every a there is an α H such that α m WT |a|O :Forσ<ω1 we ∈ 1 ∈ 1 ∈ have WT σ ∆1, hence Y := j(WT σ) ∆1 with Y m WT σ. Putting α := χY we get α H and α m WT σ. 1 O Therefore rng(FP ) is Σ1–deﬁnable and unbounded in . This, however, contradicts Lemma 7.6.8.

8.2.3 Theorem (KLEENE’s Basis Theorem) The functions which are computable in the class of 1 1 Σ1–predicates are a basis for the collection of Σ1–classes of functions. 1 The class of ∆1–definable functions is not a basis for this collection and hence not even a basis 0 for the collection of Π1–classes of functions. Proof: The first part is Theorem 7.5.5. For the second part we define a relation P by P (α):⇔α/∈H.

1 Thus P is a nonempty Σ1–relation for which ⇔ ∈ 1 P (α) α/∆1 ∈ 1 1 holds. Obviously there is no β ∆1 with P (β). Thus the class of ∆1–deﬁnable functions is not 1 a basis for the collection of Σ1–classes of functions. The rest follows from Lemma 7.5.2. This theorem has a surprising consequence.

~~8.2.4 Theorem There is a non well–founded decidable tree without inﬁnite hyperarithmetical path (i.e. H thinks that the tree is well–founded).~~

~~0 Proof: By the second part of the last theorem there is a nonempty (0, 1)-ary Π1-relation P with (∀α ∈ H)¬P (α).~~

0 As P is Π1 there is a decidable predicate R such that P (α) ⇔ (∀x)R(α(x)) holds. The tree T := s ∈ Seq (∀s0)(s0 ( s ⇒ R(s0)) is the one we are looking for. A somehow more constructive proof of the last theorem is given on page 107. One further goal of the present section is to show that the class H is a model of the scheme 1 − 01 ∀∃ ⇒ ∃ ∀ (Π1 AC )(x)( α)P (x, α) ( β)( x)P (x, (β)x ) 1 1 − 01 1 where P is a (1, 1)–ary Π1–relation. We call (Π1 AC ) the Π1–axiom of choice of type (0, 1). 1 1 By the weak Π1–uniformization theorem (Theorem 7.4.5) we get for a Π1–predicate P (∀x)(∃y)P (x, y) ⇒ (∃β ∈ H)(∀x)P (x, β(x)). (8.3)

~~1 This shows that H is a model of the Π1–axiom of choice of type (0, 0) 103 8. The Hyperarithmetical Hierarchy~~

~~1 − 00 ∀∃ ⇒ ∃ ∀ (Π1 AC )(x)( y)P (x, y) ( β)( x)P (x, β(x)).~~

~~1 − 01 8.2.5 Theorem The class H of hyperarithmetical functions is a model of (Π1 AC ),i.e.fora 1 (1, 1)–ary Π1–relation P we have~~

~~(∀x)(∃α ∈ H)P (x, α) ⇒ (∃β ∈ H)(∀x)P (x, (β)x).~~

Proof: Using indices for hyperarithmetical functions we get (∀x)(∃α ∈ H)P (x, α) ⇔ (∀x)(∃e) {e}I ∈ H ∧ P (x, {e}I ) . (i) { }I ∈ 1 { }I It follows from (8.1) that e H is a Π1–statement. But we also have for total e P (x, {e}I ) ⇔ (∀α) ((∀z)(∀y)({e}I (z)=y⇒α(z)=y)) ⇒ P (x, α)

1 which shows that the expression in square brackets in (i) is Π1. Thus starting with (∀x)(∃α ∈ H)P (x, α) we get by (i) and (8.3) a hyperarithmetical function γ such that (∀x) {γ(x)}I ∈ H ∧ P (x, {γ(x)}I ) . We deﬁne a total function β by {γ((u) )}I ((u) ) if Seq(u) ∧ lh(u)=2 β(u):= 0 1 0 otherwise and easily see I (β)x = {γ(x)} . Furthermore we obtain β(ha, bi) ' y ⇔{γ(a)}I(b)'y Π1 ⇔ 1 Uk(γ(a))(b, y) 1 ∈ which shows that β has a Π1–graph. Hence β H and

