THE VEBLEN FUNCTIONS FOR COMPUTABILITY THEORISTS
ALBERTO MARCONE AND ANTONIO MONTALBAN´
Abstract. We study the computability-theoretic complexity and proof-theo- retic strength of the following statements: (1) “If is a well-ordering, then so X is ε ”, and (2) “If is a well-ordering, then so is ϕ(α, )”, where α is a fixed X X X computable ordinal and ϕ represents the two-placed Veblen function. For the former statement, we show that ω iterations of the Turing jump are necessary + in the proof and that the statement is equivalent to ACA0 over RCA0. To prove the latter statement we need to use ωα iterations of the Turing jump, and we 0 show that the statement is equivalent to Πωα -CA0. Our proofs are purely computability-theoretic. We also give a new proof of a result of Friedman: the statement “if is a well-ordering, then so is ϕ( , 0)” is equivalent to ATR0 X X over RCA0.
1. Introduction The Veblen functions on ordinals are well-known and commonly used in proof theory. Proof theorists know that these functions have an interesting and complex behavior that allows them to build ordinals that are large enough to calibrate the consistency strength of different logical systems beyond Peano Arithmetic. The goal of this paper is to investigate this behavior from a computability viewpoint. The well-known ordinal ε0 is defined to be the first fixed point of the function ω α ωα, or equivalently ε = sup ω,ωω,ωω ,... . In 1936 Gentzen [Gen36], !→ 0 { } used transfinite induction on primitive recursive predicates along ε0, together with finitary methods, to give a proof of the consistency of Peano Arithmetic. This, combined with G¨odel’s Second Incompleteness Theorem, implies that Peano Arith- metic does not prove that ε0 is a well-ordering. On the other hand, transfinite induction up to any smaller ordinal can be proved within Peano Arithmetic. This makes ε0 the proof-theoretic ordinal of Peano Arithmetic. This result kicked offa whole area of proof theory, called ordinal analysis, where the complexity of logical systems is measured in terms of (among other things) how much transfinite induction is needed to prove their consistency. (We refer the reader to [Rat06] for an exposition of the general ideas behind ordinal analysis.) The proof-theoretic ordinal of many logical systems have been calculated. An example + that is relevant to this paper is the system ACA0 (see Section 2.4 below), whose proof-theoretic ordinal is ϕ2(0) = sup ε0,εε ,εε ,... ; the first fixed point of the { 0 ε0 } epsilon function [Rat91, Thm. 3.5]. The epsilon function is the one that given γ, α returns εγ , the γth fixed point of the function α ω starting with γ = 0. The Veblen functions, introduced in 1908 [Veb08!→ ], are functions on ordinals that are commonly used in proof theory to obtain the proof-theoretic ordinals of predicative theories beyond Peano Arithmetic. ϕ (α)=ωα. • 0 Date: last saved: July 5, 2010 Compiled: July 5, 2010. Marcone’s research was partially supported by PRIN of Italy. Montalb´an’sresearch was par- tially supported by NSF grant DMS-0901169. We thank the referees for their careful reading of the first draft of the paper and their many suggestions for improving the exposition. 1 2 ALBERTO MARCONE AND ANTONIO MONTALBAN´
ϕβ+1(α) is the αth fixed point of ϕβ starting with α = 0. • when λ is a limit ordinal, ϕ (α) is the αth simultaneous fixed point of all • λ the ϕβ for β<λ, also starting with α = 0.
Note that ϕ1 is the epsilon function. The Feferman-Sch¨utte ordinal Γ0 is defined to be the least ordinal closed under the binary Veblen function ϕ(β,α)=ϕβ(α), or equivalently
Γ0 = sup ϕ0(0),ϕϕ (0)(0),ϕϕ (0)(0),... . { 0 ϕ0(0) } Γ0 is the proof-theoretic ordinal of Feferman’s Predicative Analysis [Fef64, Sch77], 1 and of ATR0 [FMS82] . Again, this means that the consistency of ATR0 can be proved by finitary methods together with transfinite induction up to Γ0, and that ATR0 proves the well-foundedness of any ordinal below Γ0. 1 Sentences stating that a certain linear ordering is well-ordered are Π1. So, even if they are strong enough to prove the consistency of some theory, they have no set-existence implications. However, a sentence stating that an operator on linear 1 orderings preserves well-orderedness is Π2, and hence gives rise to a natural reverse mathematics question. The following theorems answer two questions of this kind. Theorem 1.1 (Girard, [Gir87, p. 299]). Over RCA , the statement “if is a 0 X well-ordering then ωX is also a well-ordering” is equivalent to ACA0.
Theorem 1.2 (H. Friedman, unpublished). Over RCA0, the statement “if is a well-ordering then ϕ( , 0) is a well-ordering” is equivalent to ATR . X X 0 Let F be an operator on linear orderings. We consider the statement WOP(F): ( is a well-ordering = F( ) is a well-ordering). ∀X X ⇒ X We study the behavior of F by analyzing the computational complexity of the proof of WOP(F) as follows. The statement WOP(F) can be restated as “if F( ) has a descending sequence, then has a descending sequence to begin with”. GivenX F, the question we ask is: X Given a linear ordering and a descending sequence in F( ), how difficult is to build a descendingX sequence in ? X X From Hirst’s proof of Girard’s result [Hir94], we can extract the following answer for F( )=ωX . X Theorem 1.3. If is a computable linear ordering, and ωX has a computable X descending sequence, then 0! computes a descending sequence in . Furthermore, there exists a computable linear ordering with a computable descendingX sequence X in ωX such that every descending sequence in computes 0!. X The first statement of the theorem follows from the results of Section 3, which includes the upper bounds of the computability-theoretic results and the “forward directions” of the reverse mathematics results. We include a proof of the second statement in Section 4, where we modify Hirst’s idea to be able to apply it on our other results later. In doing so, we give a new definition of the Turing jump which, although computationally equivalent to the usual jump, is combinatorially easier to manage. This allows us to define computable approximations to the Turing jump, and we can also define a computable operation on trees that produces trees whose paths are the Turing jumps of the input tree. Furthermore, our definition of the Turing jump behaves nicely when we take iterations. In Section 5 we use these features of our proof of Theorem 1.3. First, in Section 5.1 we consider finite iterations of the Turing jump and of ordinal exponentiation.
1 for the definition of ATR0 and of other subsystems of second order arithmetic mentioned in this introduction see Section 2.4 below. THE VEBLEN FUNCTIONS FOR COMPUTABILITY THEORISTS 3
n, (We write ω" X# for the nth iterate of the operation ωX ; see Definition 2.2.) In Theorem 5.3, we prove:
n, Theorem 1.4. Fix n N. If is a computable linear ordering, and ω" X# has a computable descending∈ sequence,X then 0(n) computes a descending sequence in . Conversely, there exists a computable linear ordering with a computable X n, X descending sequence in ω" X# such that the jump of every descending sequence in computes 0(n). X From this, in Section 5.4, we obtain the following reverse mathematics result.
