TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 362, Number 10, October 2010, Pages 5283–5319 S 0002-9947(10)05037-3 Article electronically published on May 4, 2010 ON THE DEGREE SPECTRUM OF A Π01 CLASS THOMAS KENT AND ANDREW E. M. LEWIS Abstract. For any P ⊆ 2ω , define S(P), the degree spectrum of P, to be the set of all Turing degrees a such that there exists A ∈ P of degree a. We prove a number of basic properties of the structure which is the degree spectra of Π01 classes ordered by inclusion and also study in detail some other phenomena relating to the study of Π01 classes from a degree theoretic point of view, which are brought to light as a result of this analysis. 1. Introduction The study of the degree theoretic complexity of members of Π01 classes, dating back as far as Kleene [SK], has resulted in a rich and well developed theory. Two fundamental papers in this area are [JS1] and [JS2] by Jockusch and Soare, in which they proved, amongst other things, the much cited low basis theorem as well as a number of basic results regarding special Π01 classes, i.e. non-empty Π01 classes which have no computable members. For a detailed account of work in the area we refer the reader to [DC]. In this paper we shall be concerned with a natural structure regarding the study of Π01 classes from a degree theoretic point of view which, while many known theorems can be seen to relate, has not been studied explicitly in the previous literature. Unless stated otherwise, by a Π01 class we shall mean a Π01 class of sets of natural numbers. We let D denote the set of all Turing degrees. For any P ⊆ 2ω we define S(P), the degree spectrum of P, to be the set of all degrees a such that there exists A ∈ P of degree a. We define P = {S(P) : P is a Π01 class}, and we consider the elements of P to be ordered by inclusion. While this structure has not been studied directly before, it does relate to recent work by Simpson [SS], [SS2], Binns, Cole [SC] and others who have been interested in the Muchnik degrees of Π01 classes. As pointed out by Simpson in a private correspondence, it is not difficult to see that the latter structure is dually isomorphic to the substructure of (P, <) given by the non-empty degree spectra of Π01 classes which are upward closed. Let us begin, then, by making some basic observations concerning the structure (P, <). The following facts are easily derived: (i) (P, <) has a greatest element 1P = D and a least element 0P = ∅. (ii) (P, <) is an uppersemilattice. Received by the editors June 13, 2008. 2010 Mathematics Subject Classification. Primary 03D28. Key words and phrases. Π01 classes. The first author was supported by Marie-Curie Fellowship MIFI-CT-2006-021702. The second author was supported by a Royal Society University Research Fellowship. c 2010 American Mathematical Society Reverts to public domain 28 years from publication 5283 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 5284 THOMAS KENT AND ANDREW E. M. LEWIS (iii) We shall say that α > 0P in P is minimal if there does not exist β ∈ P with 0P < β < α. (P, <) has at least one minimal element α = {0}. In order to prove (ii) it suffices to observe that if P0 and P1 are Π01 classes, then P0 ∪ P1 is a Π01 class with degree spectrum S(P0 ) ∪ S(P1 ). Some well known examples of degree spectra will prove to be useful tools in analyzing the structure. Recall that a degree is PA if it contains a set which codes a complete and consistent extension of Peano Arithmetic according to some computable bijection between sentences in the first order language of arithmetic and the natural numbers, and that a degree is (Martin-Löf) random if it contains a random set. For a detailed account of the theory of algorithmic randomness, we refer the reader to [DH] and [AN]. Definition 1.1. We let p denote the set of all PA degrees, and we let r denote the set of all Martin-Löf random degrees. It is well known that p and r are both degree spectra of Π01 classes. For degree spectra α and β, we shall write α ∩ β in order to denote their intersection and, when α and β are both elements of P, we shall write α ∧ β in order to denote their greatest lower bound in P when this exists. Stephan has shown [FS] that the degrees which are random and PA are precisely the degrees ≥ 0′ . Since every non-empty Π01 class contains a member of low degree [JS1], this suffices to show that there exist α, β ∈ P such that α ∩ β ∈ / P. Note, however, that r ∧ p is defined and is equal to 0P , so that we cannot immediately conclude that (P, <) is not a lattice. In section 8 we shall show that, in fact, the structure is a lattice, since the intersection of two degree spectra of Π01 classes is the degree spectrum of a Π01 class if it is the superset of the degree spectrum of a non-empty Π01 class. Since any special Π01 class is uncountable, it immediately follows that any minimal element of (P, <) other than {0} has uncountably (and so continuum) many elements. The following theorem suffices to show that there are no maximal elements amongst the degree spectra of special Π01 classes. Theorem 1.1 (Jockusch, Soare [JS2]). If P is a special Π01 class, then there exists a non-zero c.e. degree a ∈ / S(P). On the other hand, for every degree a with 0 < a ≤ 0′ there exists a special Π01 class P with a ∈ S(P). In the following sections we shall prove a number of basic structural properties of (P, <), and in doing so we shall come across some phenomena relating to the study of Π01 classes which are worthy of examination in their own right. In section 2 we shall describe notation and recall basic definitions required for what is to follow. In section 3 we shall give some easy examples of elements of the structure. In section 4 we shall prove, as well as some corresponding results for the Borel hierarchy, that there exist degrees a such that no Σ03 class contains a member of degree a unless it contains a member of every degree. One immediate consequence of these proofs is a characterization, for sufficiently low levels of the arithmetical and Borel hierarchies, of those sets which contain a member of every degree in an upper cone. Much of this section will be devoted to a study of the invisible degrees—degrees, that is, which do not belong to any member of P other than 1P . The existence of invisible degrees suffices to show that there do not exist α, β ∈ P such that α < 1P and β < 1P with α ∨ β = 1P . In section 5 we shall give a characterization of the minimal elements of the structure and use this to show, for example, that both r and p are minimal. We shall also show that for any α < 1P there exists β which License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ON THE DEGREE SPECTRUM OF A Π01 CLASS 5285 is minimal and such that β ≤ α. In section 6 we shall consider pairs of degrees a and b such that, whenever a Π01 class contains a member of degree a, it contains a member of degree b. We define the Π01 -mates of a to be the intersection of all α in P which contain a, and we show that for any α < 1P there exists a degree a∈ / α such that the Π01 -mates of a are precisely α ∪ {a}. This suffices to show that every element of the structure other than 1P has a strong minimal cover. We also show that there exists a non-computable set A which is (Cantor-Bendixson) ranked, such that every B ≤T A is ranked, and such that no non-computable B ≤T A is a member of any rank faithful class. In section 7 we prove two anti-basis theorems for Π01 classes. We show first that any Π01 class that contains a member of every low degree contains a member of every degree. We then go on to show that if b is non-GL2 , then, for any Π01 class P which does not contain a member of every degree, there exists some non-zero a ≤ b such that P does not contain any member of non-zero degree below a. 2. Basic definitions and notation Some frequently used variables. We shall use the variables e, i, j, k, l, n, m, s, t for elements of ω; σ, τ for elements of 2<ω ; φ for elements of ω <ω ; f, g for elements of ω ω ; h for (possibly) partial functions ω → ω; A, B, C for subsets of ω (which we identify in the standard way with infinite binary strings); and a, b for elements of D. We shall use the variables P and Q for subsets of 2ω , and we shall use the variable R for subsets of ω ω . We shall use the variables α, β and γ to range over the elements of P. We write φ ⊆ h in order to denote that φ is an initial segment of h. We write h0 ⊕ h1 in order to denote that (possibly partial) function h2 such that for all n, h2 (2n) ↓= m iff h0 (n) ↓= m and h2 (2n + 1) ↓= m iff h1 (n) ↓= m. Turing functionals. We let {Ψi }i∈ω be an effective listing of the Turing functionals, and we shall use the variables Ψ and Φ to range over the Turing functionals. We say that a Turing functional Φ is total if Φ(A) is total for all A. The following convention is worth singling out for attention: we assume that, for any σ and any n, Ψ(σ; n) is not defined unless σ is of length at least n, this computation converges in at most |σ| steps, and Ψ(σ; n′ ) is defined for all n′ < n. Thus at any stage of a computable construction we can decide any value Ψ(σ; n). Strings and sets of strings. For any φ ∈ ω <ω and any φ′ ∈ ω <ω (f ∈ ω ω ) we let φ ⋆ φ′ (φ ⋆ f ) denote the concatenation of φ and φ′ (φ and f ). For i ∈ ω we may sometimes write i in order to denote the string φ of length 1 such that φ(0) = i. We may also write 01, for example, in order to denote the string τ of length 2 such that τ (0) = 0 and τ (1) = 1. We shall use the variables Λ, Υ, T and U to range over subsets of 2<ω . We say that Λ is downward closed if, whenever τ ∈ Λ, all initial segments of τ are also in this set. Generally we shall use the variables Λ, Υ to range over downward closed subsets of 2<ω , and we shall use the variables T and U to range over subsets of 2<ω which may not be downward closed. We say that τ and τ ′ are Ψ-splitting if Ψ(τ ) and Ψ(τ ′ ) are incompatible. We say T is Ψ-splitting if every pair of incompatible strings in T is Ψ-splitting. We shall use the variable Π to range over subsets of ω <ω . We define [Π] = {f : ∃∞ φ(φ ⊂ f ∧ φ ∈ Π)}. A leaf of Π is a string φ ∈ Π such that no proper extension of this string is in Π. If φ ∈ Π, then the level of φ in Π is defined to be the number of proper initial segments of φ in Π. If φ, φ′ ∈ Π, φ ⊂ φ′ and there does not exist φ′′ with φ ⊂ φ′′ ⊂ φ′ , then we License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 5286 THOMAS KENT AND ANDREW E. M. LEWIS say that φ′ is a successor of φ in Π and that φ is the immediate predecessor of φ′ in Π. We say that non-empty Π is perfect if every element has two incompatible extensions in the set and that Π is 2-branching if it has a single element of level 0 and every element has precisely two successors. Π is finitely 2-branching if Π is finite, it has a single element of level 0, all leaves are of the same level, and every element which is not a leaf has precisely two successors. We say that Π is of level n if Π is finite and all leaves of Π are of level n in Π. Infinite Π is pointed if every element of [Π] computes Π. We let λ denote the string of length 0. Listing the Π01 classes. We let {Λi }i∈ω be a uniformly computable sequence of downward closed subsets of 2<ω such that, for any Π01 class P, there exists an infinite number of i with P = [Λi ]. Cantor-Bendixson rank. We consider the Cantor topology on 2ω . A subset of 2ω is Fσ if it is a countable union of closed sets, and it is Gδ if it is a countable intersection of open sets. The Cantor-Bendixson derivative of P is the set of nonisolated points of P according to the Cantor topology and is denoted D(P). The iterated derivative Dα (P) is defined for all ordinals α by transfinite recursion: (i) D0 (P) = P; = D(Dα (P)); (ii) Dα+1 (P)  λ (iii) D (P) = α<λ Dα (P) for any limit ordinal λ. A set A has rank α relative to P if α is the least ordinal such that A ∈ Dα (P) and A∈ / Dα+1 (P). The (Cantor-Bendixson) rank of any set (should it have a rank) is its least rank relative to any Π01 class. Some standard theorems and facts. We shall use the standard Lebesgue measure on Cantor space. It’s a theorem of Kucera [AK] that every Π01 class of positive measure contains a member of every random degree, and it is also easily observed that any Π01 class containing a random set is of positive measure. The low basis theorem [JS1] states that every non-empty Π01 class contains a member of low degree. Recall that a degree a is hyperimmune-free iff every f of degree below a is majorized by a computable function; i.e. there exists a computable function g such that g(n) ≥ f (n) for all n. The hyperimmune-free basis theorem [JS1] states that every non-empty Π01 class contains a member of hyperimmune-free degree. If A is of hyperimmune-free degree and B ≤T A, then there exists a total Turing functional Φ such that Φ(A) = B, i.e. B ≤tt A. 3. Examples In order to get a better picture of what (P, <) looks like, it seems a good idea to describe some easy examples of elements of the structure. Since every nonzero degree below 0′ is hyperimmune and the hyperimmune degrees are upward closed, it follows immediately from the low basis theorem and the hyperimmunefree basis theorem that no non-trivial upper cone can be the degree spectrum of a Π01 class. The situation becomes more interesting, however, if we consider the union of upper cones with {0}. For every computable ordinal α and every degree a with 0(α) ≤ a ≤ 0(α+1) , the sets {0} ∪ {b : b ≥ a} and {0, a} are each the degree spectrum of a Π01 class. This follows (as pointed out by Jockusch in a private correspondence) since Jockusch and McLaughlin [JM] have shown that any such degree contains some function which is a Π01 singleton, i.e. which is the only element License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ON THE DEGREE SPECTRUM OF A Π01 CLASS 5287 of some Π01 class of functions. From this it is easily seen that the cone above a is the degree spectrum of a Π01 class of functions. Then it follows from the proof of Theorem 1 of [JS2] that if R is a Π01 class in ω ω , there is a Π01 class P ⊆ 2ω such that S(P) = S(R) ∪ {0}. By taking a suitable increasing sequence of degrees, it follows that (P, <) is not well-founded. Combined with the hyperimmune-free basis theorem, it also follows that there exists an uncountable α such that all β ≤ α contain 0. For every a which contains a set of rank 1, {0, a} is the degree spectrum of a Π01 class. Upon realizing this, it is natural to ask whether for every such degree a, {0} ∪ {b : b ≥ a} is the degree spectrum of a Π01 class. Later, in section 4, we shall be able to give a negative solution. Recall that a degree is fixed point free iff it contains a DNC (diagonally noncomputable) function, where a function f is defined to be DNC if f (n) = Ψn (n) for all n. Definition 3.1. We let f denote the set of all fixed point free degrees. Simpson [SS], [SS2] has shown that f is an element of P. Another very natural example of an element of the structure is the set of c.e. degrees. Theorem 3.1. The set of all c.e. degrees is the degree spectrum of a Π01 class. Proof. We let Wi denote the ith c.e. set according to some fixed effective listing. It is not difficult to see that there exists a computable function f such that, for any i ∈ ω, S([Λf (i) ]) = {0, ai }, where ai is the degree of Wi . Now let Λ be the computable set of strings which contains a copy of Λf (i) above each string 0i ⋆ 1, where 0i is the sequence of i many zeros.  