(∀x)P (x, (β)x ). 1 − 01 The next goal is to show the class of hyperarithmetical functions is characterized by (Π1 AC ). This needs some preparation. Our ﬁrst observation is that the stages Ha can be deﬁned implicitly. Let a ∈O.Then

~~x∈Ha ⇔ (a=1∧x=6 x) ∨ ∃ z 6 ∧ ∈ ( z)[a=2 =1 x j(Hz)] ∨ ∃ · z ∧ 0 ∧ ∈ ∧ ∧ ( z) a =3 5 (x)1~~

~~104 8.2. Hyperarithmetical functions~~

(∀x)(α(x) ≤ 1 ∧ (α(x)=0⇒Seq(x) ∧ lh(x)=2)) (8.5) 0 ∧(∀a)(¬a ≤O b ⇒ (∀x)(α(hx, ai)=1)) 0 ∧(∀a)(∀z)(a ≤O b ⇒ [a =1⇒(∀x)(α(hx, ai)=1) ∧(a=2z =16 ⇒(∀x)(α(hx, ai)=0 ⇔(∃u)(∃s)(Seq(s) ∧ lh(s)=u

~~∧(∀i~~

~~Hyp(b, H≤b). (8.6) On the other hand if b ∈Othen we have~~

~~Hyp(b, α) ⇒ α = H≤b. (8.7) To prove (8.7) we show~~

~~α(hx, ai)=0 ⇔ a≤O b∧x∈Ha (8.8) by induction on |a|O. But (8.8) is more or less obvious from the induction hypothesis, the deﬁni- tion (8.5) and (8.4). Summarizing we get~~

~~8.2.6 Lemma There is an (1, 1)–ary arithmetical relation Hyp such that for b ∈Owe have~~

~~Hyp(b, α) ⇔ α = H≤b.~~

~~8.2.7 Lemma Let M be a nonempty collection of functions which is closed under ≤T and satis- 1 − 01 ∈O ∈M ﬁes (∆0 AC ).Thenb implies H≤b . Proof: We prove~~

b ∈O⇒H≤b ∈M by induction on |b|O. For b =1we have H≤b = χ∅. Hence H≤b is computable. But since M is nonempty and closed under ≤T it contains all computable functions. Let b =2c =16 .Then c H≤b(hx, ai)=0 ⇔ (a=2 ∧x∈j(Hc)) ∨ (H≤c(hx, ai)=0). (i)

~~It follows from (i) that H≤b is semi–decidable in H≤c. Therefore there is a decidable relation R such that~~

~~H≤b(x)=0 ⇔ (∃z)R(H≤c,x,z). Deﬁne Q(α, x, y):⇔Hyp(c, α) ∧ y ≤ 1 ∧ [y =0 ⇔ (∃z)R(α, x, z)] . By Lemma 8.2.6 and b ∈Owe obtain~~

~~(∀x)(∀y)[(∃α)Q(α, x, y) ⇒ H≤b(x)=y]. (ii)~~

~~105 8. The Hyperarithmetical Hierarchy~~

~~Since |c|O < |b|O we get by the induction hypothesis and Lemma 8.2.6 (∃α ∈M)[Hyp(c, α)] which implies (∀x)(∃y)(∃α ∈M)Q(α, x, y).~~

~~Since M is closed under ≤T we get by contraction of quantiﬁers~~

~~(∀x)(∃β ∈M)Q((β)0,x,(β)1(0)). (iii) M| 1− 01 As =(∆0 AC ) and Q is arithmetical we obtain from (iii)~~

~~(∃γ ∈M)(∀x)Q((γ)x0,x,(γ)x1(0)) (iv) andby(ii)and(iv)~~

~~(∀x)[H≤b(x)=(γ)x1(0)] .~~

~~Hence H≤b = λx. (γ)x1 (0) and H≤b ∈Msince M is closed under ≤T . Let b =3·5e.Then h i ⇔ ∧ ∈ ∨ ∧ ∈ H≤b(z,a )=0 [a=b z Hb] [a~~