n, Theorem 1.5. Over RCA , n WOP( ω" X#) is equivalent to ACA! . 0 ∀ X!→ 0 The first main new result of this paper is obtained in Section 5.2 and analyzes the complexity behind the epsilon function. Theorem 1.6. If is a computable linear ordering, and ε has a computable de- scending sequence,X then 0(ω) can compute a descending sequenceX in . Conversely, there is a computable linear ordering with a computable descendingX sequence in ε such that the jump of every descendingX sequence in computes 0(ω). X X We prove this result in Theorems 3.4 and 5.21. Then, as a corollary of the proof, we obtain the following result in Section 5.4. + Theorem 1.7. Over RCA0, WOP( ε ) is equivalent to ACA0 . X!→ X Our proof is purely computability-theoretic and plays with the combinatorics of the ω-jump and the epsilon function. By generalizing the previous ideas, we obtain a new definition of the ω-Turing jump, which we can also approximate by a computable function on finite strings and by a computable operator on trees. An important property of our ω-Turing jump operator is that it is essentially a fixed point of the jump operator: for every real Z, the ω-Turing jump of Z is equal to the ω-Turing jump of the jump of Z, except for the first bit (we mean equal as sequences of numbers, not only Turing equivalent). Notice the analogy with the ε and ω operators. After a draft of the proof of Theorem 1.7 was circulated, Afshari and Rathjen [AR09] gave a completely different proof using only proof-theoretic methods like cut-elimination, coded ω-models and Sch¨utte deduction chains. They prove that WOP( ε ) implies the existence of countable coded ω-models of ACA0 con- X!→ X + taining any given set, and that this in turn is equivalent to ACA0 . To this end they prove a completeness-type result: given a set Z, they can either build an ω-model of ACA0 containing Z as wanted, or obtain a proof tree of ‘0=1’ in a suitable logical system with formulas of rank at most ω. The latter case leads to a contradiction as follows. The logical system where we get the proof tree has cut elimination, increasing the rank of the proof tree by an application of the ε operator. Using WOP( ε ), being the Kleene-Brouwer ordering on the proof tree of ‘0=1’, they obtainX!→ aX well-foundedX cut-free proof tree of ‘0=1’. In Section 6, we move towards studying the computable complexity of the Veblen functions. Given a computable ordinal α, we calibrate the complexity of WOP( ϕ(α, )) with the following result, obtained by extending our definitions toXω!→α- TuringX jumps. Theorem 1.8. Let α be a computable ordinal. If is a computable linear or- α dering, and ϕ(α, ) has a computable descending sequence,X then 0(ω ) computes a descending sequenceX in . Conversely, there is a computable linear ordering such that ϕ(α, ) has a computableX descending sequence but every descending sequenceX X α in computes 0(ω ). X 4 ALBERTO MARCONE AND ANTONIO MONTALBAN´
This result will follow from Theorem 3.6 and Theorem 6.15. In Section 6.3, as a corollary, we get the following result.
Theorem 1.9. Let α be a computable ordinal. Over RCA0, WOP( ϕ(α, )) 0 X!→ X is equivalent to Πωα -CA0. Exploiting the uniformity in the proof of Theorem 1.8, we also obtain a new purely computability-theoretic proof of Friedman’s result (Theorem 1.2). Before our proof, Rathjen and Weiermann [RW] found a new, fully proof-theoretic proof of Friedman’s result. They use a technique similar to the proof of Afshari and Rathjen mentioned above. Friedman’s original proof has two parts, one computability- theoretic and one proof-theoretic. The table below shows the systems studied in this paper (with the exception of ACA0! ). The second column gives the proof-theoretic ordinal of the system, which were calculated by Gentzen, Rathjen, Feferman, and Sch¨utte. The third column gives the operator F on linear orderings such that WOP(F) is equivalent to the given system. The last column gives references for the different proofs of these equivalences in historical order ([MM] refers to this paper). System p.t.o. F( ) references X ACA0 ε0 ωX Girard [Gir87]; Hirst [Hir94] + ACA0 ϕ2(0) ε [MM]; Afshari-Rathjen [AR09] 0 X Π α -CA ϕ (0) ϕ(α, ) [MM] ω 0 α+1 X ATR Γ ϕ( , 0) Friedman [FMW]; Rathjen-Weiermann [RW]; [MM] 0 0 X Notice that in every case, the proof-theoretic ordinal equals sup F(0), F(F(0)), F(F(F(0))),... . { } 2. Background and definitions 2.1. Veblen operators and ordinal notation. We already know what the ω, ε and ϕ functions do on ordinals. In this section we define operators ω, ε and ϕ, that work on all linear orderings. These operators are computable, and when they are applied to a well-ordering, they coincide with the ω, ε and ϕ functions on ordinals. To motivate the definition of ωX we use the following observation due to Cantor [Can97]. Every ordinal below ωα can be written in a unique way as a sum
β0 β1 βk 1 ω + ω + + ω − , ··· where α>β0 β1 βk 1. ≥ ≥···≥ − Definition 2.1. Given a linear ordering , ωX is defined as the set of finite strings <ω X x0,x1, . . . , xk 1 (including the empty string) where x0 x1 − X X X ' (∈X x0 x1 x≥k 1 ≥ ···≥ xk 1. We think of x0,x1, . . . , xk 1 ωX as ω +ω + +ω − . The ordering − ' − (∈ ··· on ωX is the lexicographic one: x0,x1, . . . , xk 1 ωX y0,y1, . . . , yl 1 if either k l and x = y for every i n, 0, n+1, ω" X# Definition 2.2. Given a linear ordering , let ω" X# = and ω" X# = ω . X X To motivate the definition of the ε operator we start with the following obser- vations. On the ordinals, the closure of the set 0 under the operations + and t ωt, is the set of the ordinals strictly below ε .{ The} closure of 0,ε under the !