Corollary 3.1. There exists α < 1P such that, for every β > 0P , α ∩ β = ∅. Proof. Every element of P other than 0P contains a c.e. degree (the degree of the leftmost element of any Π01 class is the same as the degree of the set of finite binary strings strictly to its left, which is clearly a computably enumerable set).  Note, however, that for any countable α ∈ P and any β which is the degree spectrum of a special Π01 class, α ∧ β = 0P (since any countable Π01 class has a computable element), so that the only α < 1P for which it is not obviously the case that there exists β > 0P with α ∧ β = 0P are those uncountable α which contain 0. The following theorem suffices to show that the equivalent of Corollary 3.1 does not hold for the degree spectra of special Π01 classes. Although the result has already been proved by Cole and Simpson [CS], we include a proof here which serves as a good introduction to techniques which will be used later in the paper. Theorem 3.2 (Cole and Simpson [CS]). For any special Π01 class P0 there exists a special Π01 class P1 such that no member of P1 computes any member of P0 . Proof. Suppose we are given downward closed and computable Λ such that P0 = [Λ]. We define an approximation to a 2-branching T such that P1 = [T ] satisfies the statement of the theorem. Those τ in T of level 2i + 1 will be defined so as to satisfy the requirement Θi : If A ∈ P1 , then A = Ψi (∅). Those τ in T of level 2i + 2 will be defined so as to satisfy the requirement Ξi : If A ∈ P1 and Ψi (A) is total then Ψi (A) ∈ / P0 . License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 5288 THOMAS KENT AND ANDREW E. M. LEWIS Stage 0. Enumerate λ into T . Stage s > 0. Consider all the strings in T to be ordered first according to their level in T and then lexicographically. Find the least string τ ∈ T (if any) such that either: (1) τ is of level 2i + 1 and τ ⊂ Ψi (∅)[s]. In this case let τ0 be the immediate predecessor of τ in T and let τ1 be a leaf of T extending τ0 and incompatible with τ . Remove all strings properly extending τ0 from T and then enumerate in two incompatible extensions of τ1 . (2) τ is of level 2i+2, Ψi (τ ) is compatible with some string in Λ of length s and there exists a leaf τ ′ of T extending τ such that Ψi (τ ′ ) properly extends Ψi (τ ). In this case remove all strings extending (and including) τ from T , other than τ ′ . Once these instructions are completed, choose two incompatible strings extending each leaf of T , and enumerate these strings into T . It is clear that P1 is a Π01 class. In order to see that our approximation to T converges, suppose that for some least i there exists a sequence {τj }j≥0 of strings such that each τj is a string of level  2i + 2 in T at some stage of the construction and τj ⊂ τj+1 for all j. Let A = j τj . Then Ψi (A) is computable and is in P0 , which gives the required contradiction.  4. Invisible degrees Before going on to consider the issue of which α ∈ P can be cupped to 1P , in this section we present a number of theorems concerning lower levels of the arithmetical and Borel hierarchies, which will give us valuable information regarding the structure (P, <) when applied to the countable sequence of Π01 classes. We prove Theorems 4.1 and 4.2 in a way that may initially seem more complicated than is necessary, so that we can later use these proofs to deduce certain corollaries. Theorem 4.1. For any countable sequence of Fσ sets, {Qk }k∈ω say, there exists a degree a such that, for any k ∈ ω, if a ∈ S(Qk ), then S(Qk ) = D. In fact a can be chosen to be hyperimmune-free and minimal. Proof. We will in fact show that the conclusion holds for a countable sequence of closed sets. That this is sufficient to imply the full theorem is immediate, since any Fσ set is the union of a countable sequence ofclosed sets. More specifically, let {Qk }k∈ω be a sequence of Fσ sets, with Qk = j∈ω Pk,j with Pk,j closed for each k, j ∈ ω, and let a be such that if a ∈ S(Pk,j ), then S(Pk,j ) = D. If a ∈ S(Qk ), then there exists a j such that a ∈ S(Pk,j ), giving S(Qk ) = D as required. So suppose we are given a sequence {Pk }k∈ω of closed sets. In order to show that any P contains a member of every degree, it suffices to show that [T ] ⊂ P for some 2-branching computable T . In order to see that this suffices, suppose that P contains all paths through T ofthis kind. Then we can define, given any set B, a set C ∈ [T ] such that C = s σs and is of the same degree as B. Define σ0 to be the string of level 0 in T . Given σs , define σs+1 to be the leftmost successor of σs in T if B(s) = 0, and define σs+1 to be the rightmost successor otherwise. Note also, for future reference, that the same argument suffices to show that any P containing [T ] for some perfect pointed b-computable T contains a member of every degree above b. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ON THE DEGREE SPECTRUM OF A Π01 CLASS 5289 For every k, let Pk = [Υk ] for some downward closed Υk ⊆ 2<ω . We shall use a  simple forcing argument in order to construct A = s σs , which is of hyperimmunefree minimal degree. Initially we define σ0 = λ, and we define T0 to be the identity tree. At every stage s, given computable 2-branching Ts and σs which is the string of level 0 in Ts , we define σs+1 ⊃ σs and Ts+1 ⊂ Ts . At stage s we act according to the first of the following situations which applies. Let s = i, j, k. (1) There exists σ ∈ Ts such that no two strings extending σ in Ts are Ψi splitting. In this case we define σs+1 to be the first such σ properly extending σs , and we define Ts+1 to be the set of strings in Ts extending σ. In so doing we have ensured that Ψi (A) is either computable or partial. (2) Since the previous case does not apply, we may let Ts0 be a computable, 2-branching and Ψi -splitting subset of Ts containing σs (we shall eventually define Ts+1 to be a subset of Ts0 ). There exists σ ∈ Ts0 such that either Ψj (Ψi (σ)) is incompatible with σ or else for no string τ ⊃ σ in Ts0 is it the case that Ψj (Ψi (τ )) properly extends Ψj (Ψi (σ)). In this case we define σs+1 to be the first such σ properly extending σs , and we define Ts+1 to be the set of strings in Ts0 extending σ. In so doing we have ensured it is not the case that Ψj (Ψi (A)) = A. (3) Since the previous case does not apply, we may let Ts1 be a computable and 2-branching subset of Ts0 containing σs such that, whenever τ, τ ′ ∈ Ts1 and τ ′ ⊃ τ , we have that Ψj (Ψi (τ ′ )) properly extends Ψj (Ψi (τ )). There exists σ ∈ Ts1 such that Ψi (σ) is not in Υk . In this case we define σs+1 to be the first such σ properly extending σs , and we define Ts+1 to be the set of / Pk . strings in Ts1 extending σ. In so doing we have ensured that Ψi (A) ∈ (4) Since none of the previous cases apply, we have that, for all B ∈ [Ts1 ], Ψi (B) ∈ Pk and Ψj (Ψi (B)) = B. Since Ts1 is 2-branching and computable, it follows that Ps contains a member of every Turing degree. In this case we define σs+1 to be some proper extension of σs in Ts1 , and we define Ts+1 to be the set of strings in Ts1 which extend σs+1 . The verification that A is of minimal degree is standard (that A is non-computable follows from Posner’s lemma); see for example [BC]. In order to see that A is of hyperimmune-free degree, observe that whenever Ψi (A) is total and noncomputable, A lies on a computable 2-branching Ψi -splitting tree T . Thus, for any n, Ψi (A; n) is amongst the values Ψi (τ ; n) such that τ is of level n + 1 in T .  If we restrict ourselves to the arithmetical hierarchy, then we can do a little better: Theorem 4.2. There exists a hyperimmune-free minimal degree a below 0′′ , such that no Σ03 class contains a member of degree a unless it contains a member of every degree. Proof. For the same reasons that it sufficed to prove Theorem 4.1 for countable sequences of closed sets, it suffices to prove this result for Π02 classes. If U is a set of finite binary strings, then let I(U ) be the set of all A which extend some member 0 of U . Let Pk be the kth Π 2 class according to the standard indexing of such classes, and for each k let Pk = l I(Uk,l ), where {Uk,l }l∈ω is a uniformly c.e. sequence of sets of strings. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 5290 THOMAS KENT AND ANDREW E. M. LEWIS The construction works in exactly the same way as that of Theorem 4.1, except that now case (3) is as follows: (3) Since the previous case does not apply, we may let Ts1 be a computable and 2-branching subset of Ts0 containing σs such that, whenever τ, τ ′ ∈ Ts1 and τ ′ ⊃ τ , we have that Ψj (Ψi (τ ′ )) properly extends Ψj (Ψi (τ )). Now let Ts2 be the subset of Ts1 enumerated as follows: Step t = 0. Enumerate the string of level 0 in Ts1 into Ts2 . Step t > 0. For each leaf τ of the present value Ts2 , if τ is of level l, then search for two incompatible extensions τ0 , τ1 in Ts1 such that Ψi (τ0 ) and Ψi (τ1 ) extend strings in every Uk,l′ such that l′ ≤ l. If two such extensions are found, then enumerate them into Ts2 . It is easily seen that if Ts2 is 2-branching then it is computable. If not then there exists σ which is a leaf of Ts2 . In this case we define σs+1 to be an element of Ts1 properly extending the first such σ and we define Ts+1 to be the set of strings in Ts1 extending σ. In so doing we have ensured that Ψi (A) ∈ / Pk . In case (4) we must now replace Ts1 with Ts2 .  An analysis of the proofs of Theorems 4.1 and 4.2 gives a characterization, for sufficiently low levels of the arithmetical and Borel hierarchies, of those sets which contain a member of every degree in an upper cone: Theorem 4.3. An Fσ set Q contains a member of every degree above b iff there exists some b-computable perfect and pointed T with [T ] ⊆ Q.  Proof. Let Q = k Pk , where each Pk is closed. Given B of degree b, consider running the construction of Theorem 4.1, but beginning with T0 as the set of strings of the form τ0 ⊕ τ1 such that τ0 ⊂ B—so that all the Ts are now computable in B. Let the set A constructed be of degree a. If Q contains a member of every degree above b, then, in particular, some Pk must contain a member of degree a. Therefore, for some s = i, j, k it must be that case (4) applies. Then T = {Ψi (σ) : σ ∈ Ts+1 } is B-computable perfect and pointed, and [T ] ⊆ Q.  Theorem 4.4. A Σ03 class P contains a member of every degree above b iff there exists some b-computable perfect and pointed T with [T ] ⊆ P. Proof. Almost exactly the same as the proof of Theorem 4.3.  It follows easily from the fact that the class of all non-computable sets is Π03 and Gδ , that Theorems 4.1, 4.2, 4.3 and 4.4 are the best we can do—Theorems 4.1 and 4.3 do not hold for Gδ sets, and Theorems 4.2 and 4.4 do not hold for Π03 classes. Definition 4.1. We say that a degree a is invisible if any Π01 class which contains a member of degree a contains a member of every degree. The existence of invisible degrees immediately suffices to give the following corollary. Corollary 4.1. There do not exist α < 1P and β < 1P with α ∨ β = 1P . Proof. If neither α nor β contain any invisible degrees, then neither does their union.  