~~∧ ([a = b ∧ Seq(z) ∧ lh(z)=2∧(∃n)((α)n(z)=0)] (v) 0 ∨ [a~~

~~Q(α, x, y):⇔(∀n)[Hyp({e}(n), (α)n)] ∧ y ≤ 1 ∧ (y =0⇔R(α, x)). By Lemma 8.2.6 and (v) we get~~

~~(∀x)(∀y)[(∃α)Q(α, x, y) ⇒ H≤b(x)=y]. (vi) The induction hypothesis yields (∀n)(∃α ∈ H)[Hyp({e}(n),α)] (vii) M| 1− 01 which entails by =(∆0 AC )~~

~~(∃α ∈ H)(∀n)[Hyp({e}(n), (α)n)] . (viii) From (viii), however, we get (∀x)(∃α ∈M)(∃y)Q(α, x, y) which, analogous to the previous case, yields~~

~~(∀x)(∃β ∈M)Q((β)0,x,(β)1(0)). M| 1− 01 Using =(∆0 AC ) we obtain~~

~~(∃γ ∈M)(∀x)Q((γ)x0,x,(γ)x1(0)) and ﬁnally we get from (vi)~~

~~(∀x)[H≤b(x)=γx1(0)],~~

~~106 8.3. The hyperarithmetical quantiﬁer theorem i.e.~~

~~H≤b = λx. (γ)x1 (0).~~

~~Hence H≤b ∈M. Summing up we have shown~~

8.2.8 Theorem The collection H of hyperarithmetical functions is the with respect to set inclusion 1 − smallest nonempty class of functions which is closed under “computable in” and satisﬁes (∆0 01 | 1− 01 AC ). We even have H =(Π1 AC ).

~~8.3 The hyperarithmetical quantiﬁer theorem~~

CK CK If we regard all ordinals below ω1 as given, i.e. we allow bounded search over ω1 ,thenall arithmetical predicates are decidable and so are all the sets Ha. In that sense we may regard the 1 collection H of hyperarithmetical functions as computable and ∆1–sets as decidable. The aim of 1 the present section is to show that in that interpretation the Π1–sets play the role of semi–decidable sets. We introduce some notations. If ϕ is an analytical formula we denote by ϕH the formula which is obtained from ϕ by restricting all function quantifiers to functions in H.Then 1,H H ∈ 1 Σn = ϕ ϕ Σn and dually 1,H H ∈ 1 Πn = ϕ ϕ Πn . It is quite easy to see that 1,H ⊆ 1 Σ1 Π1. (8.9) This follows by induction from (∃α ∈ H)(∀x)P (a,α,x) ⇔ (∃e)(∀x) {e}I ∈ H ∧ P (a,λy.{e}I(y),x) ⇔ (∃e)(∀α)(∀x) {e}I ∈ H ∧ ((∀y)(∀z)({e}I (y)=z⇒α(y)=z) ⇒P(a,α,x))] . We can now give an alternative proof of Theorem 8.2.4 where we showed that there is a non well–founded decidable tree without infinite hyperarithmetical path. Proof: We show that there is a decidable predicate R such that (∃α)(∀x)R(α(x)) ∧¬(∃α∈H)(∀x)R(α(x)). Putting T := s (∀s0)[s0 (s⇒R(s0)] we have a tree as desired. To construct R we define 1 Σ1,1,0 1,1 K 1 := x x ∈ Ux = x (∃α) (α, x) ∈/ W . Σ1 x Now let ∃ ∈ ∈ 1,1 M := x ( α H) (α, x) / Wx .