→ 0 { 0} same operations, is the set of the ordinals strictly below ε1. In general, if we take the closure of 0 ε : β<α we obtain all ordinals strictly below ε . { }∪{ β } α Definition 2.3. Let be a linear ordering. We define ε to be the set of formal terms defined as follows:X X THE VEBLEN FUNCTIONS FOR COMPUTABILITY THEORISTS 5 0 and εx, for x , belong to ε , and are called “constants”, • ∈X X if t1,t2 ε , then t1 + t2 ε , • if t ε ∈, thenX ωt ε . ∈ X • ∈ X ∈ X Many of the terms we defined represent the same element, so we need to find normal forms for the elements of ε . The definition of the ordering on ε is what one should expect when is an ordinal.X We define the normal form of aX term and X the relation ε simultaneously by induction on terms. We say that≤ X a term t = t + + t is in normal form if either t = 0 (i.e. k =0 0 ··· k and t0 = 0), or the following holds: (a) t0 ε t1 ε ε tk > 0, and (b) each X X X si ≥ ≥ ···≥ ti is either a constant or of the form ω , where si is in normal form and si = εx for any x. * Every t ε can be written in normal form by applying the following rules: ∈ X + is associative, • s + 0 = 0 + s = s, • s r r if s<ε r, then ω + ω = ω , • X ωεx = ε . • x Given t = t0 + + tk and s = s0 + + sl in normal form, we let t ε s if one of the following··· conditions apply ··· ≤ X t = 0, • t = εx and, for some y x, εy occurs in s, X • t$ ≥ t = ω , s0 = εy and t! ε εy, X • t$ s$ ≤ t = ω , s0 = ω and t! ε s!, • ≤ X k>0 and t0 <ε s0, • X k>0, t0 = s0, l>0 and t1 + + tk ε s1 + + sl. • ··· ≤ X ··· The observation we made before the definition shows how the ε operator coin- cides with the ε-function when is an ordinal (this includes the case = , when X X ∅ 0 is the only constant and we obtain ε0 as expected). n,t Definition 2.4. In analogy with Definition 2.2, for t ε we use ω" # to denote the term in ε obtained by applying the ω function symbol∈ X n times to t. X Definition 2.5. If is a linear ordering and x , let !x be the linear ordering with domain y X : y< x . ∈X X { ∈X X } The following lemma expresses the compatibility of the ω and ε operators. Lemma 2.6. If is a linear ordering, then for every t ε and n N X ∈ X ∈ n,ε !t n,t ω" X # = ε !ω" # ∼ X ε !t t via a computable isomorphism. In particular, ω X = ε !ω . ∼ X Proof. The proof is by induction on n. When n = 0 the identity is the required n,ε !t n,t isomorphism. If ψ : ω" X # ε ! ω" # is an isomorphism, then the function X → ψ(t0) ψ(tk) mapping the empty string to 0 and t0, . . . , tk to ω + + ω witnesses n+1,ε !t n+1,t ' ( ··· ω" X # = ε !ω" #. " ∼ X To define the ϕ operator we start with the following observations. If we take t the closure of the set 0 under the operations +, t ω and t εt, we get all the ordinals up to ϕ{ (0).} If we take the closure of !→0 ϕ (β):!→ β<α we 2 { }∪{ 2 } get all the ordinals below ϕ2(α). In general, we obtain ϕγ (α) as the closure of 0 ϕ (β):β<α under the operations +, and t ϕ (t), for all δ<γ. { }∪{ γ } !→ δ Definition 2.7. Let and be linear orderings. We define ϕ( , ) to be the set of formal terms definedX as follows:Y Y X 0 and ϕ ,x, for x , belong to ϕ( , ), and are called “constants”, • Y ∈X Y X 6 ALBERTO MARCONE AND ANTONIO MONTALBAN´ if t1,t2 ϕ( , ), then t1 + t2 ϕ( , ), • if t ϕ(∈ , Y) andX δ , then ϕ∈ (t)Y Xϕ( , ). • ∈ Y X ∈Y δ ∈ Y X We define the normal form of a term and the relation ϕ( , ) simultaneously by ≤ Y X induction on terms. We write ϕ instead of ϕ( , ) to simplify the notation. ≤ ≤ Y X We say that a term t = t0 + + tk is in normal form if either t = 0, or the following holds: (a) t t ··· t > 0, and (b) each t is either a constant 0 ≥ϕ 1 ≥ϕ ···≥ϕ k i or of the form ϕδ(si), where si is in normal form and si = ϕδ$ (si! ) for δ! >δ. Every t ϕ( , ) can be written in normal form by applying* the following rules: ∈ Y X + is associative, • s + 0 = 0 + s = s, • if ϕ (s) < ϕ (r), then ϕ (s)+ϕ (r)=ϕ (r). • δ$ ϕ δ δ$ δ δ if δ! >δ, then ϕ (ϕ (r)) = ϕ (r). • δ δ$ δ$ if δ then ϕδ(ϕ ,r)=ϕ ,r. • ∈Y Y Y The motivation for the last two items is that if δ! >δ, anything in the image of ϕδ$ is a fixed point of ϕδ. Given t = t0 + + tk and s = s0 + + sl in normal form, we let t ϕ s if one of the following conditions··· apply ··· ≤ t = 0, • t = ϕ ,x and, for some y x, ϕ ,y occurs in s, • Y ≥X Y δ<δ! and t! ϕ (s!), or ≤ϕ δ$ t = ϕδ(t!), s0 = ϕδ (s!) and δ = δ! and t! s!, or • $ ≤ϕ δ>δ! and ϕδ(t!) ϕ s!, ≤ k>0 and t < s , 0 ϕ 0 • k>0, t = s , l>0 and t + + t s + + s . • 0 0 1 ··· k ≤ϕ 1 ··· l 2.2. Notation for strings and trees. Here we fix our notation for sequences (or strings) of natural numbers. The Baire space NN is the set of all infinite sequences of natural numbers. As usual, an element of NN is also called a real. If X NN and ∈ n N, X(n) is the (n + 1)-st element of X. N Definition 2.8. If σ N Definition 2.9. A tree is a set T N < Definition 2.10. If T is a tree and σ N N we let Tσ = ρ T : ρ σ σ ρ . ∈ { ∈ ⊆ ∨ ⊆ } < < Definition 2.11. KB is the usual Kleene-Brouwer ordering of N N: if σ,τ N N, ≤ ∈ we let σ KB τ if either σ τ or there is some i such that σ ! i = τ ! i and σ(i) <τ(i≤). ⊇ The following is well-known (see e.g. [Sim99, Lemma V.1.3]). Lemma 2.12. Let T N ACA0: RCA0+ Y X (X = Y !) ∀ ∃ (n) ACA! : RCA + Y n X (X = Y ) 0 0 ∀ ∀ ∃ ACA+: RCA + Y X (X = Y (ω)) 0 0 ∀ ∃ Π0 -CA : RCA + β well-ordered Y X (X = Y (β)), β 0 0 ∧∀ ∃ where β is a presentation of a computable ordinal2 (α) ATR0: RCA0+ α(α well-ordered = Y X (X = Y )) 0 ∀ 0 ⇒∀ ∃ + 0 Notice that Π1-CA0 is ACA0 and Πω-CA0 is ACA0 .Πβ-CA0 is strictly stronger 0 (α) than Πγ -CA0 if and only if β γ ω. In fact the ω-model α<γ ω X : X T 0 0 ≥ · 0 · { ≤ } satisfies Πα-CA0 for all α<γ ω, but not Πγ ω-CA0. Each theory in the above list · · % is strictly stronger than the preceding ones if we assume β ω2. ≥ 2 0 0 The system Πβ -CA0 is sometimes denoted by (Π1-CA0)β in the literature. 8 ALBERTO MARCONE AND ANTONIO MONTALBAN´ ACA0 and ATR0 are well-known and widely studied: [Sim99] includes a chapter devoted to each of them and their equivalents. (The axiomatization of ATR0 given + above is equivalent to the usual one by [Sim99, Theorem VIII.3.15].) ACA0 was introduced in [BHS87], where it was shown that it proves Hindman’s Theorem in + combinatorics (to this day it is unknown whether ACA0 and Hindman’s Theorem + + are equivalent). ACA0 has also been used in [Sho06] (where it is proved that ACA0 is equivalent to statements asserting the existence of invariants for Boolean alge- + bras) and in [MM09] (where ACA0 is used to prove a restricted version of Fra¨ıss´e’s conjecture on linear orders). ACA0! is also featured in [MM09]. The computation of its proof-theoretic ordinal, which turns out to be εω, is due to J¨ager(unpublished notes, a proof appears in [McA85], and a different proof is included in [Afs08]). The 0 + theories Πβ-CA0 are natural generalizations of ACA0 . 