License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ON THE DEGREE SPECTRUM OF A Π01 CLASS 5291 A simple analysis tells us something about the distribution of the invisible degrees. Since f ∈ P no fixed point free degree is invisible. We say a degree is completely unranked if it does not contain a ranked set. Since no degree containing a ranked set is invisible—any ranked set is a member of a countable Π01 class— the invisible degrees are a subclass of the completely unranked degrees and are all incomparable with 0′ . Since there are only countably many ranked sets, the completely unranked degrees are of measure 1. Since the fixed point free degrees are of measure 1, the invisible degrees are of measure 0 and so are a proper subclass of the completely unranked degrees. It was shown in [AL] that any hyperimmune-free degree which is not fixed point free has a strong minimal cover (and where b is a strong minimal cover for a if the degrees strictly below b are precisely the degrees below and including a). Thus all invisible degrees which are hyperimmune-free have a strong minimal cover. Recall that T ⊆ 2<ω is dense if for every τ there exists σ ⊇ τ in T , and that A is weakly 2-generic if for every dense set of strings T which is c.e. in 0′ there exists σ ⊂ A which is an element of T . A degree is weakly 2-generic if it contains a weakly 2-generic set. Theorem 4.5 (With Jockusch). Every weakly 2-generic degree is invisible. Proof. For every i, j, τ we try to define a string σ(i, j, τ ). In order to do so, let T be a Ψi -splitting set of strings (enumerated in some uniform way) which has τ as the single element of level 0, such that each element which is not a leaf has precisely two successors, such that for any σ which is a leaf of T there does not exist a Ψi splitting set of strings above σ, and such that at each stage of the enumeration of T we only enumerate in strings which properly extend leaves of the set of strings already enumerated into T . Let the strings in T be ordered according to their level and then from left to right. If there exists a least string σ in T such that either σ / Λj , then define σ(i, j, τ ) to be that string. If there is a leaf of T or else Ψi (σ) ∈ exists no such string, then σ(i, j, τ ) is undefined. Now if [Λj ] does not contain a member of every degree, then for each i the set of strings Ti = {σ(i, j, τ ) : τ ∈ 2<ω } is dense and c.e. in 0′ . If A is weakly 2-generic, then for each i there exists some σ ⊂ A which is in Ti , so that either Ψi (A) is partial or computable, or else is not an element of [Λj ]. The result follows since no weakly 2-generic is computable.  The reader may have noticed that the proof of Theorem 4.5 actually suffices to show that every weakly 2-generic degree is strongly invisible: Definition 4.2. We say that a degree a > 0 is strongly invisible if any Π01 class which contains a member of non-zero degree below a contains a member of every degree. Also, in the proofs of Theorems 4.1 and 4.2 we did not really need to consider ordered triples i, j, k. It would have sufficed to consider ordered pairs i, k in order to deduce according to this modified proof of Theorem 4.2, for example, that the degree constructed is strongly invisible. Of course the fact that the degree constructed is minimal suffices to show it is strongly invisible anyway. This extra level of complication was used so that we could immediately apply the proofs of Theorems 4.1 and 4.2 in order to deduce Theorems 4.3 and 4.4 (and also for later applications). Since the constructions of invisible degrees that we have described License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 5292 THOMAS KENT AND ANDREW E. M. LEWIS so far automatically produce strongly invisible degrees, however, it seems natural to ask whether all invisible degrees are strongly invisible. Note that this holds iff the invisible degrees are downward closed avoiding 0, i.e. iff any non-zero degree bounded by an invisible degree is invisible. In order to answer this question we need to consider a third category of invisibility: Definition 4.3. We say that a degree a is weakly invisible if any Π01 class which contains a member of every degree above a contains a member of every degree. Theorem 4.6. There exists a non-zero weakly invisible degree which is not invisible. Proof. We fix some effective bijection from ω to the finite subsets of 2<ω , and we let {Φi }i∈ω be an effective listing of the Turing functionals Φ such that: (i) for any σ, n, m if Φ(σ; n) ↓= m, then m codes some T which is finitely 2-branching and of level n; (ii) for any σ, m, m′ and n′ > n if Φ(σ; n) ↓= m and Φ(σ; n′ ) ↓= m′ , m codes T and m′ codes T ′ , then T ⊂ T ′ and all strings in T are of the same level in T as they are in T ′ . We construct downward closed computable Υ and A which is the only noncomputable element of [Υ], so as to satisfy requirements: Θi : A = Ψi (∅). Ξi : Let i = j, k. If Φj (A) is total and [Φj (A)] ⊆ [Λk ], then [Λk ] contains a member of every degree. It follows from Theorem 4.3 that this suffices to prove the result. The construction is most simply described using modules. The role of the module θi placed on σ will be to ensure that if σ ⊂ A, then Θi is satisfied, and the role of the module ξi placed on σ will be to ensure that if σ ⊂ A, then Ξi is satisfied. When any module is passed control at a stage s, it will either have outcome 0 or outcome 1 at this stage. If the outcome of the module placed on σ is d (∈ {0, 1}) at stage s, it may then pass control to its d-successor module which will be placed on a string extending σ ⋆ d, or else define its d-successor module if it has not already done so. If the module is passed control at an infinite number of stages and has outcome 1, then σ ⋆ 1 will be an initial segment of A. If the module is passed control at an infinite number of stages but has outcome 1 at only finitely many of these stages, then σ ⋆ 0 will be an initial segment of A. At the end of each stage s, if any string σ is a leaf of the present value Υ which has not been declared terminal, then we shall enumerate σ ⋆ 0 into Υ, and if a module is placed on σ, then we shall also enumerate the string σ ⋆ 1 into Υ. We use the variable υ to range over the modules. The instructions for the module θi placed on σ, when passed control at stage s. If Ψi (∅)[s] ⊇ σ ⋆ 0, then the module has outcome 1, and otherwise it has outcome 0. Suppose the outcome at stage s is d. If the d-successor module is already defined, then pass control to it. Otherwise let σ ′ be the unique leaf of the present value Υ extending σ ⋆ d, place the module ξi on σ ′ , and define this to be the d-successor module for this module (without passing control to it at this stage). The instructions for the module ξi placed on σ. Let υ be the module ξi placed on σ. The module uses a parameter τυ , which is initially defined to be σ ⋆ 0 and which may be redefined at any given stage at which the module is passed License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ON THE DEGREE SPECTRUM OF A Π01 CLASS 5293 control to be some extension of its previous value. If there is some stage after which this value is never redefined, then its final value will be an initial segment of A, and otherwise σ ⋆ 1 will be an initial segment of A (providing υ is passed control at an infinite number of stages). The basic idea is that the module is prepared to keep A looking computable on a long initial segment (the final value τυ , should this exist) extending σ ⋆ 0, so that if it cannot ensure that either Φj (A) is partial or [Φj (A)] ⊆ [Λk ] by making this initial segment τυ sufficiently long, then actually [Λk ] must contain [T ] for some 2-branching computable T . In this latter case, the requirement is automatically satisfied and σ ⋆ 1 will be an initial segment of A. When the module is passed control at stage s it performs instructions according to the first case below which applies. Let n be the greatest such that Φj (τυ ; n) ↓, or if there exists no such, then let n = −1. (1) Either: (i) n ≥ 0 and there exists some leaf of Φj (τυ ; n) which has no extension in Λk of length s, or (ii) there does not exist σ ′ which is a leaf of the present value Υ extending τυ which has not been declared terminal, and such that Φj (σ ′ ; n′ ) ↓ for some n′ > n. In this case the module has outcome 0. Pass control to the 0-successor module if this is already defined. Otherwise let σ ′′ be the unique leaf of the present value Υ extending τυ , place the module θi+1 on σ ′′ , and define this to be the 0-successor module for υ. (2) Since (1) does not apply there must exist σ ′ as in (1)(ii), so let σ ′ be such. Remove all modules from strings extending σ ⋆ 0, declare all leaves of Υ extending σ ⋆ 0 to be terminal except for σ ′ , and redefine τυ = σ ′ . Make the 0-successor of υ undefined. The outcome of the module is 1 at stage s. If the 1-successor module for υ is already defined, then pass control to it. Otherwise let σ ′′ be the unique leaf of the present value Υ extending σ ⋆ 1, place the module θi+1 on σ ′′ , and define this to be the 1-successor module for υ. The construction. Stage 0. Enumerate λ, 0 and 1 into Υ, and place the module θ0 on λ. Stage s > 0. Pass control to the module θ0 placed on λ, passing control to other modules and carrying out their instructions according to the instructions for each module detailed above, until some module defines a successor module. For each leaf σ of the present value Υ which has not been declared terminal, enumerate σ ⋆ 0 into Υ, and if a module is placed on σ, then also enumerate the string σ ⋆ 1 into Υ.  We define A = i σi as follows. We let υ0 be the module θ0 placed on λ. Suppose we are given υi , and suppose that υi is placed on σ. We define σi = σ. If there exists an infinite number of stages at which υi is passed control and has outcome 1, then let υi+1 be the 1-successor of υi . Otherwise there exists υ which is defined to be the 0-successor of υi at some stage of the construction after which the 0-successor is never made undefined. Define υi+1 = υ. The verification. It is clear that Υ is downward closed and computable, that A ∈ [Υ] and that all requirements Θi are satisfied. It is also clear that A is the only non-computable element of [Υ], since all other elements are either a finite string concatenated with an infinite string of zeros or else are the infinite string extending License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 5294 THOMAS KENT AND ANDREW E. M. LEWIS all values taken by τυ for some fixed υ. In order to see that all requirements Ξi are satisfied, consider the action of υ2i+1 (as in the definition of A). Let i = j, k. If Φj (A) is total and [Φj (A)] ⊆ [Λk ], then it must be the case that this module has outcome 1 at an infinite number of stages. Let B be the infinite string extending all values taken by τυ2i+1 . Then B is computable, Φj (B) is total and [Φj (B)] ⊆ [Λk ].  Theorem 4.7. A degree is weakly invisible iff it is bounded by an invisible degree. Proof. Any degree bounded by an invisible degree is clearly weakly invisible, so it suffices to show that any weakly invisible degree is bounded by an invisible degree. In order to see this suppose given B of degree b which is weakly invisible, let Pk be the kth Π01 class and consider running the construction of Theorem 4.1, but beginning with T0 as the set of strings of the form τ0 ⊕ τ1 such that τ0 ⊂ B. Now whenever case (4) applies we may deduce that Pk contains [T ] for some perfect pointed b-computable T . Since b is weakly invisible, Pk therefore contains a member of every degree.  Corollary 4.2. There exist invisible degrees which are not strongly invisible. Proof. By Theorem 4.6 there exists a weakly invisible degree which is non-zero and which is not invisible. By Theorem 4.7 this degree is bounded by an invisible degree which cannot be strongly invisible.  We conclude this section by showing, as promised, that there exist degrees a containing a set of rank 1, and such that {0} ∪ {b : b ≥ a} is not the degree spectrum of a Π01 class (in fact this theorem can also be deduced from an analysis of the proof of Theorem 4.6, but the following lemmas give a more direct proof). The proof of the following theorem is essentially a generalization of Kucera’s proof that every degree above 0′ is random. Theorem 4.8. Let P be a Π01 class. If there exists a 2-branching T and a computable function f such that [T ] ⊂ P and such that, for all l, all strings in T of level l are of length ≤ f (l), then P contains a member of every degree above 0′ . Proof. Given P, T and f as in the statement of the theorem, we prove first that there exists some computable function f ∗ such that the set of all T ′ satisfying the following condition is non-empty: (∗) T ′ is 2-branching, [T ′ ] ⊆ P and, for all l, all strings in T ′ of level l are of length f ∗ (l). We may suppose that f (0) is the length of the string of level 0 in T . Define f ∗ (0) = f (0) and f ∗ (1) = f (1), and for all l ≥ 1 define f ∗ (l + 1) = f (f ∗ (l) + 1). It is clear that the set of T ′ satisfying (∗) above is non-empty, since for any A ∈ [T ] and any l there exist at least two incompatible strings in T extending A ↾ f ∗ (l) and of length at most f ∗ (l + 1). Note that the set of T ′ satisfying (∗) can be thought of as a Π01 class when effectively coded in the appropriate way (and that the existence of f was essential in order that this should be the case). Next we define T ∗ ≤T ∅′ satisfying (∗) above by recursion. Let the string of level 0 in T ∗ be the string of level 0 in T . Suppose we have defined all the strings of level l in T ∗ . For each σ of level l let σ0 be the leftmost string extending σ of length f ∗ (l + 1) and which is in some T ′ satisfying (∗). Let σ1 be the rightmost License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ON THE DEGREE SPECTRUM OF A Π01 CLASS 5295 string extending σ of length f ∗ (l + 1) and which is in some T ′ satisfying (∗). Let σ0 and σ1 be the two successors of σ in T ∗ .  Given A of degree above 0′ we define B = l σl such that B ∈ P and B ≡T A. Define σ0 to be the string of level 0 in T ∗ . Given σl , define σl+1 to be the leftmost successor of σl in T ∗ if A(l) = 0, and define σl+1 to be the rightmost successor otherwise. That B ≤T A is clear and that A ≤T B follows from the fact that any σl+1 can compute whether it is the leftmost or rightmost string extending σl of  length f ∗ (l + 1) and which is in some T ′ satisfying (∗). Corollary 4.3. If a is hyperimmune-free, then any Π01 class which contains a member of every degree above a must also contain a member of every degree above 0′ . Proof. If P contains a member of every degree above a, then by Theorem 4.3 there exists 2-branching T such that [T ] ⊆ P and which is of degree a. If a is hyperimmune-free there exists computable f such that, for all l, all strings in T of level l are of length ≤ f (l). Thus by Theorem 4.8, P contains a member of every  degree above 0′ . Corollary 4.4. There exists a degree a containing a set of rank 1 for which it is not the case that {0} ∪ {b : b ≥ a} is the degree spectrum of a Π01 class. Proof. This follows immediately from Corollary 4.3, since Downey [RD] has shown that there exists a hyperimmune-free degree a containing a set of rank 1 (and which is therefore non-zero).  