~~1,H 1 1 Then M ⊆ K 1 and M ∈ Σ ⊆ Π by (8.9). Let e be a Π –index for M. Then we obtain Σ1 1 1 1 107 8. The Hyperarithmetical Hierarchy~~

~~1 Π1 e ∈ Ue ⇔ e ∈ M~~

~~⇒ e ∈ K 1 Σ1 1 Σ1,1,0 ⇔ e ∈ Ue 1 Π1,1,0 ⇔ e/∈Ue~~

~~1 1 1 1 Π1 Σ1 Π1 Σ1 since we deﬁned Ue as the complement of Ue . Hence e/∈Ue which entails e ∈ Ue . Therefore~~

e/∈M∧e∈K 1. Σ1 Let P be a decidable predicate such that ∈ 1,1 ⇔ ∃ (α, e) We ( z)P (α(z)). From e/∈Mit follows ¬(∃α ∈ H)(∀z)¬P (α(z)) and from e ∈ K 1 Σ1 (∃α)(∀z)¬P (α(z)). Choosing R := ¬P the proof is terminated. In order to obtain also the opposite inclusion in (8.9) we need some preparations. It is obvious that Lemma 8.2.2 relativizes. I.e. we introduce the class A ∈ 1 H := α Gα ∆1[A] and obtain A ∈ 1 \ 1 H Π1[A] Σ1[A]. (8.10) Another obvious observation is that HA is closed under relativizations, i.e. α ∈ HA ⇒ HA,α = HA. (8.11) A ⊆ A,α ∈ A,α 1 This holds since we have H H and for β H the graph Gβ is a ∆1[A, α]–predicate. 1 1 But α has a ∆1[A] graph and the ∆1[A]–predicates are closed under substitution with functions 1 ∈ A having ∆1[A] graphs. Hence β H . Let 1,HA HA ∈ 1 Σ1 [A]:= ϕ ϕ Σ1[A] and 1,HA HA ∈ 1 Π1 [A]:= ϕ ϕ Π1[A] . There are universal predicates

~~A Σ1,H [A] 1 ∃ ∈ A ∈ A,1,1 Ue := x ( α H )[(x, α) / We ] A Π1,H [A] 1 ∀ ∈ A ∈ A,1,1 Ue := x ( α H )[(x, α) We ]~~

~~1,HA 1,HA 1,HA and we introduce ∆1 [A]–indices as pairs of Σ1 [A]–andΠ1 [A]–indices which describe the same sets. We show the following lemma.~~

~~8.3.1 Lemma For a ∈ WTA B B TreeB TreeA WT A := x ∈ WT otyp (x) ≤ otyp (a) otypTree (a) 108 8.3. The hyperarithmetical quantiﬁer theorem~~

~~1,HA,B is a ∆1 [A, B] set. This holds uniformly, i.e. there is a computable function g such that g(a) A,B is a ∆1,H [A, B]–index for WTB . Moreover, g is independent of A and B. 1 otypTreeA (a)~~

CK A Proof: We use the Recursion Lemma along ω1 [A].Letrest (a, n) be an A–index of the restric- tion of the tree {a}A to the node hni,i.e. {restA(a, n)}A = χ . s {a}A(hni_s)=0 We obtain x ∈ WT B ⇔ x ∈ Tree B otypTreeA (a) ∧ (∀z )[{x}B(hzi)=0 (i) ⇒(∃m)({a}A(hmi)=0∧restB (x, z) ∈ WT B )]. otypTreeA (restA(a,m))

~~A A Because of otypTree (restA(a, m)) < otypTree (a) we get by the recursion hypothesis~~

A,B Σ1,H [A,B] ∈ B ⇔ ∈ 1 u WT A u U A otypTree (restA(a,m)) ({e}(rest (a,m)))0 A,B Π1,H [A,B] ⇔ ∈ 1 u U { } A ( e (rest h(a,m)))1 i (ii) ⇔ ∃ ∈ A,B ∈ A,B,1,1 ( α H ) (u, α) / W { } A h ( e (rest (a,m)))0 i ⇔ ∀ ∈ A,B ∈ A,B,1,1 ( α H ) (u, α) W A . ({e}(rest (a,m)))1 Inserting (ii) into (i) and remembering that HA,B is a model of (Π1−AC01) shows that WT B 1 otypTreeA (a) 1,HA,B is a ∆1 [A, B] set whose index can be computed from e and a. Note that the computable function g given by the Recursion Lemma is independent of A and B. As a consequence of Lemma 8.3.1 we obtain