3. Forward direction In this section we prove the “forward direction”of Theorems 1.1, 1.5, 1.7, 1.9, and 1.2. The results in this section are already known (though often written in different settings) but we include them as our proofs illustrate how the iterates of the Turing jump relate with the epsilon and Veblen functions. The following theorem is essentially contained in Hirst’s proof [Hir94] of the closure of well-orderings under exponentiation in ACA0. Theorem 3.1. If is a Z-computable linear ordering, and ωX has a Z-computable X descending sequence, then Z! can compute a descending sequence in . X Proof. Let (ak : k N) be a Z-computable descending sequence in ωX . We can ∈ x x x write a in the form ω k,0 m + ω k,1 m + + ω k,lk m where each k · k,0 · k,1 ··· · k,lk mk,0 N is positive and xk,i > xk,i+1 for all i + below ϕ2(0) can be proved well-founded in ACA0 , and that every ordinal below Γ0 can be proved well-ordered in Predicative Analysis [Fef64, Sch77]. Theorem 3.4. If is a Z-computable linear ordering, and ε has a Z-computable descending sequence,X then Z(ω) can compute a descending sequenceX in . X Proof. Let (ak : k N) be a Z-computable descending sequence in ε . If no X ∈ n0,0 constant term εx appears in a0, then a0 <ω" # for some n0 so that we essentially n0,0 have a descending sequence in ω" #. Then, applying n0 times Theorem 3.1, we have that Z(n0) computes a descending sequence in 0, a contradiction. Thus we can let x be the largest x such that ε appears in a . It is not 0 ∈X x 0 n0,εx0 +1 hard to prove by induction on terms that εx0 a0 <ω" # for some n0 N. ≤ ∈ n0,εx0 +1 n0,ε !(εx0 +1) By Lemma 2.6, ε ! ω" # is computably isomorphic to ω" X # and X we can view the ak’s as elements of the latter. Using Theorem 3.1 n0 times, we (n0) obtain a Z -computable descending sequence in ε !(εx0 + 1). Noticing that the proof of Theorem 3.1 is uniform, we can apply thisX process again to the sequence we have obtained, and get an x1 < x0 and a descending sequence in ε !(εx1 + 1) X X (n0+n1) computable in Z for some n1 N. Iterating this procedure we obtain a (ω) ∈ Z -computable descending sequence x0 > x1 > ... in . " X X X + Corollary 3.5. ACA0 WOP( ε ). ; X!→ X + Proof. The previous proof can be formalized within ACA0 . " Theorem 3.6. Let α be a Z-computable well-ordering. If is a Z-computable α linear ordering, and ϕ(α, ) has a Z-computable descendingX sequence, then Z(ω ) can compute a descendingX sequence in . X Proof. By Z-computable transfinite recursion on α, we define a computable proce- dure that given a Z-computable index for a linear ordering and for a descending α sequence in ϕ(α, ), it returns a Z(ω )-computable indexX for a descending se- X quence in . Let (ak : k N) be a computable descending sequence in ϕ(α, ). Let x beX the largest x ∈ such that the constant term ϕ appears in a X(if 0 ∈X α,x 0 no ϕα,x appears in a0, just use 0 in place of ϕα,x0 in the argument below). It is not hard to prove by induction on terms that ϕ a <ϕn0 (ϕ + 1) for α,x0 0 β0 α,x0 n0 ≤ some β0 <α and n0 N, (where ϕβ (z) is obtained by applying the ϕβ function ∈ n0 symbol n0 times to z). It also not hard to show that ϕ(α, ) ! ϕ (ϕα,x0 + 1) is X β0 computably isomorphic to ϕn0 (β , ϕ(α, x )+1) (where ϕn0 (β, ) is obtained by 0 X ! 0 Z applying the ϕ(β, )-operator on linear orderings n0 times to ). Using the induc- · β0 Z (ω n0) tion hypothesis n0 times, we obtain a Z · -computable descending sequence in ϕ(α, x ) + 1. Then, we apply this process again to the sequence we have ob- X ! 0 tained, and get x1 < x0 and a descending sequence in ϕ(α, !x1)+1 computable β0 β1 X X (ω n0+ω n1) in Z · · for some β1 <α and n1 N. Iterating this procedure we obtain (ωα) ∈ a Z descending sequence x0 > x1 > ... in . " X X X 0 Corollary 3.7. Let α be a computable ordinal. Then Πωα -CA0 WOP( ϕ(α, )). ; X!→ X 0 Proof. The previous proof can be formalized within Πωα -CA0 for a fixed computable α. " Corollary 3.8. ATR WOP( ϕ( , 0)). 0; X!→ X Proof. Let α be a well-ordering and assume, towards a contradiction, that there exists a descending sequence in ϕ(α, 0). Let Z be a real such that both α and the α descending sequence are Z-computable. By Theorem 3.6 Z(ω ) (which exists in ATR0) computes a descending sequence in 0, which is absurd. " 10 ALBERTO MARCONE AND ANTONIO MONTALBAN´ 4. Ordinal exponentiation and the Turing Jump In this section we give a proof of the second part of Theorem 1.3. Our proof is a slight modification of Hirst’s proof, and prepares the ground for the generalizations in the following sections. We start by defining a modification of the Turing jump operator with nicer combinatorial properties. We will then define two computable approximations to this jump operator, one from strings to strings, and the other one from trees to trees. Definition 4.1. Given Z NN, we define the sequence of Z-true stages as follows: ∈ Z tn = max tn 1 +1, µt( n t (n) ) , { − { } ↓ } Z starting with t 1 = 1 (so that tn n+2). If there is no t such that n t (n) , then − ≥ { } ↓ the above definition gives tn = tn 1 + 1. So, tn is a stage where Z can correctly − Z t guess Z! n + 1 because m n(m Z! m ! n (m) ). With this in mind, ! ∀ ≤ ∈ ⇐⇒ { } ↓ we define the Jump operator to be the function : NN NN such that for every J → Z NN and n N, ∈ ∈ (Z)(n)=Z t , J ! n or equivalently (Z)= Z t ,Z t ,Z t ,Z t ,... J ' ! 0 ! 1 ! 2 ! 3 ( Here is a sample of this definition: t0 t1 t2 t3 Z = Z(0),Z(1),Z(2),Z(3),Z(4),Z(5),Z(6),Z(7),Z(8),Z(9),Z(10),Z(11),Z(12), ! ···" (Z)(0) J ! "# $ (Z)(1) J ! "#(Z)(2) $ J ! "# (Z)(3)$ J ! "# $ (Z)(n) Of course, (Z) T Z! for every Z as n Z! n J (n) . So, from a computabilityJ viewpoint,≡ there is no essential∈ diff⇐⇒erence { between} (↓Z) and the J usual Z!. Definition 4.2. The Jump function is the mapping J : N < < Lemma 4.4. For every σ,τ ! N N and τ J(N N), ∈ ∈ (P1) J(σ)= if and only if σ 1. (P2) K(J(σ))∅ = σ when σ | 2|.