5. Minimal degree spectra Definition 5.1. We say that α = 0P is subclass invariant if for any Π01 class P with S(P) = α and any non-empty Π01 class P ′ ⊆ P, S(P ′ ) = α. We say that α = 0P is weakly subclass invariant if there exists a Π01 class P with S(P) = α and for any non-empty Π01 class P ′ ⊆ P, S(P ′ ) = α. Clearly any α which is minimal must be subclass invariant. Now suppose that α is subclass invariant, let P be a Π01 class with S(P) = α and suppose that P ′ is a non-empty Π01 class with S(P ′ ) ⊂ α. Then {0⋆A : A ∈ P}∪{1⋆A : A ∈ P ′ } is a Π01 class with degree spectrum α and witnesses the fact that α is not subclass invariant, a contradiction. Thus being subclass invariant is equivalent to minimality. Theorem 5.1. Suppose that α is weakly subclass invariant. If a Π01 class contains any member of any hyperimmune-free degree in α, then it contains a member of every degree in α. Proof. Let P0 be a Π01 class with S(P0 ) = α and such that for any non-empty Π01 class P ⊆ P0 , S(P) = α. Let P1 be a Π01 class which contains A of hyperimmunefree degree in α. Then there exists B ≡tt A and which is in P0 . Let i, j be such that the total functionals Φi and Φj satisfy Φi (A) = B and Φj (B) = A. Now let P2 be the Π01 class {C : C ∈ P0 & (∃D ∈ P1 )[Φi (D) = C & Φj (C) = D]}. Then S(P2 ) ⊆ S(P1 ). Since P2 is non-empty and P2 ⊆ P0 , S(P2 ) and S(P1 ) contain every degree in α.  Corollary 5.1. α is minimal iff it is weakly subclass invariant. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 5296 THOMAS KENT AND ANDREW E. M. LEWIS Proof. If α is weakly subclass invariant, then, by the hyperimmune-free basis theorem, any non-empty Π01 class which contains only members of degree in α contains a member of hyperimmune-free degree in α and therefore, by Theorem 5.1, contains a member of every degree in α.  Corollary 5.2. r is minimal in (P, <). Proof. Any non-empty Π01 class containing only random sets is witness to the fact that r is weakly subclass invariant, since any Π01 class containing a random set is of positive measure and any Π01 class of positive measure contains a member of every random degree [AK].  The random degrees, then, are an example of a degree spectrum α which satisfies this nice property that there exists a Π01 class P such that S(P) = α and such that any non-empty Π01 class P ′ ⊆ P has degree spectrum α. Corollary 5.1 says that the minimal elements of P are precisely those degree spectra which satisfy this property. Recall that a degree is PA iff it contains a {0, 1}-valued DNC function. If P0 and P1 are Π01 classes and there exist A ∈ P0 , B ∈ P1 with B ≤tt A, then it is easy to see that there exists a non-empty Π01 class P2 ⊆ P0 such that every element of P2 tt-computes a member of P1 . We may immediately deduce that no random set tt-computes a {0, 1}-valued DNC function, since no hyperimmune-free random does (if a is random and hyperimmune-free, then every non-zero b < a is random [OD] and is not PA [FS]). Theorem 5.2 below was first proved by Simpson. A sketch of the following short proof was provided by Kucera in a private correspondence. Theorem 5.2 (Simpson [SS2]). Any non-empty Π01 class containing only {0, 1}valued DNC functions contains a member of every PA degree. Proof. If Λ is computable and downward closed, then consider Ψi (∅) such that Ψi (∅; i) ↓= n iff there exists some l > i such that τ (i) = n for all τ ∈ Λ of length l. By the uniformity of the recursion theorem it follows that there exists computable f such that, whenever [Λj ] is non-empty and contains only {0, 1}-valued DNC functions, there exist A, B ∈ [Λj ] with A(f (j)) = 0 and B(f (j)) = 1. Now suppose we are given j0 such that [Λj0 ] is non-empty and contains only {0, 1}-valued DNC functions. Let A be a {0, 1}-valued DNC function. We construct  B = s σs which is in [Λj0 ] and is of the same degree as A. Stage 0. Define σ0 = λ. Stage s > 0. We have already decided js−1 and σs−1 . There exists C ∈ [Λjs−1 ] with C(f (js−1 )) = A(s − 1). Using the oracle for A we can therefore compute σ of length f (js−1 )+1 such that σ(f (js−1 )) = A(s−1) and which is an initial segment of some C ∈ [Λjs−1 ] (this follows using the standard argument that any {0, 1}-valued DNC function computes a member of any non-empty Π01 class). Define σs = σ and define js so that [Λjs ] is the set of all C ∈ [Λjs−1 ] which extend σ. That B computes A follows from the fact that an oracle for B allows us to retrace every step of the construction defining B.  Corollary 5.3 (Simpson [SS2]). p is minimal in (P, <). Proof. By Theorem 5.2 and Corollary 5.1.  Corollary 5.3 tells us that, while there are sentences which are independent of PA, from a degree theoretic point of view it is already ‘saturated’, in the sense that License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ON THE DEGREE SPECTRUM OF A Π01 CLASS 5297 any consistent and axiomatizable extension of the theory has complete and consistent extensions of precisely the same degrees as PA. The latter result was originally proved by Kripke and Pour-El [KP], who proved, in fact, that the set of consistent completions of any consistent axiomatizable extension of PA is recursively homeomorphic to the set of consistent complete extensions of PA. In fact, since every Π01 class is recursively homeomorphic to the class of complete extensions of some finitely axiomatizable theory (a result due to Hanf and Peretyakin [HP]), the minimal elements of P are precisely the degree spectra corresponding to axiomatizable theories which are saturated in this sense. We shall say that T is homogenous if all strings of the same level in T are of the same length and, whenever σ0 and σ1 are of the same level in T and σ0 ⋆ τ ∈ T , σ1 ⋆ τ is also in T . Recall that a Π01 class P is thin if for any Π01 class P ′ ⊂ P there exists a clopen set Q such that P ′ = P ∩ Q. Theorem 5.3. For any α < 1P there exists β which is minimal and such that β ≤ α. Proof. We suppose given downward closed computable Λ such that S([Λ]) = α is not the set of all Turing degrees, and we approximate a 2-branching and homogenous T such that P = [T ] is a thin Π01 class. At every stage of the construction our approximation to T will be a finitely 2-branching set of strings. In order to ensure that P is a Π01 class it suffices that, if at any stage we redefine the strings in T of level l, we define each of these strings to extend leaves of the previous version of T , and we enumerate strings into T properly extending each of its leaves at the end of each stage. The fact that T is 2-branching and homogenous and that P is thin suffices to show that β = S(P) is minimal and does not contain 0. We shall also approximate A ∈ P such that any non-computable B ≤T A is not in [Λ]. Since P does not have any computable members, this suffices to ensure that β ≤ α. We must satisfy all requirements: Θi : If [Λi ] ⊂ P, then there exists a clopen Q such that [Λi ] = P ∩ Q. Ξi : If Ψi (A) is total and non-computable, then Ψi (A) ∈ / [Λ]. There is a difficulty inherent in looking to satisfy the requirements Ξi . The basic idea behind the module to satisfy the requirement is that, given some string σ which is potentially an initial segment of A, we should search for Ψi -splittings above this string. Either we can find some string above which there is no Ψi -splitting, or we can find some string τ such that Ψi (τ ) is not in Λ, or else we can enumerate a perfect Ψi -splitting tree whose image under Ψi is a perfect and computable tree with all infinite paths in [Λ]—contradicting the fact that the latter set does not contain a member of every degree. The complication, however, is that carrying out this procedure requires us to search for Ψi -splittings inside some Q which is a Π01 subclass of P. It is quite possible to have a Π01 class Q and A ∈ Q such that Ψi (A) is total and non-computable, but such that no two initial segments of elements of Q are Ψi -splitting. The solution we shall adopt is to construct A which is of hyperimmune-free degree. Then if Ψi (A) = B, there exists a total Turing functional Φ such that Φ(A) = B. The totality of Φ means that {Φ(C) : C ∈ Q} is a Π01 class, and so cannot consist of a single non-computable member. We therefore construct A which is of hyperimmune-free degree and act in order to satisfy all requirements Ξi . Since we succeed for all i such that Ψi is total, this suffices to show that, in fact, we succeed for all i. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 5298 THOMAS KENT AND ANDREW E. M. LEWIS Ultimately, then, we shall have to ensure for every i that either Ψi (A) is partial or else A lies on some computable Ti satisfying the property that, for any string σ of level l + 1, Ψi (σ; l) ↓. We shall ensure that each string in Ti eventually either has two successors defined or else is irrevocably declared terminal. If A lies on Ti of this kind, then it is not difficult to see that Ψi (A) is majorized by a computable function. Rather than immediately presenting the full construction, however, we shall first describe a simplified version which satisfies all requirements Θi and which satisfies all requirements Ξi such that Ψi is a total functional, but which does not act in order to ensure that A is of hyperimmune-free degree. The module θi . In order to satisfy requirement Θi we may place at some stage of the construction and for some l, a module θi on all strings of level l in our present approximation to T . At every stage strictly after this, until it has been declared satisfied (or else is removed), the module θi placed on σ searches for some string σ ⋆ τ which is a leaf of T and which is not in Λi . If it finds such a string, then we say that the module requires attention. If we then act for this module, it will remove all strings of level > l from T together with any modules or submodules placed on these strings, and for each string σ ′ of level l it will enumerate σ ′ ⋆ τ ⋆ 0 and σ ′ ⋆ τ ⋆ 1 into T before declaring itself satisfied. Any module θi placed on σ of level l will subsequently be removed iff our approximation to T changes at a level ≤ l. Now suppose that, subsequent to placing at some stage of the construction a module θi on each string in T of level l, these modules are never removed. Suppose also that for each of these modules there is either a stage after which it never requires attention or else a stage at which it is declared satisfied. Then for each σ of level l in T , [Λi ] either contains every element of P extending σ, or else contains no such element of P. The module ξi . In order to satisfy requirement Ξi in the case that Ψi is total, we utilize the fact that [Λ] cannot contain the set of all paths through any 2-branching computable T ′ . In this simplified version of the construction we can define a string σs at each stage s, so that the sequence {σs }s∈ω approximates A. Suppose that, at some stage s, σ is a string of level l in our present approximation to T and that σ extends σs . If we place the module ξi on σ, then we also place a submodule ξi (0) on this string. It is the role of the module ξi and all of its submodules to ensure that Ξi is satisfied. At every stage strictly after being placed on σ, until it has been declared satisfied (or else is removed), the module ξi searches for σ ⋆ τ which is a leaf of our present approximation to T and such that Ψi (σ ⋆ τ ) ∈ / Λ. If it finds such a string, then we say that the module requires attention. If we then act for this module, it will remove all strings of level > l from T together with any modules or submodules placed on these strings, and for each string σ ′ of level l it will enumerate σ ′ ⋆ τ ⋆ 0 and σ ′ ⋆ τ ⋆ 1 into T , before declaring itself satisfied. It is the role of the submodules to ensure that if Ψi is total and Ψi (A) is noncomputable and if the module ξi placed on σ is never declared satisfied, then [Λ] must contain [T ′ ] for some 2-branching computable T ′ , a contradiction. As the construction progresses we may have to place more submodules ξi (j) on strings in T extending σ, but this process will only continue for a finite number of stages— only a finite number of submodules of this form will be placed on strings in T . If the submodule ξi (j) is placed on σ ′ ⊇ σ, then this will mean that σ ′ is of level License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ON THE DEGREE SPECTRUM OF A Π01 CLASS σ ′′ ∗ τ0 and σ ′′ ∗ τ1 are Ψi -splitting for all σ ′′ ∈ U . τ0 τ1 U ξi (2) ξi (2) ξi (2) ξi (1) σ′ 5299 This submodule is the least not to be declared satisfied. ξi (2) ξi (1) ξi (0) σ ξi Figure 1. The placement of submodules. l′ = l + j in T and that the set of strings of the form Ψi (σ ′′ ) such that some submodule ξi (j ′ ) for j ′ ≤ j is placed on σ ′′ is finitely 2-branching and of level j. At every stage strictly after being placed on σ ′ until it has been declared satisfied (or else is removed), the submodule ξi (j) acts as follows. Let σ ′ ⋆ τ0 and σ ′ ⋆ τ1 be the leftmost and rightmost successors of σ ′ in T , respectively. Let U be the set of all strings σ ′′ strictly to the left of σ ′ on which submodules ξi (j) have been placed (all these strings will also be of level l′ ). If it is not the case that all submodules ξi (j) placed on strings in U have already been declared satisfied, then the submodule performs no action at this stage. Otherwise, it performs one of the instructions below according to the state that it is presently in. The submodules ξi (j) placed on strings in U will have chosen τ0 and τ1 so that for each string σ ′′ ∈ U , σ ′′ ⋆ τ0 and σ ′′ ⋆ τ1 are Ψi -splitting. Now this submodule aims to extend this process by finding a splitting above σ ′ ⋆ τ0 , a splitting above σ ′ ⋆ τ1 and then choosing