8.3.2 Theorem 1,HA 1 ∆1 [A]=∆1[A] 1,HA ⊆ 1 Proof: The inclusion ∆1 [A] ∆1[A] follows from (8.9). The converse inclusion is a conse- 1 quence of Lemma 8.3.1 and the relativization of Theorem 7.4.2 which says that every∆1[A]–set is A A CK 1,H m–reducible to WT σ for some σ<ω1 [A]. The result now follows from the fact that ∆1 [A] is closed under m–reducibility. Now we have all the ingredients for one of the main results of this lecture.

~~8.3.3 Theorem (Hyperarithmetical Quantiﬁer Theorem)~~

1 1,HA Π1[A]=Σ1 [A]. Proof: The easy direction from right to left is (8.9). A 1 Because WT is Π1[A]–complete it sufﬁces to show A ∈ 1,HA WT Σ1 [A] (i) 1,HA to obtain also the converse inclusion, as Σ1 [A] is obviously closed under m-reducibility. Since A ∈ 1 ∈ N H Π1[A] there is a computable function f and an e such that α ∈ HA ⇔ λx. f(α, x) ∈ WTA (ii) ⇔ e ∈ WT A,α, 109 8. The Hyperarithmetical Hierarchy

A,α where e is a uniform A, α–index for λx. f(α, x). Note that otypTree (e) varies with α ∈ HA. We show ∀ CK ∃ ∈ A ≤ TreeA,α ( σ<ω1 [A])( α H )[σ otyp (e)] (iii) indirectly and assume ∃ CK ∀ ∈ A TreeA,α ( σ<ω1 [A])( α H )[otyp (e) <σ]. But this entails A ∈ WTA H = α λx. f(α, x) σ WTA 1 which contradicts (8.10) since σ is a ∆1[A]–relation. A,α Note that for α ∈ HA we have otypTree (e) <ωCK[A]. Thus we obtain by (iii) and Lemma 8.3.1 1 ∈ A ⇔ ∃ CK ∈ A a WT ( σ<ω1 [A]) a WT σ ⇔ (∃α ∈ HA)[a ∈ WT A ] otypTreeA,α (e) (iv) ⇔ (∃α ∈ HA)(∃β ∈ HA,α[(β,a) ∈/ WA,α ]). (g(e))0

~~A ∈ A A,α A 1,H A But since α H we have H = H and (iv) yields a Σ1 [A] deﬁnition for WT .~~

~~110 9. Appendix~~

~~1 O 9.1 A Π1–path through~~

1 From the Boundedness Principle we inferred in Corollary 7.6.10 that there are no Σ1–paths O 1 O through . We want to show that there are indeed Π1–paths through and present a construction which is – as far as we know – due to SPECTOR and FEFERMAN.

~~1 O 9.1.1 Theorem There is a Π1–path through . Proof: We introduce the set 0 0 Pd~~

~~⇔ ∀∈ 0 ∨ ∃ c ψ(a): (x Pd~~

~~∈ 0 ⇒ ∈O b Pd~~

~~0 ∩O P := Pd~~

~~111 9. Appendix~~

~~ ⊆O ∈O | | ≤| | P |b|O := c cO bO (iii)~~

~~∈O 0 ∩O 1 for some b .ButthenP=Pd~~

~~112 Bibliography~~

~~Bibliography~~

~~[1]P.G.HINMAN, Recursion-theoretic hierarchies, Perspectives in Mathematical Logic, Springer-Verlag, Heidelberg/New York, 1978.~~

~~[2] Y. N. MOSCHOVAKIS, Elementary induction on abstract structures, Studies in Logic and the Foundations of Mathematics, vol. 77, North-Holland Publishing Company, Amsterdam, 1974.~~

~~[3] P. ODIFREDDI, Classical recursion theory, Studies in Logic and the Foundations of Math- ematics, vol. 125, North-Holland Publishing Company, Amsterdam, 1989.~~