≤ (P3) J(K(τ)) = τ. | |≥ (P4) If σ = σ! and at least one has length 2, then J(σ) = J(σ!). (P5) J(σ)* < σ and K(τ) > τ except when≥ τ = . * | | | | | |< | | ∅ (P6) If τ ! τ then τ ! J(N N) and K(τ !) K(τ). ⊂ ∈ ⊂ THE VEBLEN FUNCTIONS FOR COMPUTABILITY THEORISTS 11 Proof. (P1) is obvious from the definition. (P2) follows from the fact that, when σ 2, tk 1 = σ (using the notation of | |≥ − | | Definition 4.2). In fact tk 1 σ by definition of k, and if tk 1 < σ then we have − − σ tk ≤| | | | either k ! (k) (and hence tk σ ) or tk = tk 1 +1 σ , against the definition of k. { } ↓ ≤| | − ≤| | (P3) follows from (P2) and K( )= . (P4) follows immediately from∅ (P1)∅ and (P2). The first part of (P5) follows from tn n + 2. The second part is a consequence of the first, (P1) and (P2). ≥ (P6) is obvious when τ ! = , using the second part of (P5). Otherwise we have ∅ τ ! = σ !t0,σ !t1,...,σ !tj for some j < k 1, so that K(τ !)=σ !tj σ !tk 1 = ' ( − ⊂ − K(τ). It is easy to check that τ ! = J(σ !tj). " The following Lemma explains how the Jump function approximates the Jump operator. Lemma 4.5. Given Y,Z NN, the following are equivalent: ∈ (1) Y = (Z); (2) for everyJ n there exists σ Z with σ >nsuch that Y n = J(σ ). n ⊂ | n| ! n Proof. Suppose first that Y = (Z). When n = 0 let σn = Z ! 1, which works by J Z (P1). When n>0 let σn = K(Y !n)=K(Y !n)= (Z)(n 1) Z. If 0 (0) Y (0) J − ⊂ σn{ } ↓ then Y (0) Z is such that 0 (0) and Y (0) σn so that also 0 (0) and ⊂ Z { } ↓ ⊆ { } ↓ J(σn)(0) = Y (0). If 0 (0) then Y (0) = Z ! 2=σn ! 2=J(σn)(0). This is the base step of an induction{ } ↑ that, using the same argument, shows that Y (i)= < J(σn)(i) for every i Corollary 4.6. For every m>0, given Y,Z NN, the following are equivalent: ∈ (1) Y = m(Z); (2) for everyJ n there exists σ Z with σ n+m such that Y n = J m(σ ). n ⊂ | n|≥ ! n The Jump function leads to the definition of the Jump Tree. Definition 4.7. Given a tree T N Proof. First let Z [T ]. Since by Lemma 4.5 for every n N, (Z) n = J(σ) for ∈ ∈ J ! some σ Z, so (Z)!n (T ). This implies (Z):Z [T ] [ (T )]. To prove⊂ theJ other inclusion,∈JT fix Y [ (T )],{J notice that∈ Y (n})⊆ JTY (n + 1) Definition 4.10. Let T be a tree and order (T ) by KB. Define h: T (T ) JT ≤ → ωJT by h(σ)= ωJ(σ)!i + ωJ(σ) 2 · i< J(σ) i)|σ (i) | { } ↑ for σ = and h( )=ω∅ 3. * ∅ ∅ · The sum above is written in KB-decreasing order, so that indeed h(σ) (T ) ≤ ∈ ωJT . Since J is computable, h is computable as well. The proof below should help the reader understand the motivation for the defi- nition above. Lemma 4.11. h is ( ,< (T ) )-monotone. ⊃ ωJT Proof. Suppose ρ,σ T are such that ρ σ; we want to show that h(ρ) < (T ) ωJT h(σ). ∈ ⊃ If σ = then ω∅ occurs with multiplicity 3 in J(σ) and with multiplicity at most ∅ < 2 in J(ρ). Since is the KB-maximum element in N N(and hence also in (T )), ∅ ≤ JT this implies h(ρ) < (T ) h(σ). ωJT If σ = then J(σ) = J(ρ) by (P4). Since σ ρ, if i ρ(i) then i σ(i) as well. Thus there* ∅ are two possibilities.* If for all i<⊂ J(σ){, } i σ↑(i) whenever{ } ↑ i ρ(i) then J(σ) J(ρ) and the first difference between| h(|σ){ and} h↑(ρ) is the coe{ffi}cient↑ ⊂ of ωJ(σ), which in h(σ) is 2 and in h(ρ) is either 1 or 0 (depending on whether ρ J(σ) ( J(σ) ) or not). In any case, h(ρ) < (T ) h(σ). If instead for some ωJT i<{| J(|}σ) ,| i σ(|i↑) and i ρ(i) let i be the least such i. Then the first difference | | { } ↑ { } ↓ 0 between h(σ) and h(ρ) occurs at ωJ(σ!i0), which appears in h(σ) but not in h(ρ). Again, we have h(ρ) < (T ) h(σ). ωJT " We can now finish offthe proof of the second part of Theorem 1.3. The sequence Z h(Z !n) n is Z-computable and strictly decreasing in ωX by Lemma 4.11. ' ( ∈N THE VEBLEN FUNCTIONS FOR COMPUTABILITY THEORISTS 13 Obviously our proof yields the following generalization of the second part of Theorem 1.3. Theorem 4.12. For every real Z there exists a Z-computable linear ordering X with a Z-computable descending sequence in ωX such that every descending se- quence in computes Z!. X 5. The ε function and the ω-Jump In this section we extend the construction of Section 4. To iterate the construc- tion, even only a finite number of times, requires generalizing the definition of h. Then we tackle the issue of extending the definition at limit ordinals by considering the ω-Jump. 5.1. Finite iterations of exponentiation and Turing Jump. We start by defining a version of the function h used in the previous section that we can it- erate. Definition 5.1. Let be a linear ordering, T a tree and X g : (T ) JT →X a function. Define h : T ωX g → by g(J(σ)!i) g(J(σ)) hg(σ)= ω + ω 2 · i< J(σ) i)|σ (i) | { } ↑ g( ) for σ = and h ( )=ω ∅ 3. * ∅ g ∅ · Note that when g is the identity, then hg = h of the previous section. Also, hg is g-computable. Lemma 5.2. If g is ( ,< )-monotone, then hg is ( ,<ω )-monotone. ⊃ X ⊃ X Proof. Notice that g ( ,< )-monotone implies that the sum in the definition of ⊃ X hg(σ) is written in decreasing order. The proof is the same as the one for Lemma 4.11. " We can now prove the analogue of Theorem 1.3 for iterations of the exponential n, (recall the notation ω" X# introduced in Definition 2.2). Theorem 5.3. For every n N and Z NN, there is a Z-computable linear n ∈ ∈ n (n) ordering Z such that the jump of every descending sequence in Z computes Z , X n, n X but there is a Z-computable descending sequence in ω" XZ #. Proof. Letting again TZ = Z !n : n N , we define a sequence Ti i n of trees as follows: let T = T and T { = (∈T )} for every i ω(Z) = Y J ! $. . . . . 