one string from each of these splittings in order to give τ0⋆ and τ1⋆ such that σ ′′ ⋆ τ0⋆ and σ ′′ ⋆ τ1⋆ are Ψi -splitting for each σ ′′ ∈ U ∪ {σ ′ }. The submodule is initially in state 0. State 0. The submodule searches for σ ′ ⋆ τ0 ⋆ τ2 and σ ′ ⋆ τ0 ⋆ τ3 which are leaves of the present approximation to T and which are Ψi -splitting. If it finds such strings then we say that the submodule requires attention. If we then act for the submodule it will remove all strings from T of level > l′ + 1 together with any modules or submodules placed on these strings, and for each string σ ′′ of level l′ + 1 it will enumerate σ ′′ ⋆ τ2 and σ ′′ ⋆ τ3 into T , before declaring itself to be in state 1. State 1. The submodule searches for σ ′ ⋆ τ1 ⋆ τ4 and σ ′ ⋆ τ1 ⋆ τ5 , which are leaves of the present approximation to T and which are Ψi -splitting. If it finds such strings, then we say that the submodule requires attention. If we then act for the submodule, it will proceed as follows. Choose τ0⋆ such that σ ′ ⋆ τ0⋆ is a leaf of the present approximation to T extending one of {σ ′ ⋆ τ0 ⋆ τ2 , σ ′ ⋆ τ0 ⋆ τ3 } and License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 5300 THOMAS KENT AND ANDREW E. M. LEWIS τ1⋆ ∈ {τ1 ⋆ τ4 , τ1 ⋆ τ5 }, such that σ ′ ⋆ τ0⋆ and σ ′ ⋆ τ1⋆ are Ψi -splitting. Remove all strings of level > l′ from T , together with any modules or submodules placed on these strings and for each σ ′′ of level l′ enumerate σ ′′ ⋆ τ0⋆ and σ ′′ ⋆ τ1⋆ into T , before declaring the submodule to be satisfied. Hopefully it is reasonably clear how we should place modules and submodules on strings in T in order to define this simplified version of the construction. Since this is not the final version of the construction, we shall not describe it in absolute detail. On each level of T we place only one kind of module or submodule (except in the case of modules ξi when a module ξi (0) will also be placed on the same string). If we have placed modules θi on all strings of the present approximation to T of level l, then we place a module ξi , together with the submodule ξi (0), on some string of level l + 1 which extends σs , our present approximation to A. If all submodules ξi (j) placed on strings of level l′ have been declared satisfied, then above each of these we place two submodules ξi (j + 1) on strings of level l′ + 1. If, on the other hand, one of these submodules is yet to be declared satisfied, we place modules θi+1 on all strings of level l′ + 2. If ξi has been placed on a string in T and is satisfied, then we place modules θi+1 on all strings of the next level. The modules are prioritized first according to the level of the string in our present approximation to T on which they are placed (with lower levels being of higher priority) and then in order from left to right. A module ξi placed on σ has higher priority than a submodule ξi (0) placed on the same string. At each stage, if any module or submodule requires attention, then we act for that of highest priority. The modules ξi and the submodules ξi (j) determine our approximation to A at each stage in the obvious way. At the end of each stage, for each leaf σ of the present T , we enumerate σ ⋆ 0 and σ ⋆ 1 into T . Modifying the construction. In order to ensure that A is of hyperimmune-free degree, the entire construction must be carried out within nested trees of the kind described previously. Computable sets of strings Ti , that is, such that for any string σ of level l + 1, Ψi (σ; l) ↓. Each string in Ti must eventually either be declared terminal or else have two successors defined. We begin, then, by attempting to enumerate an infinite tree T0 . Responsibility for this task is given to a supermodule υ0 . At each stage of the construction this supermodule considers the leaves of T0 to be ordered first according to their level in the tree and then from left to right, and the supermodule searches for strings to enumerate into the tree extending the least leaf which has not been declared terminal, σ say. If the supermodule never finds such a pair of strings, then we define A so as to extend σ, and in doing so we ensure that Ψ0 (A) is partial. While searching for strings to enumerate into T0 extending σ, we must continue with a construction which works above σ and which assumes that Ψ0 (A) will be partial. Above σ, then, we work to satisfy requirements Θ0 and then Ξ0 before beginning construction of a tree T1 . As the tree T0 continues to be enumerated, on the other hand, we can proceed with a construction which takes place inside T0 , i.e. which assumes that A will lie on T0 . After placing modules θ0 and ξ0 , however, we cannot proceed just as we did for the simplified construction but now working inside T0 , since so far we have only ensured (as long as A lies on T0 ) that Ψ0 (A) is majorized by a computable function. Next we must deal with Ψ1 . We therefore begin enumeration of a tree T1 inside T0 , and this task is assigned to another supermodule. Note, however, License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ON THE DEGREE SPECTRUM OF A Π01 CLASS 5301 that as observed in the discussion above, we also had to enumerate a tree T1 above σ which was the least leaf of T0 not to be declared terminal, while we searched for strings to enumerate into T0 extending σ. In fact, then, we need to construct various candidates Ti,k for each Ti , where Ti,k may be regarded as the kth such candidate. Suppose that T0,0 (our only candidate for T0 ) turns out to be infinite. If T1,k , the candidate for T1 that we construct inside T0 , turns out to be finite, then A will extend the least leaf of this tree which is not declared to be terminal. If T1,k turns out to be infinite, then we will continue a construction which works within this tree, A will be a path through this tree, and so on. There are some small technical issues that are worth discussing before describing the construction precisely. Up until this point, we have described modules which search for strings satisfying certain properties amongst the leaves of T . As previously described, the module ξi placed on σ of level l in T , for example, searches for σ ⋆ τ , which is a leaf of our present approximation to T and such that Ψi (σ ⋆ τ ) ∈ / Λ. If such a string is found, then we remove all strings from T of level > l and redefine the strings of level l + 1 in some appropriate way. Now, however, suppose that we are constructing T1,k inside T0,0 . Suppose that we are searching for a pair of strings to enumerate into T1,k extending σ, which is also a string in T0,0 but which is not a leaf of this tree. If we only search amongst strings which are leaves of T0,0 , then these strings may not be leaves of T . Upon enumerating these stings into T1,k we would have to be careful about the way in which we redefine T in order to accommodate this enumeration. While this would not be very difficult to do, a slicker approach is actually just to search for strings to enumerate into T1,k amongst the leaves of T as before (which will extend leaves of T0,0 ). Upon enumerating such strings into T1,k we may then have to redefine some of the leaves of T0,0 extending σ so as to be extensions of their previous values, so that T1,k should remain a subtree of T0,0 . Since it is basically only the leaves of any of the Ti,k which may be redefined a finite number of times each in this way, each of the Ti,k will still be computable. In order to avoid much repetition in what follows, then, we agree now to the following conventions which dictate action which is to be automatically taken whenever our approximation to T is modified. Some conventions. Suppose that all strings of level > l are removed from T and σ ⋆ τ0 and σ ⋆ τ1 are enumerated into T for every string σ of level l. C1 We remove all supermodules, modules or submodules placed on strings of level > l in T . C2 Suppose σ is of level l in T , σ ′ ⊇ σ is in Ti,k , τ2 is a leaf of Ti,k , and that τ2 and τ3 are the successors of σ ′ in Ti,k . • If τ2 ⊂ σ ⋆ τ0 and τ2 ⊂ σ ⋆ τ1 , then we remove τ2 and τ3 from Ti,k (together with any strings extending τ3 ) and enumerate σ⋆τ0 and σ⋆τ1 into Ti,k . • If τ2 ⊂ σ⋆τ0 but τ2 ⊂ σ⋆τ1 , then we remove τ2 from Ti,k and enumerate σ ⋆ τ0 into Ti,k . C3 Any leaf of any tree which is not compatible with any leaf of T is declared terminal. All submodules, modules and supermodules are prioritized just as before; first according to the level of the string in our present approximation to T on which they are placed, and then in order from left to right. A module ξi placed on σ License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 5302 THOMAS KENT AND ANDREW E. M. LEWIS has higher priority than a submodule ξi (0) placed on the same string. All trees are considered to be prioritized according to the supermodule with which they are associated. When we pass control to a module it is useful to have some tidy way of recording the nested set of trees that this module works within. At any point in the construction and for any σ, we define T (σ) to be the set of all Ti,k (associated with supermodules placed on strings in T and which have not been removed) such that σ ∈ Ti,k but is not a leaf of this tree. T (σ), then, is the nested set of trees which any module placed on σ believes to be infinite. Any module, submodule or supermodule placed on the leaf of any tree believes that tree to be finite. Let T ′ be the lowest priority tree in T (σ) when the supermodule υi is first passed control, having been placed on σ (or if this set is empty, then let T ′ = T ). T ′ is the tree within which the supermodule must build its candidate for Ti . The supermodule υi . When placed on the string σ the supermodule υi will find the least k such that Ti,k is not yet associated with any supermodule (in other words it will choose some fresh tree to begin enumerating), and we will enumerate the string σ into Ti,k . We shall say that Ti,k is the tree associated with the supermodule. Let T ′ be the lowest priority tree in T (σ) when the supermodule is first passed control, or if this set is empty, then let T ′ = T . T ′ is the tree within which the supermodule must build Ti,k . At each stage strictly after that at which it is placed on σ (until any point at which it is removed), the supermodule will proceed as follows. Let σ ′ be the least leaf of Ti,k which has not been declared terminal, and suppose this string is of level l in Ti,k and of level l′ in T . If σ ′ is not a leaf of T ′ , then the supermodule checks to see if there exists σ ′ ⋆ τ which is a leaf of T and such that Ψi (σ ′ ⋆ τ ; l) ↓ (if σ ′ is a leaf of T ′ , then we cannot yet continue to build Ti,k above this string and must wait for further strings to be enumerated into T ′ ). If it finds such a string, then we say that the supermodule requires attention. If we then act for the supermodule, it will perform the following instructions: (1) Remove all strings of level > l′ from T . For each string σ ′′ of level l′ in T , enumerate σ ′′ ⋆ τ ⋆ 0 and σ ′′ ⋆ τ ⋆ 1 into T . (2) Enumerate σ ′ ⋆ τ ⋆ 0 and σ ′ ⋆ τ ⋆ 1 into Ti,k (note that (1) may already have caused changes to Ti,k according to C2). In what follows, we say that Ti,k extends to level l if all leaves of this tree which have not been declared terminal are of level at least l in T . Given any T (σ), let T ′ be the lowest priority tree in this set, or if this set is empty, then let T ′ = T . We say that T (σ) extends to level l if T ′ does. The instructions for the modules θi and ξi , as well as for the submodules ξi (j), are exactly as previously described, except for one small modification. Suppose that the submodule ξi (j) is placed on σ of level l in T . The submodule does not act before the first stage at which it is the least submodule (for the corresponding module ξi ) not to be declared satisfied, and T (σ) extends to level l + 2. In the description of the module ξi we discussed being able to give an approximation to A at each stage of the construction. This situation no longer holds, but that fact does not have any effect on the instructions defined in that section. Guessing initial segments of A. In the simplified version of the construction we could approximate A. Of course, now that A must be hyperimmune-free (and noncomputable) it cannot be of degree below 0′ . Rather than approximate A during License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ON THE DEGREE SPECTRUM OF A Π01 CLASS 5303 the construction, we simply define it at the end of the construction to be the only infinite path through all trees Ti,k which turn out to be infinite. Nevertheless we do need to have some way of defining guesses as to initial segments of A during the course of the construction, based on guesses as to which of the trees Ti,k will be infinite. Let T = {Ti0 ,k0 , · · · , Tin ,kn } (= T (σ ′ ) for some string σ ′ ) be such that, for j < j ′ , the supermodule associated with Tij ′ ,kj ′ is placed on a string in Tij ,kj which is not a leaf. The indices, then, order the trees according to the priority of their associated supermodules, and, according to this ordering, each tree is built inside its immediate predecessor. We shall call a set of trees of this kind a cluster of trees. At each point of the construction we define a value σ(T ) which is a finite binary string that should be an initial segment of A, should all the trees in T turn out to be infinite and should all other trees associated with supermodules placed on strings in these trees at this point in the construction turn out to be finite. We define σ(T ) by a finite iterated process, where at each stage j in this process we define a value σj extending all previous values. At even stages in this process we look to see which string the next ξi module placed on a string extending σj inside this cluster of trees presently believes should be an initial segment of A. At odd stages we consider the next supermodule which tries to build a tree inside this cluster of trees, but which is not in T . The least non-terminal leaf of this tree determines σj+1 . Let σ0 be the string on which Tin ,kn is placed and let T ′ = Tin ,kn , or if T is empty, then let σ0 = λ and T ′ = T . Given σj such that j is even: if there is a least module ξi (for any i and ordered according to their level in T ) placed on a string σ ∈ T ′ which is not a leaf and which extends σj , then proceed as follows, and otherwise define σ(T ) = σj and terminate the iterated process. If the module ξi placed on σ has been declared satisfied, then define σj+1 = σ. Otherwise let ξi (j) placed on σ ′ be the lowest priority submodule of this module which has not been declared satisfied and define σj+1 = σ ′ . If there is no such submodule, define σ(T ) = σ and terminate the iterated process. Given σj such that j is odd: if there is a least supermodule υi placed on a string σ ∈ T ′ which is not a leaf of T ′ and which extends σj , then proceed as follows, and otherwise define σ(T ) = σj and terminate the iterated process. Let σ ′ be the least leaf of the tree associated with υi placed on σ which has not been declared terminal. Define σj+1 = σ ′ . The construction. At each stage s we proceed as follows. Stage s = 0. Enumerate λ, 0 and 1 into T . Place the supermodule υ0 on λ. Stage s > 0. If any supermodule requires attention, then act for that of highest priority. Then, if any other (different) supermodule requires attention, act for that of highest priority, and so on, until no supermodule requires attention. If any module or submodule then requires attention, act for that of highest priority. Next we place sufficient supermodules, modules and submodules on strings in T (and no more), in order to guarantee that all of the following conditions are satisfied. These instructions basically just say that the previous simplified version of the construction should be carried out inside each cluster of trees and that every so often we must begin building new trees. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 5304 THOMAS KENT AND ANDREW E. M. LEWIS (1) We continue to build the simplified construction inside a cluster. (a) If the module θi is placed on σ of level l in T , then let T (σ) = T . If T extends to level l + 2, then the module ξi and the submodule ξi (0) should both be placed on σ ′ if this string is the leftmost string of level l + 1 in T compatible with σ(T ). (b) Suppose that ξi is placed on a string σ of level l in T and has not been declared satisfied. If, for some j, all submodules ξi (j) have been declared satisfied and if T (σ) extends to level l+j+2, then a submodule ξi (j +1) should be placed on each string in T of level l+j +1 extending σ. (2) We begin building a new tree inside the cluster. If either: (a) the module ξi is placed on σ of level l in T and has been declared satisfied or (b) the module ξi is placed on a string in T and has not been declared satisfied, and ξi (j) placed on σ of level l − 1 in T is the leftmost submodule of this module not to have been declared satisfied, then for σ ′ which is the leftmost string in T of level l + 1 extending σ, the supermodule υi+1 should be placed on σ ′ if T (σ) extends to level l + 2. (3) We continue to build the simplified construction within an extended cluster. If the supermodule υi is placed on σ of level l in T and if the tree associated with this supermodule extends to level l + 2, then the module θi should be placed on all strings in T of level l + 1. (4) We continue to build the simplified construction within the cluster, after a failed attempt to build another tree within the cluster. If σ is of level l in T and is the least non-terminal leaf of some Ti,k , then let T (σ) = T . Let i′ be the least such that no module ξi′ is placed on any string σ ′ of level < l in T such that T (σ ′ ) ⊆ T . The module θi′ should be placed on all strings in T of level l + 1 if T extends to level l + 2. Once these instructions are completed, enumerate σ ⋆ 0 and σ ⋆ 1 into T for each leaf σ of T . A is defined to be the infinite string extending all σ on which a supermodule is placed and such that the tree associated with that supermodule is infinite. The verification. The instructions for the supermodule υi implicitly assume that any leaf of its associated tree Ti,k which has not been declared terminal will also be a string in T . That this assumption is correct can easily be seen by induction on stage s using convention C2. It is then clear that our approximation to T is finitely 2-branching at each stage. That our approximation to T converges and is 2-branching follows since: (1) each module or submodule can only redefine T a finite number of times; (2) each supermodule can only define any given level of T a finite number of times; (3) any module, submodule or supermodule can only redefine levels of T strictly above that on which they are placed; (4) only a finite number of modules, submodules or supermodules are ever placed on any given level of T . That P = [T ] is a Π01 class follows immediately from the fact that, if at any stage we redefine the strings in T of level l, we define each of these strings to extend leaves License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ON THE DEGREE SPECTRUM OF A Π01 CLASS 5305 of the previous version of T , and we enumerate strings into T properly extending each of its leaves at the end of each stage. That each Ti,k is computable follows from the fact that if Ti,k is infinite and if σ ∈ Ti,k has successors in this set at any stage of the construction, then there exists a stage s such that at this stage: • σ has successors τ0 and τ1 ; • each τi has either been declared terminal or else has successors in Ti,k . Neither of the τi will be removed from Ti,k subsequent to stage s, and any τi which has been declared terminal will remain a leaf of Ti,k . Suppose that some module (or some submodule or supermodule) is placed on σ and that T ′ is the lowest priority tree in T (σ) when this module is first passed control, or that T ′ = T if this set is empty. We call T ′ the parent tree for the module. If a module (or a submodule or supermodule) is placed on some string in T and is never subsequently removed, then we say that the module is permanent. The fact that A is properly defined and is hyperimmune-free and that all other requirements are satisfied as required then follows easily from the discussion prior to the precise description of the construction and from the existence of a sequence of strings τυ0 ⊂ τθ0 ⊂ τξ0 ⊆ · · · ⊆ τυi ⊂ τθi ⊂ τξi ⊆ · · · such that the following conditions are satisfied: (i) Each string in this sequence is a string in the limit value T . (ii) No permanent supermodule, whose associated tree is infinite, is placed on a string incompatible with any member of the sequence. Any member of the sequence τ is eventually always compatible with σ(T (τ )). (iii) Either: (a) a permanent supermodule υi is placed on τυi and the tree associated with this supermodule is infinite or (b) τυi is a leaf of a tree Ti,k associated with a permanent supermodule whose parent tree is infinite. Furthermore, τυi is the least leaf of Ti,k which is never declared terminal. (iv) The string τθi is of some level, l say, in the limit value of T . Permanent modules θi are placed on all strings in T of level l. The parent tree of the permanent module θi placed on τθi is infinite. (v) Either: (a) a permanent module ξi , with infinite parent tree, is placed on some string σ in T and is eventually declared satisfied. The leftmost successor of σ in the limit value of T is τξi or (b) a permanent submodule ξi (j), with infinite parent tree, is placed on some string σ of level l in T and is never declared satisfied, although all permanent submodules ξi (j) placed on strings strictly to the left of σ are declared satisfied. The leftmost string extending σ of level l + 2 in the limit value of T is τξi . In order to see that there exists a sequence of this kind, suppose that we have already defined τυ0 ⊂ τθ0 ⊂ τξ0 ⊆ · · · ⊆ τυi−1 ⊂ τθi−1 ⊂ τξi−1 satisfying all of the required conditions, or if i = 0 suppose that τξ−1 = λ. If i = 0, then the instructions for stage 0 of the construction ensure that a permanent supermodule υi is placed on τξi−1 . If i > 0, then either (a) or (b) from (v) above applies to τξi−1 . Since the parent tree associated with the permanent module ξi−1 (and with the permanent submodule ξi−1 (j) if (b) applies) is infinite, (2) from the description of License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 5306 THOMAS KENT AND ANDREW E. M. LEWIS the construction will ensure that a permanent supermodule υi is placed on τξi−1 , which then has an infinite parent tree. First suppose that the tree associated with the permanent supermodule υi placed on τξi−1 is infinite. In this case we can define τυi = τξi−1 , and (i), (ii) and (iii) will remain satisfied. Let τυi be of level l in the limit value of T . Since there will be a stage after which the tree associated with υi always extends to level l + 2, we can define τθi to be the leftmost successor of τυi in the limit value of T and (3) from the description of the construction will ensure that (iv) remains satisfied for the induction step. The parent tree for the permanent module θi placed on τθi will be the tree associated with υi . Next suppose that the tree associated with the permanent supermodule υi placed on τξi−1 is finite. Let Ti,k be the tree associated with this supermodule. Then the parent tree of υi is infinite, so that if we define τυi to be the least leaf of Ti,k which is never declared terminal, then (i), (ii) and (iii) will be maintained for the induction step. Let τυi be of level l in the limit value of T . Clause (4) from the description of the construction will place a permanent module θi on all strings of level l + 1 in the limit value of T . We can therefore define τθi to be the leftmost successor of τυi in the limit value of T , maintaining satisfaction of (iv) for the induction step. The parent tree for the permanent module θi placed on τθi will be the parent tree of the supermodule υi . Whether or not the tree associated with the permanent supermodule υi placed on τξi−1 is finite, we have described how to define τθi so that all required conditions are satisfied. Since the parent tree for the permanent module placed on this string is infinite, and by (ii) of the induction hypothesis, clause (1) from the description of the construction will place a permanent module ξi and a submodule ξi (0) on the leftmost successor of τθi in the limit value of T . If the permanent module ξi is eventually declared successful, then we can define τξi to be its leftmost successor in the limit value of T in order to complete the induction step. Otherwise, if there exists a least submodule ξi (j) for this module which is placed on a string σ of level l (say) in the limit value of T and which is not declared satisfied, then we can define τξi to be the leftmost successor of σ of level l + 2 in the limit value of T . So suppose that there exists no such submodule. In this case it is easily seen, by using an induction on j, that the strings on which the submodules are placed form a perfect Πi -splitting tree. Note that the modification of the instructions for the simplified construction—that any submodule ξi (j) placed on any string σ of any level l in T should not act before the first stage at which it is the least submodule for the corresponding module ξi not to be declared satisfied and that T (σ) extends to level l + 2—was necessary in order that action taken to extend the parent tree of ξi did not interfere with the process by which the submodules build splittings. Then the set of strings of the form Ψi (τ ), such that τ is in this perfect Ψi -splitting tree, forms a perfect and computable tree with all paths in [Λ], which gives the required contradiction. That this set of strings is computable follows from the fact that no submodule for the permanent module ξi will ever be removed from a string in T once it has been placed upon it.  Corollary 5.4. For every α < 1P there exists β > 0P such that α ∧ β = 0P . Corollary 5.5. (P, <) has an infinite number of minimal elements. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ON THE DEGREE SPECTRUM OF A Π01 CLASS 5307 Proof. Suppose that (P, <) has only a finite number of minimal elements, α0 , · · · , αk , say. Then α0 ∪ · · · ∪ αk is an element α of P and by Theorem 5.3 there exists β which is minimal and such that β ≤ α, a contradiction.  Following Martin and Pour-El [MP] and Cholak, Coles, Downey and Herrmann [CC], define a Martin-Pour-El class to be a thin special Π01 class of the form {C|C separates A, B}, where A, B is a disjoint computably enumerable pair of c.e. sets. One can prove that, given any special Π01 class P, there exists a Martin-Pour-El class Q such that no member of Q computes any member of P. This has been noted by Simpson (unpublished). Moreover, it is easy to see that the degree spectrum of a Martin-Pour-El class is minimal in (P, <). Combining these two results we get that given any α ∈ P not containing 0 there exists β ∈ P not containing 0 which is minimal and such that α ∩ β = ∅. As pointed out by Simpson in a private correspondence, and also by the anonymous referee, this suffices in order to give an alternative proof of Corollary 5.5. Theorem 5.4. There exists α with no minimal predecessor. Proof. Jockusch and Soare [JS2] have constructed a special Π01 class P in which all elements are Turing incomparable. Let α = S(P) and suppose that S(P ′ ) ⊂ α for some non-empty Π01 class P ′ . Then, by the hyperimmune-free basis theorem there exist i, j, A and B such that Ψi and Ψj are total Turing functionals, A ∈ P, B ∈ P ′ , and such that Ψi (A) = B and Ψj (B) = A. Consider P ′′ , which is the set of all C ∈ P such that Ψi (C) is in P ′ and Ψj (Ψi (C)) = C. Then P ′′ is a Π01 class and S(P ′′ ) ⊆ S(P ′ ). In order to show that S(P ′′ ) is not minimal, let σ0 and σ1 be incomparable strings, each of which have infinite extensions in P ′′ . The set of all elements in P ′′ which extend σ0 is a Π01 class, with degree spectrum a proper subset of S(P ′′ ).  