~~[4] W. POHLERS AND T. GLASS, An introduction to mathematical logic. Institut f¨ur Mathema- tische Logik und Grundlagenforschung, M¨unster, 1992.~~

~~[5] H. ROGERS JUN., Theory of recursive functions and effective computability,McGraw- Hill Book Company, New York, 1967.~~

~~[6]G.E.SACKS, Higher recursion theory, Perspectives in Mathematical Logic, Springer- Verlag, Heidelberg/New York, 1990.~~

~~[7] H. SCHWICHTENBERG, Rekursive Ordinalzahlen, hyperarithmetische Mengen und minimale Modelle der Analysis. Institut f¨ur Mathematische Logik und Grundlagenforschung, M¨unster, 1974.~~

~~[8]J.R.SHOENFIELD, Mathematical logic, Addison-Wesley Publishing Company, Reading, 1967.~~

~~[9] R. I. SOARE, Recursively enumarable sets and degrees, Perspectives in Mathematical Logic, Springer-Verlag, Berlin/Heidelberg/New York, 1987.~~

~~113 Index~~

~~Index~~

~~Notations~~

m,n F : N −→ p N ,5 otyp(≺),43 a(k),5 On,43 (α, ~y,β~),5 z,43 F(a)'G(a),6 σ<τ,43 ↑,6 ω,45 G ,6 PO(α),45 F LO χ ,7 (α),46 A WF sk,9 (α),46 WO Ind(P ),11 (α),46 ⊆ Tm,11 s t,46 T Wn,11 (B),46 e T {e}m,n,11 (α),, 46 ≤∗ Wm,n,12 B,48 e <∗,48 Rec(G, H),13 B Bs,48 A≤ B,15 m sx,49 A≤ B,15 T

~~114 Index~~

σ IΓ ,63 <σ IΓ ,64 ||Γ||,64 |n|Γ,64 Iϕ,66 ||ϕ||,66 |n|ϕ,66 Iϕ,Iϕ(α~),66 ≤∗ ~x ϕ,ψ ~y,71 ∗ ~x<ϕ,ψ ~y,71 ≤∗,<∗,72 |~x|S,73 |(α,~ ~x)|S,73 ϕT(X, x),76 IT,76 κA,78 WT σ,84 Ha,97 Hyp,97 {e}I, 102 {e}I, 102 H, 102 H≤b, 105 ϕH, 107 1,H Σn , 107 1,H Πn , 107 HA, 108 1,HA Σ1 [A], 108 1,HA Π1 [A], 108 1,HA ∆1 [A], 108

~~115 Index~~

Key–words acceptable universal class, 32 Formula accessible part, 62 analytic, 35 0 Application, 9 ∆0–, 27 0 Arithmetical Hierarchy Theorem, 31 Π1–, 27 1 arithmetical inductive, 66 Π1–, 35 1 Πk+1–, 35 0 Bar induction, 47 Πn+1–, 27 0 Basis, 90 Σ1–, 27 1 Boundedness Principle, 82 Σ1–, 35 Branching bound 1 Σk+1–, 35 0 of a tree, 49 Σn+1–, 27 FRIEDBERG–MUCHNIK Theorem, 19 1 Characterization Theorem for ∆1–sets, 102 Function, 5 Clauses, 61 hyperarithmetical, 102 closed implicitly defined, 92 under a system of clauses, 61 Functional, 5 under an operator, 62 characteristic, 7 Closure ordinal computable, 6 inductive, 78 partial–computable, 6 of an inductive definition, 64 total, 5 Closure Properties of arithmetical relations, 28 Graph of a functional, 6 Closure Theorem, 78 complement of a set, 15 Hierarchy computable, 6 hyperarithmetical, 97 in a set, 14 hyperarithmetical, 97 in α,14 Hyperarithmetical Quantifier Theorem, 109 Contraction of a binary predicate, 45 Index Course of values, 5 A–, 14 α–, 14 decidable hyperarithmetical, 102 in a set, 14 K–, 32 in α,14 of a functional, 11 decidable in of a relation, 12 for sets, 16 Induction Definition by cases, 9 on inductive definitions, 61 Degree, 18 inductive, 66 m–, 15 inductive definition TURING–, 18 generalized, 62 inductive norm, 64, 73 Equivalence inductively definable, 63 of well–orders, 42 inductively defined by a system of clauses, 61 Field infix notation, 41 of a predicate, 41 Initial segment Finiteness Theorem, 49 of a sequence number, 46 Fixed–point of an operator, 62 Jump the — of an operator, 63 of a set, 18