4(Z)(0) = Y (3) . J ⊂ ··· f J 3(Z)(0) = Y (2) K(Y (3)) 2(Z) ⊂ ⊂ ⊂ J g J g J ········· J O J 2(Z)(0) = Y (1) K(Y (2)) K2(Y (3)) (Z) ⊂ ⊂ ⊂ ⊂ J g J g J g J ······ J O J (Z)(0) = Y (0) K(Y (1)) K2(Y (2)) K3(Y (3)) Z J ⊂ ⊂ ⊂ ⊂ ··· ⊂ % ! ! ! ! ! Kω(Y 1) Kω(Y 2) Kω(Y 3) Kω(Y 4) ω(Y ) ! ⊂ ! ⊂ ! ⊂ ! ⊂ ··· ⊂ K Figure 1. Assuming Y = ω(Z). J Lemma 5.2, it is immediate that each g is ( ,< m,T )-monotone and com- m ω" n# ⊃ n, n putable. Hence the sequence gn(Z ! j) j in ω" XZ # is Z-computable and de- ' ( ∈N scending. " 5.2. The ω-Jump. Now we define the iteration of the Jump operator at the first limit ordinal ω. Again, our definition is slightly different than the usual one so that it has nicer combinatorial properties. The difference being that instead of pasting all the i(Z) together as columns, we will take only the first value of each. Later we will showJ that this is enough. We will also define two computable approximations to this ω-jump operator, one from strings to strings, and the other one from trees to trees, and a computable inverse function. Definition 5.4. We define the ω-Jump operator to be the function ω : NN NN J → such that for every Z NN ∈ ω(Z)= (Z)(0), 2(Z)(0), 3(Z)(0),... , J 'J J J ( or, in other words, ω(Z)(n)= n+1(Z)(0). J J Notice that ω( (Z)) equals ω(Z) with the first element removed. Before J J J showing that ω(Z) Z(ω) it is convenient to define the inverse of ω. J ≡T J Definition 5.5. Given Y ω(NN) we define ∈J ω(Y )= Kn(Y (n)). K n - We need to show that the union above makes sense. Assume Y = ω(Z). It might help to look at Figure 1. Notice that for each n, Y (n) n(Z) becauseJ for every X, (X)(0) X and Y (n)= ( n(Z))(0). We also know⊂J that if σ (X), J ⊂ n J J n ⊂J then K(σ) X, so that K (Y (n)) Z. It follows that n K (Y (n)) Z. Applying (P5⊂) n times we get that Kn⊂(Y (n)) >n, and therefore the union above⊆ does actually produce Z. We have just| proved| the following lemma.% Lemma 5.6. For every Z NN, ω( ω(Z)) = Z. ∈ K J ω (ω) Lemma 5.7. For every Z NN, (Z) T Z . ∈ J ≡ (n) n Proof. We already know that Z T (Z) uniformly in n and hence that (ω) (n) n ≡ J ω (ω) Z = n Z T n (Z). It immediately follows that (Z) T Z . ∈N ≡ ∈N J J ≤ . . THE VEBLEN FUNCTIONS FOR COMPUTABILITY THEORISTS 15 For the other direction we need to uniformly compute all the reals n(Z) from ω n J (Z). We do this as follows. Given Y NN, let Y − be Y with its first n ele- J ω n ω ∈ n ments removed. Then, ( (Z)) = (Z)− . By the lemma above we get that n ω ω n J J J ω (Z)= ( (Z)− ), which we can compute uniformly from (Z). J K J J " As in section 4, where we computably approximated the jump operator, we will now approximate the ω-Jump operator with a computable operation on finite strings. Definition 5.8. The ω-Jump function is the map J ω : N τ $ τ $ -(τ !)= τ( τ ! 1) = J | |(σ)(0) J | |(σ) ' | |− ( ' (⊂ τ $ τ $ because τ ! < τ and hence J | |(σ) > 1. Let σ! = K| |(-(τ !)). Using (P6) | | | | | τ| $ τ ! times we know that σ! σ and J | |(σ!)=-(τ !). Now, we need to show that | ω| ⊂ ω τ $ J (σ!)=τ !. First notice that J (σ!) = τ ! because J | |(σ!) = -(τ !) = 1. By | | | | | | | | induction on i τ ! we can show, using (P6) and (P2), that ≤| | τ $ i i i τ $ τ $ i (5.1) J | |− (σ!)=K (-(τ !)) K (J | |(σ)) = J | |− (σ). ⊂ ω j+1 j+1 Now for j< τ ! , J (σ!)(j)=J (σ!)(0) = J (σ)(0) = τ(j)=τ !(j). ω | | ω For (P 7) let τ ! = J (σ!). Then, if i = τ ! m, equation (5.1) shows that m m | |− J (σ!) J (σ). ⊆ " 16 ALBERTO MARCONE AND ANTONIO MONTALBAN´ τ = ω(σ) J ! ! ∅O $$ J τ τ τ( τ 1) = J| |(σ)(0) τ( τ 1) = J| |(σ) | |− ! | |− " c J ...... b J τ( τ 1) = J τ $ (σ)(0) τ( τ 1) = J τ $ (σ ) J τ $ (σ) | "|− | | ! | "|− " | | " ⊂··· ⊂ | | b J . . K # ...... ` J K # τ(0) = J(σ)(0) J(σ") J(σ) ⊂ ···⊂ J %% a K % σ σ " ⊂ ···⊂ ω Figure 2. Assuming τ = J (σ) and τ ! τ. ⊂ As we did in Section 4, we now explain how the ω-Jump function approximates the ω-Jump operator. Lemma 5.11. Given Y,Z NN, the following are equivalent: ∈ (1) Y = ω(Z); (2) for everyJ n there exists σ Z with σ >nsuch that Y n = J ω(σ ). n ⊂ | n| ! n ω Proof. First assume Y = (Z). When n = 0 let σ0 = Z ! 1, which works by ω J ω (P 1). When n>0 let σn = K (Y ! n). We recommend the reader to look at ω ω n Figure 1 again. By (P 3) we have Y ! n = J (σn). Since σn = K ( Y (n 1) )= n 1 ' ω− ( K − (Y (n 1)), σn is one of the strings occurring in the definition of (Y ) and − ω K hence σn (Y )=Z by Lemma 5.6. We get that σn >nby (P5) applied n ⊂K n | | times to σn = K ( Y (n 1) ). ' − ( ω m Suppose now that (2) holds. By (P 7), we get that for all m and n, J (σn) m ⊆ J (σn+1). Using Corollary 4.6, it is straightforward to show that for all m 5.3. ω-Jumps versus the epsilon function. Our goal now is to generalize Def- inition 5.1 with an operator that uses ε rather than ω. We thus wish to define an operator hω that, given an order preserving function g : ω(T ) (where is ω JT →X X a linear order), returns an order preserving function hg : T ε . To do so we will → X iterate the h operator of Definition 5.1 along the elements of ω(T ). Let us give the rough motivation behind the definition of theJT operator hω below. Suppose we are given an order preserving function g : ω(T ) . For each i, we i JT →X would like to define a monotone function fi : (T ) ε such that fi = hf , JT → X i+1 where hfi+1 is as in Definition 5.1. Notice that the range of this function is correct, ε using the fact that ω X is computably isomorphic to ε . However, we do not have X a place to start as to define such fi we would need fi+1, and this recursion goes the wrong way. Note that if τ = J ω(σ) ω(T ), then τ(i) i(T ), and we ∈JT ' (∈JT could use g to define fi at least on the strings of length 1, of the form τ(i) . (This is not exactly what we are going to do, but it should help picture the construction.)' ( The good news is that to calculate fi 1 = hf on strings of length at most 2, we − i only need to know the values of fi on strings of length at most 1. Inductively, this would allow us to calculate f : T on strings of length at most i. Since this 0 →X would work for all i, we get f0 defined on all T . We now give the precise definition. First, we need to iterate the Jump Tree operator along any finite string. Definition 5.15. If T is a tree we define ω τ +1 ω (T )= J | | (σ):σ T τ J (σ) . JTτ { ∈ ∧ ⊆ } ω τ +1 ω Notice that τ (T ) | | (T ) and that τ (T ) is empty when τ/ ω JT ⊆JT JT ω ∈ (T ). The following Lemma provides an alternative way of defining τ (T ) byJT an inductive definition. JT Lemma 5.16. Given a tree T N Remark 5.19. First of all notice that we are really doing a recursion on σ . In | | fact, to compute h (σ) when σ = we use f ! on strings of the form fτ! σ(0) τ σ(0) " # * ∅ " # J(σ)!i, which have length J(σ) < σ by (P5). Let us notice the functions≤| have the| | right| domains and ranges. The proof is done ω ω simultaneously for all τ (T ) by induction on σ . Take σ τ (T ) with ∈JT ω | | ω ∈JT σ = n. Suppose that for all τ ! (T ) and all σ! (T ) with σ! ω + a real Y such that Y = (Z). Therefore, we get that ACA0 is equivalent to RCA + Z Y (Y = ω(Z)).J 0 ∀ ∃ J We already know, from Girard’s result Theorem 1.1 that over RCA0, the state- ment “if is a well-ordering then ωX is a well-ordering” is equivalent to ACA0. We now startX climbing up the ladder. n, Theorem 5.22. Over RCA , n WOP( ω" X#) is equivalent to ACA! . 