~~116 Index~~

operator, 18, 25 Predicate, 5 coinductive, 69 KLEENE’s Basis Theorem, 103 hyperelementary, 69 KONIG¨ ’s Lemma, 48 irreflexive, 41 Weak, 49 strict, 41 KREISEL’s Basis Theorem, 92 well–founded, 42 Projection Lemma first order, 7 KONIG¨ ’s, 48 second order, 7 Recursion, 56 Substitution, 8 Recursion Lemma, 56 Weak KONIG¨ ’s, 49 Recursion operator, 13 Recursion Theorem, 12 Normal form for semi–decidable predicates, 83 1 for Π1–relations, 75 Relativized, 14 Normal–form Theorem, 11 recursively enumerable, 5 Relativized, 14 Reducibility 1–, 15 Operator m–, 15 arithmetically definable, 65 many–one, 15 definable, 65 one–one, 15 elementary definable, 65 Reduction Theorem, 85 monotone, 62 Reflection positively definable, 66 0 Σ1–, 83 order–type, 42 Relation, 5 Ordering, 41 analytical in A,36 countable partial, 41 arithmetical, 28 partial, 41 arithmetical in A,28 strict, 41 coinductive, 69 well–, 42 decidable, 9 Ordinal, 42 1 ∆k,36 0 finite, 45 Πn[A],28 LEENE 1 K –recursive, 54 Πn,36 1 limit–, 44 Πn[A],36 recursive, 51 recursive, 9 in A,51 recursively enumerable, 5 successor–, 44 semi–decidable, 5 0 Σn[A],28 partial–computable, 6 1 Σn,36 in, 13 1 Σn[A],36 in a set, 14 universal, 31 Path Root in a tree, 46 of a tree, 46 in O,96 through a tree, 46 Satisfaction through O,96 of clauses, 61 1 Π1–complete, 76 Search operator, 8 positively decidable, 5 bounded, 8 POST’s Theorem, 17 Segment pre–well–ordering, 79 induced by an element, 43 Predecessor of an ordering, 43 immediate in a tree, 46 proper, 43

117 Index semi–decidable, 5 recursion along R,42 in a set, 14 Tree, 46 in α,14 binary, 49 semi–recursive, 5 boundedly branching, 49 Separation Theorem, 86 ﬁnitely branching, 48 Set well–founded, 46 Γ–closed, 62 tree of unsecured sequences, 75 singleton, 92 TURING–equivalent, 18 Stage TURING–reducible, 16 Comparison predicate, 70 Uniformization 1 Theorem, 71 Π1–, 88 1 of a ﬁxed–point, 63 weak Π1–, 87 strictly order–preserving, 42 Weak KONIG¨ ’s Lemma, 49 Substitution Lemma, 8 Substitution operator, 8 Substitution Theorem, 69 Subtree above a node, 48 Successor immediate in a tree, 46 of an ordinal, 44 System of clauses, 61

Theorem Arithmetical Hierarchy, 31 Bar induction, 47 1 Characterization of ∆1–sets, 102 Closure, 78 Finiteness, 49 Hyperarithmetical Quantiﬁer, 109 KLEENE’s Basis, 103 KREISEL’s Basis, 92 of FRIEDBERG and MUCHNIK,19 POST,17 Recursion, 12 for semi–decidable predicates, 83 Reduction, 85 Relativized Normal–form, 14 Relativized Recursion, 14 m,n Relativized Sk ,14 Separation, 86 Stage Comparison, 71 Substitution, 69 Uniformization 1 strong Π1–, 88 1 weak Π1–, 87 Transﬁnite induction along R,42

~~118~~