0 ∀ X!→ 0 n, Proof. We showed, in Corollary 3.3, that ACA0! n WOP( ω" X#). n, ;∀ X!→ Suppose now that n WOP( ω" X#) holds. Consider Z NN and n ω; we ∀n X!→ ∈ ∈ want to show that (Z) exists. By Girard’s theorem we can assume ACA0. Let n J n Z = Tn, KB , where Tn = (TZ ) as in the proof of Theorem 5.3. The proof X ' ≤ ( JT n, n that there is a Z-computable descending sequence in ω" XZ # is finitary and goes n, n through in RCA0. So, by WOP( ω" X#) we get a descending sequence in Z . X!→ n i X By Lemma 2.12, using ACA , we get Y [T ]. For each i n, let Y = − (Y ). 0 n ∈ n ≤ i K n Lemma 4.9 shows that for each i, Yi [Ti] and Yi = (Yi 1). Since Z is the only path through T , we get that Y = Z∈, and so Y = Jn(Z).− Z 0 n J " + Theorem 5.23. Over RCA0, WOP( ε ) is equivalent to ACA0 . X!→ X + Proof. We already showed that ACA0 proves WOP( ε ) in Corollary 3.5. X!→ X Assume RCA0+WOP( ε ). Let Z NN; we want to show that there exists ω X!→ X ω ∈ Y with Y = (Z). Build = (TZ ), KB as in Theorem 5.21. The proof that ε has aJZ-computableX descending'JT sequence≤ ( is completely finitary and can be X carried out in RCA0. By WOP( ε ), we get that has a descending sequence. X!→ X X Since we have ACA0 we can use this descending sequence to get a path Y through ω (TZ ). Now, the proof of Lemma 5.14 translates into a proof in RCA0 that JT ω Y is of some path through TZ . Since Z is the only path through TZ , we get Y = Jω(Z) as wanted. J " 6. General Case In this section we define the ωα-Jump operator, the ωα-Jump function, and the ωα-Jump Tree, for all computable ordinals α. The constructions of Sections 4 and 5, where we considered α = 0 and α = 1 respectively, are thus the simplest cases of what we will be doing here. The whole construction is by transfinite recursion, and the base case was covered in Section 4. If α> 0 is a computable ordinal, we assume that we have a fixed non- decreasing computable sequence of ordinals αi : i N such that α = supi (αi + { ∈ } ∈N 1). (So, if α = γ +1, we can take α = γ for all i.) Notice that we have ωαi = i i N ωα. In defining the ωα-Jump operator, the ωα-Jump function, and the ω∈α-Jump Tree we make use the ωαi -Jump operator, the ωαi -Jump function, and0 the ωαi - Jump Tree for each i. 6.1. The iteration of the jump. Our presentation here is different from the one of previous sections, where we defined the operator first. Here we start from the ωα-Jump function, prove its basic properties, then use it to define the ωα-Jump Tree, and eventually introduce the ωα-Jump operator. Let α> 0 be a computable ordinal and αi : i N be its canonical sequence as described above. To simplify the notation in{ the definition∈ } of the ωα-Jump function, α α α ω i ω i ω In other words: α αn 1 αn 2 α0 J ω = J ω − J ω − J ω , n ◦ ◦···◦ α α0 α1 αn 1 Kω = Kω Kω Kω − . n ◦ ◦···◦ α Definition 6.1. The ωα-Jump function is the map J ω : N < ωα < Lemma 6.2. For σ,τ ! N N, τ J (N N), α α ∈ ∈ (Pω 1) J ω (σ)= if and only if σ 1. α α α ∅ | |≤ (Pω 2) Kω (J ω (σ)) = σ for σ 2. α α α | |≥ (Pω 3) J ω (Kω (τ)) = τ. ωα ωα ωα (P 4) If σ = σ! and at least one has length 2, then J (σ) = J (σ!). α α* α ≥ * (Pω 5) J ω (σ) < σ and Kω (τ) > τ except when τ = . ωα | | | | | ωα < | | | ωα ωα ∅ (P 6) If τ ! τ then τ ! J (N N) and K (τ !) K (τ). ωα ω⊂α ωα∈ ⊂ ωα ωα (P 7) If J (σ!) J (σ) and α> 0 then for every m, J (σ!) J (σ). ⊆ m ⊆ m Proof. The proof is by transfinite induction on α. The case α = 0 is Lemma 4.4. α α α Since J ω (σ)= if and only if J ω 0 (σ)= ,(Pω 1) follows from the same ∅ ∅ property for α0. ωα ωα ωα ωα To prove (P 2) let J (σ) = n 1 > 0. Then -(J (σ)) = Jn 1(σ)(0) = ωα ωα | | − ωα ωα ' − ωα ω(α Jn 1(σ) because Jn 1(σ) = 1 as noticed above. Since K (J (σ)) = Kn 1(Jn 1(σ)), ω−α ωα | − | ωα − − K (J (σ)) = σ follows from (P 2) for αn 2,αn 3,...,α0. − − α α α As in the proof of the case α = 1 in Lemma 5.10,(Pω 3), (Pω 4) and (Pω 5) follow from the properties we already proved. α The proof of (Pω 6) is also basically the same as the proof of (Pω6). We rec- ommend the reader to have Figure 3 in mind while reading the proof. The non- ωα trivial case is when τ ! = . Let σ be such that τ = J (σ). The idea is to * ∅ ωα define σ! σ as in the picture and then show that τ ! = J (σ!). Notice that ⊂ α α α σ > τ 2 by (Pω 5), and that σ = Kω (τ) by (Pω 2). Notice also that | | | |≥ ωα ωα -(τ !)= τ( τ ! 1) = J (σ)(0) J (σ), where the strict inclusion is because τ $ τ $ ' | |− ( 'ωα| | (⊂ | | τ ! < τ and hence J (σ) > 1. By induction on i τ ! we can show, using τ $ | ω| α | | ωα | | | | ≤| | (P 6) and (P 2) for α τ 1,...,α1,α0, that | $|− α α α α τ i τ 1 τ i τ 1 α ω | $|− ω | $|− ω | $|− ω | $|− ω (K K )(-(τ !)) (K K )(J τ (σ)) ◦···◦ ⊂ ◦···◦ | $| ωα = J τ i(σ) | $|− THE VEBLEN FUNCTIONS FOR COMPUTABILITY THEORISTS 21 α τ = ω (σ) J ! ! ∅O $$ α τ Jω | | α α = ω = ω τ( τ 1) J τ (σ)(0) τ( τ 1) J τ (σ) | |− ! | |− " α τ 1 | | | | b Jω | |− ...... α b τ Jω | $| α α α = ω = ω ω τ( τ " 1) J τ (σ)(0) τ( τ " 1) J τ (σ") J τ (σ) α | |− $ ! | |− " $ ⊂··· ⊂ $ τ 1 | | | | | | b Jω | $|− α . . ω τ 1 K | $|− " ...... α b Jω 1 ωα1 α K # α α τ(0) = Jω (σ)(0) Jω (σ ) Jω (σ) 1 1 " ⊂ ⊂ 1 α ··· Jω 0 %% c ωα0 K $ σ σ " ⊂ ···⊂ ωα Figure 3. Assuming τ = J (σ) and τ ! τ. ⊂ α α α ω τ$ i ω τ$ 1 ω < and (K | |− K | |− )(-(τ !)) J ( N). In particular, when i = τ ! , τ $ i N ◦···ωα ◦ ∈ | |− ωα | | if we set σ! = K τ (-(τ !)), we obtain σ! σ. Furthermore, by (P 2) applied to | $| ⊂ α0,...,α τ i 1, we also get | $|− − α α α τ i τ 1 α ω ω | $|− ω | $|− ω (6.1) J τ i(σ!)=(K K )(-(τ !)) J τ i(σ). | $|− ◦···◦ ⊂ | $|− Therefore, for every j< τ ! | | ωα ωα ωα J (σ!)(j)=Jj+1(σ!)(0) = Jj+1(σ)(0) = τ(j)=τ !(j). ωα ωα Since J (σ!)=-(τ !) which has length 1, we get that J (σ!) has length τ ! as τ $ 1 | | wanted.| |− ωα ωα For (P 7) let τ ! = J (σ!). Then, if i = τ ! m, equation 6.1 shows that ωα ωα | |− J (σ!) J (σ). m ⊆ m " We can now introduce the ωα-Jump Tree and prove its computability. Definition 6.3. Given a tree T N α α We first show that ω is indeed the inverse of ω . K J α α Lemma 6.6. If Y = ω (Z) then Z = ω (Y ). J K Proof. The proof of the lemma is by transfinite induction. Let αi : i N be the ωα{ ∈ωα } fixed canonical sequence fo α. Recall from the definition of K that K (Y !n)= ωα ωα ωα Kn ( Y (n 1) ). Since Y (n 1) = n (Z)(0) n (Z), by the induction ' − ( ' − ( 'J (⊆ωα J hypothesis applied to αn 1,...,α0, we get that Kn ( Y (n 1) ) Z. So Z ωα ωα − ' ωα− ( ⊆ ⊇ (Y ). By (P 5) applied to α0,...,αn 1 we get that Kn ( Y (n 1) ) >n+1 K α − | ' − ( | and hence Z = ω (Y ). K " ωα (ωα) Lemma 6.7. For every Z NN and computable ordinal α, (Z) T Z . ∈ J ≡ ωαi Proof. This is again proved by transfinite induction. Assuming that (Z) T α α (ω i ) ω J (β≡n) Z for every i, and uniformly in i, we immediately obtain n (Z) T Z , α α n 1 αi α ω J(ω ) ≡ where βn = i=0− ω , for every n. Since βn <ω , (Z) T Z is immediate. J ≤ ωα ωα For the other reduction we need to uniformly compute n (Z) from (Z). 0 α α J J α The same way we compute Z from ω (Z) applying ω , we can compute ω (Z) J K Jn by forgetting about α0,...,αn 1. In other words, by the same proof as Lemma 6.6 we can show that for every m − α αm αm+1 αn 1 ω (Z)= Kω (Kω (... (Kω − ( Y (n 1) )) ... )) Jm ' − ( n>m - αm αm+1 αn 1 α using Kω Kω Kω − instead of Kω . ◦ ◦···◦ n " α α We can now prove that J ω approximates ω , extending Lemma 5.11. J Lemma 6.8. Given Y,Z NN, the following are equivalent: α ∈ (1) Y = ω (Z); J α (2) for every n there exists σ Z with σ >nsuch that Y n = J ω (σ ). n ⊂ | n| ! n Proof. We first prove (1) = (2). When n = 0 let σ0 = Z ! 1, which works by ωα ωα ⇒ ωα ωα (P 1). Let σn = K (Y !n). Then σn (Y )=Z, and Y !n = J (σn). We ωα ⊆K ωα get that σn >nby applying (P 5) n times to σn = Kn ( Y (n 1) ). The proof| | of (2) = (1) is similar to the proof of' Lemma− 5.11( but uses ⇒ ωα ωα ωα transfinite induction. By (P 7), for all m and n, Jm (σn) Jm (σn+1), and α ω N ⊆ hence we can consider n Jm (σn) N . Then, by the induction hypothesis, α α ∈ α α J ω (σ )= ω (Z), and hence J ω (σ )(0) = ω (Z)(0) for all m < n. It n m n m % m n m follows that forJ every m and n > m J % α α α α ω (Z)(m)= ω (Z)(0) = J ω (σ )(0) = J ω (σ )(m)=Y (m). J Jm+1 m+1 n n " We are now able to show the intended connection between the ωα-Jump Tree and the ωα-Jump operator. α α Lemma 6.9. For every tree T , [ ω (T )] = ω (Z):Z [T ] . JT {J ∈ } α α Proof. To prove ω (Z):Z [T ] [ ω (T )] we can argue as in the proof of Lemma 4.9, using{J Lemma 6.8 in∈ place}⊆ ofJT Lemma 4.5. α To prove the other inclusion, fix Y [ ω (T )]. Arguing as in the proof of ωα ∈ JT ωα Lemma 5.14, we first let σn = K (Y !n) T . Let Z = (Y )= n σn [T ]. α ∈ K N ∈ We get that Y = ω (Z) from Lemma 6.8. ∈ J % " THE VEBLEN FUNCTIONS FOR COMPUTABILITY THEORISTS 23 6.2. Jumps versus Veblen. First, we need to iterate the Jump Tree operator along a finite string. α Definition 6.10. If T is a tree and τ ω (T ) we define ∈JT ωα ωα ωα τ (T )= J τ +1(σ):σ T τ J (σ) . JT { | | ∈ ∧ ⊆ } α Lemma 6.11. For τ ω (T ), ∈JT α α ω (T )= ω 0 (T ) JT∅ JT α α τ +1 α ω ω | | ω τ ! c (T )= ( τ (T ) c ). JT " # JT JT " # (T c was defined in 2.10.) " # Proof. Straightforward induction on τ . | | " Lemma 6.12. Given a tree T N Theorem 6.17. Over RCA , WOP( ϕ( , 0)) is equivalent to ATR . 0 X!→ X 0 Proof. We already showed that ATR proves WOP( ϕ( , 0)) in Corollary 3.8. 0 X!→ X For the reversal, we argue within RCA0. Let α be any ordinal. Notice that relative to the presentation of α, all the constructions of this section can be done as if α were any computable ordinal. Therefore, by the previous theorem it is enough to show that WOP( ϕ(α, )) holds. Let be a well-ordering. We now claim that ϕ(α, ) embedsX!→ in ϕ(αX+ , 0), whichX would imply that ϕ(α, ) is well-ordered too asX needed to show WOPX( ϕ(α, )). X Define f : ϕ(α, ) ϕ(α + , 0) by inductionX!→ on theX terms of ϕ(α, ), setting X → X X f(0) = 0, • f(ϕ )=ϕ (0), • α,x α+x f(t1 + t2)=f(t1)+f(t2), • f(ϕ (t)) = ϕ (f(t)). • a a The proof that f is an embedding is by induction on terms. Consider t, s ∈ ϕ(α, ). We want to show that t ϕ(α, ) s f(t) ϕ(α+ ,0) f(s). By inductionX hypothesis, assume this is≤ trueX for pairs⇐⇒ of terms≤ shorterX than t + s. Suppose that t s. Using the induction hypothesis, it is not hard to show that f(t) f(s). Suppose≤ now that t s. Then, none of the conditions of Definition ≤ *≤ 2.7 hold. If t =0,t = t1 + t2, or t = ϕa(t1), then we can apply the induction hypothesis again and get that none of the conditions of Definition 2.7 hold for f(t) and f(s) either and hence f(t) f(s). The case t = ϕα,x is the only one that deserves attention. In this case we*≤ have that for no y x, ϕ appears in s. It ≥ α,y then follows that f(s) ϕ(α + ! x, 0). Since f(t)=ϕα+x(0) is greater than all the elements of ϕ(α + ∈ x, 0) weX get f(t) f(s) a wanted. X ! *≤ " References [Afs08] Bahareh Afshari. Relative computability and the proof-theoretic strength of some theo- ries. PhD thesis, University of Leeds, 2008. [AR09] Bahareh Afshari and Michael Rathjen. Reverse mathematics and well-ordering princi- ples: A pilot study. Ann. Pure Appl. Logic, 160(3):231–237, 2009. [BHS87] Andreas R. Blass, Jeffry L. Hirst, and Stephen G. Simpson. 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