6. Π01 -mates The existence of invisible degrees serves to highlight the fact that there exist degrees a = b such that any Π01 class which contains a member of degree a must also contain a member of degree b. In this section we shall further investigate this kind of relationship between degrees.  Definition 6.1. We define the Π01 mates of a to be {α ∈ P : a ∈ α}. It is clear that there exist degrees a such that the Π01 -mates of a are just {a}. In order to see that, in fact, there exist continuum many such a, consider any nonempty Π01 class P in which all elements are Turing incomparable. Suppose A ∈ P and is of degree a. Then the Π01 -mates of a must be a subset of S(P), and for each B = A in P there exists a Π01 class P ′ ⊂ P which contains A but does not contain B. If C is of hyperimmune-free degree and B ≤T C, then B ≤tt C. Cenzer and Smith [CS] have shown that whenever B ≤tt C and C is ranked, B is also ranked and has rank less than or equal to that of C. Downey has constructed [RD] a set A of hyperimmune-free degree a and which is of rank 1, and so is a member of a Π01 class with degree spectrum {0, a}. Since a must therefore be completely ranked and a set is ranked iff it is a member of a countable Π01 class, it follows that the Π01 -mates of a are {0, a}, since if any special Π01 class contained a member B of degree a, then its intersection with any countable class with B as an element would License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 5308 THOMAS KENT AND ANDREW E. M. LEWIS be a countable special Π01 class. By Theorem 5.1, if a is any hyperimmune-free element of α which is minimal, then the Π01 -mates of a are α. So far, we have seen a for which the Π01 -mates of a are of cardinalities 1 and 2, and also a for which the Π01 -mates of a are uncountable. The following theorem, however, provides a wealth of further examples. Theorem 6.1. For any α < 1P there exists a ∈ / α such that the Π01 -mates of a are α ∪ {a} and, moreover, such that α ∪ {a} ∈ P. Proof. We write σ <L σ ′ if there exists some least n such that σ(n) ↓= σ ′ (n) and if, for this n, σ(n) = 0. We call a (possibly partial) function κ : 2<ω → 2<ω a tree function if both of the following conditions are satisfied: (i) Whenever σ ⊂ σ ′ , if κ(σ ′ ) ↓, then κ(σ) ↓ and κ(σ) ⊂ κ(σ ′ ). (ii) If σ <L σ ′ , κ(σ) ↓ and κ(σ ′ ) ↓, then κ(σ) <L κ(σ ′ ). If Λ is a set of finite binary strings we shall write κ(Λ) in order to denote {κ(σ) : σ ∈ Λ & κ(σ) ↓}. We shall say that Q is a similar copy of P if there exists a partial computable tree function κ and downward closed computable Λ such that κ(σ) ↓ for all σ ∈ Λ, P = [Λ] and Q = [κ(Λ)]. Given a downward closed computable Λ such that P = [Λ] does not contain a member of every Turing degree, we construct a downward closed computable Υ and A ∈ [Υ] of degree a such that S([Υ]) = S(P) ∪ {a} and such that the following requirements are satisfied: Θi : Either Ψi (A) is partial or computable, or Ψi (A) ∈ / P. Ξi : Let i = j, k. Either Ψj (A) is partial or computable or Ψj (A) ∈ / [Λk ], or else [Λk ] has a subset which is a similar copy of P. Having defined a construction that satisfies all of these requirements, we shall then be able to make an application of Posner’s lemma for minimal degree constructions, in order to show that A is not computable (so that the satisfaction of these requirements suffices to give the theorem). Once again, the proof is most easily described using modules. Essentially the proof is a synthesis of the techniques used in the proof of Theorem 5.3 and also those techniques used in the proof of Theorem 4.6. As in the proof of Theorem 5.3, we must deal with the difficulty that in order to satisfy the requirements we must search for splittings inside Π01 classes, and the solution is once again to ensure that A is of hyperimmune-free degree. As before, the approach we take is first to describe modules for a simplified version of the construction which acts in order to satisfy all requirements and which succeeds for those corresponding to functionals which are total, but which does not act in order to ensure that A is of hyperimmune-free degree. We shall then have to modify these simplified modules and fit them into the framework for constructing A of hyperimmune-free degree, but this should now seem straightforward given that it can be done in the same way as in the proof of Theorem 5.3. Now, in fact, things are much simpler, since we needn’t worry about maintaining homogeneity. The module θi (simplified version). We act to satisfy the requirement Θi in almost exactly the same way that we acted in order to satisfy the requirement Ξi in the proof of Theorem 5.3, except that now matters are considerably simpler since no homogeneity condition need be satisfied, so that there is no need to delegate work to submodules. It will be the role of the module θi placed on σ to ensure License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ON THE DEGREE SPECTRUM OF A Π01 CLASS 5309 that Θi is satisfied if σ ⊂ A, and the module will construct a (finite) splitting tree T (i, σ). At every stage at which it is passed control after being placed on σ, the module carries out one of the instructions below according to the first case which applies. T (i, σ) is initially defined to be {σ}. (1) The module has already been declared successful. This means that we / Λ and that we will will already have found a string σ ′ such that Ψi (σ ′ ) ∈ have placed the final successor module on this string. Pass control to this module. (2) If there exists σ ′ ⊃ σ which is a leaf of Υ which has not been declared terminal and such that Ψi (σ ′ ) ∈ / Λ, then we declare the module to be successful. In this case proceed as follows. Remove all modules from strings properly extending σ, remove any module which θi has placed on σ, and declare all leaves of Υ extending σ to be terminal other than σ ′ . Place the module ξi+1 on σ ′ and define this to be the final successor module. Enumerate the pair (σ, σ ′ ). (The enumeration of this pair (σ, σ ′ ) is just something that will be technically convenient later on when describing a new version of the convention C2, which we used in the proof of Theorem 5.3. The point is that in that proof, whenever a module acted, it would enumerate strings into T . Here this does not happen, so we need some other way of recording action which we may need to accommodate by adjusting the leaves of trees.) (3) Let σ ′ be the least leaf of the present T (i, σ) (ordered, as for all trees in the construction, according to their level in the tree and then from left to right). If there exist σ0 , σ1 extending σ ′ , which are leaves of Υ which have not been declared terminal and which are Ψi -splitting, then proceed as follows. Remove all modules (but not θi itself) from strings extending σ ′ , declare all leaves of Υ extending σ ′ other than σ0 and σ1 to be terminal, and enumerate σ0 and σ1 into T (i, σ). Enumerate the pairs (σ ′ , σ0 ) and (σ ′ , σ1 ). (4) If cases (1), (2) and (3) do not apply, then let σ ′ be defined as in (3). Pass control to the module ξi+1 placed on σ ′ , or place this module on this string if it has not already been placed there. The module ξi (simplified version). It is the role of the module ξi placed on σ to ensure that Ξi is satisfied if σ ⊂ A. This module will act in a way which is a mix between the module θi described above and the module ξi of Theorem 4.6. The basic idea is that the module will try to place a similar copy of P above σ ⋆ 0, in the form of a Ψj -splitting set of strings κi,σ (Λ) for some partial computable tree function κi,σ such that κi,σ (σ ′ ) ↓ for all σ ′ ∈ Λ. If it finds some string τ = κi,σ (σ ′ ) above which it is unable to find a Ψj -splitting, then putting this string as an initial segment of A ensures that Ψj (A) is either partial or computable. If, on the other hand, it finds some string τ in κi,σ (Λ) such that Ψj (τ ) ∈ / Λk , then it can satisfy the requirement by insisting that this string should be an initial segment of A. If neither of these possibilities holds, then [Λk ] has a subset which is a similar copy of P. In this case the requirement is automatically satisfied and σ ⋆ 1 will be an initial segment of A. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 5310 THOMAS KENT AND ANDREW E. M. LEWIS At every stage at which it is passed control, the module carries out one of the instructions below according to the first case which applies. When the module is first passed control, κi,σ (λ) is already defined to be σ ⋆ 0. (1) The module has already been declared successful. This means that we will already have found σ ′ such that Ψj (σ ′ ) ∈ / Λk and placed a final successor module on this string. Pass control to this module. The module ξi has outcome 0 at this stage. (2) If there exists σ ′ ⊃ σ ⋆ 0 which is a leaf of the present Υ which has not been / Λk , then we declare the module declared terminal and such that Ψj (σ ′ ) ∈ to be successful. In this case proceed as follows. Remove all modules from strings properly extending σ. Declare all leaves of the present Υ extending σ except σ ′ to be terminal. Place the module θi on σ ′ and define this to be the final successor module. Enumerate the pair (σ, σ ′ ). The module ξi has outcome 0 at this stage. (3) Let σ ′ be the least leaf which has not been declared terminal in κi,σ (Λ). If there exist σ0 , σ1 extending σ ′ , which are leaves of the present Υ which have not been declared terminal (in Υ) and which are Ψj -splitting, then proceed as follows. Let σ0 be to the left of σ1 . Remove all modules from strings extending σ ′ . Let τ be such that σ ′ = κi,σ (τ ), and if τ has no proper extension in Λ, then declare σ ′ to be terminal in κi,σ (Λ). Otherwise, for each d ∈ {0, 1} such that τ ⋆ d ∈ Λ, define κi,σ (τ ⋆ d) = σd and enumerate the pair (σ ′ , σd ). Declare all leaves of the present Υ extending σ ′ , except those strings which we have just enumerated into κi,σ (Λ), to be terminal. In this case the module has outcome 1 at this stage, and it passes control to the module θi placed on σ ⋆ 1 (or places this module on this string if it has not previously done so). (4) If cases (1), (2) and (3) do not apply, then the module proceeds as follows. Let σ ′ be defined as in (3). The module passes control to the module θi placed on σ ′ (or places this module on this string if it has not previously done so). The module has outcome 0 at this stage. Just as in the proof of Theorem 4.6, then, the module either has outcome 0 or 1 at each stage at which it is passed control. If it is passed control at an infinite number of stages and has outcome 1, then it places a similar copy of P above σ ⋆ 0 in the form of a Ψj -splitting tree. Since all paths of the image of this tree under Ψj are also elements of [Λk ], this means that [Λk ] contains a similar copy of P. In this case σ ⋆ 1 will be an initial segment of A. If the module is passed control at an infinite number of stages and has outcome 0 at all but finitely many of these stages, then σ ⋆ 0 will be an initial segment of A. In this case we are either able to ensure that Ψj (A) is partial or computable, or else we are able to ensure that Ψj (A) is not in [Λk ]. Our only remaining significant task is to fit modified versions of these modules into the framework for constructing A of hyperimmune-free degree. Since this can be done in the same way as in the proof of Theorem 5.3, this should now seem straightforward. There is, however, one small further point which must be considered. In the proof of Theorem 4.6, it was sometimes necessary to keep the construction ‘alive’ above strings where the construction was not presently taking place but which it might be necessary to work above later on—above the string σ⋆1, for example, when the module θi was placed on σ and had not yet observed σ ⋆ 0 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ON THE DEGREE SPECTRUM OF A Π01 CLASS 5311 to be an initial segment of Ψi (∅). In that construction, we could simply continue to concatenate these strings which we might later wish to work above with zeros in order to keep some extension in the Π01 class. Now, however, we must ensure that S([Υ]) = S(P) ∪ {a}, and S(P) may not contain 0. In order to deal with this difficulty, copies of Λ, rather than sequences of zeros, must be concatenated above strings where the construction is not presently taking place but to where it may later be necessary to return. This is basically easy but a little bit fiddly. To this end it is convenient to assume that P has elements which extend both 0 and 1. The following paragraph describes precisely the procedure by which we concatenate copies of Λ. A module υi will enumerate some tree Ti,k (in this construction we shall consider only modules rather than supermodules, modules and submodules), a module ξi placed on σ will construct a computable tree κi,σ (Λ) and a module θi placed on σ will construct some finite Ψi -splitting tree T (i, σ). Each of these is called the tree associated with the corresponding module. Suppose that at some stage of the construction σ ′ is the least leaf of Ti,k which has not been declared terminal, where Ti,k is the tree associated with the module υi . Suppose further that σ ′ is of level l in this tree, we find τ which is a leaf of Υ extending σ ′ which has not been declared terminal such that Ψi (τ ; l) ↓ and we are given the instruction to add appropriate extensions to Ti,k . In this case we proceed as follows. It is assumed that whenever we refer to a module or its associated tree, this is a module which has been placed on a string in Υ and not removed. Remove any module which υi has placed on σ ′ (unless, and this is just a small technical point, σ ′ is the leftmost string in Ti,k of level 1, in which case any module ξi on this string is initialized rather than removed, and now has parent tree Ti,k according to the definition in C2 below). Let σ ⊆ σ ′ be the longest such that either: • σ is a leaf of some tree associated with a module other than Ti,k , or • there is a module placed on σ. Let κ be the finite tree function whose image is the set of strings in Ti,k extending σ and let τ ′ be such that κ(τ ′ ) = σ ′ . The domain of κ will be an initial segment of Λ. Remove all modules from strings properly extending σ ′ . If τ ′ has no proper extension in Λ, then declare σ ′ to be terminal in Ti,k . Otherwise, for each d ∈ {0, 1} such that τ ′ ⋆ d ∈ Λ, enumerate τ ⋆ d into Ti,k and Υ (where τ is the leaf of Υ as above) and enumerate the pair (σ ′ , τ ⋆ d). Declare all leaves of the present Υ extending σ ′ , except those strings which we have just enumerated into Ti,k to be terminal. In altering the construction we fix the following conventions and modifications, all of which are necessary for an accurate proof but some of which only constitute small technical details of the kind which can reasonably be ignored upon a first reading. C1 When any module is removed, we also remove any modules placed on strings by that module. C2 Modules of the form υi will pass control to modules of the form ξi which will pass control to modules of the form θi , which in turn will pass control to modules of the form υi+1 (as opposed to modules ξi+1 as in the simplified construction), and so on. Any mention of the module ξi+1 in the instructions for the module θi , then, must now be replaced with υi+1 . The parent tree of any module is defined basically just as in the proof of Theorem 5.3. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 5312 THOMAS KENT AND ANDREW E. M. LEWIS C3 C4 C5 C6 Small technical differences, however, mean that we now need to define this term inductively. We define the parent tree for υ0 placed on λ to be Υ. Any module has the same parent tree as that which passes control to it, except for a module ξi which is passed control by υi and is placed on a string which is not a leaf of the tree associated with υi . Such a module ξi has the tree associated with υi as its parent tree. No module placed on a string σ is passed control at a stage at which the parent tree for the module does not yet contain any strings properly extending σ (so that upon receiving any instruction to do so at an earlier stage we simply terminate activity for that stage). No module will enumerate a string into its associated tree extending a leaf of that tree σ until a stage at which the parent tree for the module contains strings properly extending σ (with the same proviso for earlier stages at which instructions violating this condition are given). If σ is a leaf of Ti,k and all leaves of Υ extending σ are declared terminal in Υ, then σ is declared terminal in Ti,k . When the module ξi placed on σ is first passed control, we remove all modules from strings properly extending σ, we choose τ0 <L τ1 to be two leaves of Υ extending σ which have not been declared terminal and we enumerate the pairs (σ, τ0 ) and (σ, τ1 ). We place the module θi on τ1 and define κi,σ (λ) = τ0 . At all subsequent stages at which the module is passed control, we then carry out the first of the instructions (1)–(4) as previously described (although subject to the other conventions and modifications listed here), but with τ0 and τ1 taking the place of σ ⋆ 0 and σ ⋆ 1, respectively. This convention is essentially just a version of C2 from the proof of Theorem 5.3, but modified for the present construction. (a) Suppose that at some stage of the construction a module enumerates two pairs (σ, σ0 ) and (σ, σ1 ). Suppose further that σ ′ ⊇ σ is in Ti,k and that τ is a leaf of Ti,k and a successor of σ ′ . If both σ0 and σ1 extend τ , then we remove all proper extensions of σ ′ from Ti,k and enumerate in σ0 and σ1 . If precisely one of σ0 and σ1 extend τ , then we replace τ in Ti,k with that string. (b) Suppose that at some stage of the construction a module enumerates a single pair (σ, σ0 ). Suppose further that σ ′ ⊇ σ is in Ti,k and that τ is a leaf of Ti,k and a successor of σ ′ . If σ0 properly extends τ , then we replace τ in Ti,k with σ0 . The module υi . When placed on the string σ the module υi will find the least k such that Ti,k is not yet associated with any module, and it will enumerate the string σ into Ti,k (in other words it will choose a fresh tree to begin enumerating). Let T ′ be the parent tree for υi . As before, T ′ is the tree within which the module will try to build Ti,k . At each stage strictly after that at which it is placed on σ and at which it is passed control, the module will proceed as follows. First we look to see whether we can further build Ti,k . Let σ ′ be the least leaf of Ti,k which has not been declared terminal and let σ ′ be of level l in this tree. If there exist strings in T ′ properly extending σ ′ , then check to see if there exists τ which is a leaf of Υ extending σ ′ which has not been declared terminal and such that Ψi (τ ; l) is defined. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ON THE DEGREE SPECTRUM OF A Π01 CLASS 5313 If so: then add appropriate extensions to Ti,k as described previously. In this case, since Ti,k presently appears to be infinite, we now try to make progress with the simplified construction inside Ti,k . Pass control to the module ξi placed on the leftmost string in T of level 1, or place this module on that string if it is not already placed there. If not: then we must pass control to a module which works on the assumption that Ti,k will be finite. Let σ ′ be defined as above. If the module ξi has already been placed on σ ′ , then pass control to this module, providing this does not violate convention C3. Otherwise place ξi on σ ′ . The construction. At stage 0 we enumerate λ, 0 and 1 into Υ and place the module υ0 on λ. At all subsequent stages we proceed as follows: (1) Pass control to the module υ0 placed on λ and then proceed to pass control to other modules as directed until some module is passed control and does not pass control to another. (2) For each leaf σ of Υ which has not been declared terminal, proceed as follows. Let τ be the longest initial segment of σ which is a leaf of some tree associated with a module or which has a module placed upon it. We must continue to place a copy of Λ above τ . Let σ = τ ⋆ σ ′ . Follow the instructions for the first case below which applies. (a) If σ ′ has no proper extensions in Λ, then declare σ to be terminal. (b) For each d ∈ {0, 1} with σ ′ ⋆ d ∈ Λ, enumerate σ ⋆ d into Υ. A is defined to be the infinite string extending all σ on which a module υi is placed and such that the tree associated with that module is infinite. Some final details for verification. The modules θi and ξi function in ways which are extremely similar to modules described in previous constructions, and the way in which they operate should already be completely clear from the preceding discussion. The framework for constructing A of hyperimmune-free degree is basically just the same as that used in the proof of Theorem 5.3. It remains, however, to verify aspects of the construction which are significantly different in some respect to previous constructions. In order to show that A is not computable, we make an application of Posner’s lemma for minimal degree constructions. So suppose that A is computable. Then there exists i such that A = Ψi (∅). Consider Ψj such that, for any σ, Ψj (σ) = λ if σ ⊂ Ψi (∅) and Ψj (σ) = σ otherwise. Let τ be an initial segment of A upon which a module θj with infinite parent tree is placed at some stage of the construction and never removed. Since there are more than two strings in [Υ] extending τ , θi will eventually enumerate strings into T (i, τ ) properly extending τ . These strings will all be incompatible with A, but A must extend one of them. This gives the required contradiction. We must also show that every element of [Υ] other than A is of the same degree as an element of P. So suppose that B is an element of [Υ] other than A. There are three possibilities to consider. Case 1. There exists a module υi with infinite parent tree, which is placed on an initial segment of A and never removed, but whose associated tree Ti,k is finite. A extends the least leaf of Ti,k which is never declared terminal, but B extends another leaf of this tree which is not declared terminal, σ say. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 5314 THOMAS KENT AND ANDREW E. M. LEWIS Then the set of infinite paths through the parent tree of υi extending σ is a similar copy of P. B must be one of these paths. Case 2. There exists a module ξi with infinite parent tree which is placed on an initial segment of A, σ say, and never removed. Let σ ′ be the string upon which a module θi is placed and which the module ξi passes control to when it has outcome 1. We subdivide into three further cases. (a) A extends σ ′ , while B does not. In this case B must be an infinite path through κi,σ . (b) B extends σ ′ while A does not. In this case the module only has outcome 1 at a finite number of stages. Any infinite path through [Υ] extending σ ′ must consist of some finite initial segment τ above which the set of paths through the parent tree of ξi consists of a similar copy of P, concatenated with a path through the parent tree above this string. (c) A and B both extend different elements of κi,σ (Λ), which is finite. Let σ ′′ be the element of κi,σ (Λ) which B extends. Then the set of infinite paths through the parent tree of ξi extending σ ′′ is a similar copy of P. B must be one of these paths. Case 3. There exists a module θi with infinite parent tree which is placed on σ, which is an initial segment of A, and which is never removed. This module is never declared successful. A extends the least leaf of T (i, σ), while B extends another leaf, σ ′ say. Once again, the set of infinite paths through the parent tree of θi ex tending σ ′ is a similar copy of P. B must be one of these paths. We shall say that β is a strong minimal cover for α if the elements of P strictly below β are precisely those below and including α. Corollary 6.1. Every α < 1P has a strong minimal cover. Proof. By Theorem 6.1, let a ∈ / α be such that the Π01 -mates of a are α∪{a}. Let P 0 be a Π1 class such that S(P) ⊆ α ∪ {a}. If a ∈ S(P), then clearly S(P) = α ∪ {a}, and otherwise S(P) ⊆ α.  Corollary 6.1 suffices to show that there do not exist any maximal α < 1P . We observed already that there are no maximal elements amongst the degree spectra of special Π01 classes. If the Π01 class P is countable with rank β, then there exists a computable ordinal α > β and all members of P have degree < 0(2α) . Since there exists a ranked set of degree 0(2α) [CCS] and a set is ranked iff it is a member of a countable Π01 class, this suffices to show that there are no maximal elements amongst the countable members of P. Recall that a Π01 class is rank faithful if the rank of any member of the class relative to the class is its actual rank. Theorem 6.1 also gives an interesting corollary regarding sets which are not a member of any rank faithful class: Corollary 6.2. There exists a non-computable set A which is ranked, such that every B ≤T A is ranked, and such that no non-computable B ≤T A is a member of any rank faithful class. Proof. Let P be any countable Π01 class which is not rank faithful. The construction described in the proof of Theorem 6.1 gives Υ which is countable and A ∈ [Υ] which is of non-zero hyperimmune-free degree. Every set Turing reducible to A is therefore License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ON THE DEGREE SPECTRUM OF A Π01 CLASS 5315 ranked [CS]. No non-computable set Turing reducible to A is a member of any Π01 class unless it contains a similar copy of P and is therefore not rank faithful.  7. Two anti-basis theorems In this section we shall describe two anti-basis theorems for Π01 classes. Before doing so, however, let us describe our initial motivation for proving the first of these two theorems. From this point forward we will allow the variables α and β to range over sets of degrees which are not necessarily elements of P. Definition 7.1. We say that α is a sufficiency set for a if every Π01 class that contains a member of every degree in α also contains a member of degree a. In the previous section we considered the Π01 mates of individual degrees. The previous definition could also be phrased in terms of the Π01 mates of larger sets of 0 degrees.  If we define the Π1 mates of α (which is not necessarily an element of P) to be {β ∈ P : α ⊆ β}, then α is a sufficiency set for precisely those degrees which are an element of the Π01 mates of α. If α is a sufficiency set for a, then there exists some countable β ⊆ α which is a sufficiency set for a (for each element of P which does not contain every element of α, choose some element of α not in this set). For every countably infinite sufficiency set α for a, there exists some proper subset which is also a sufficiency set for a. In order to see this let a0 and a1 be distinct elements of α. If each α − {ai } is not a sufficiency set for a, then let each Pi be a Π01 class which contains a member of every degree in α − {ai } but does not contain a member of degree a. Then P0 ∪ P1 witnesses the fact that α is not a sufficiency set for a, a contradiction. Almost exactly the same argument suffices to show that if α is a finite sufficiency set for a, then there exists some element b of α such that {b} is a sufficiency set for a. The following theorem shows, however, that it is possible to find a and a countable α which is a sufficiency set for a, such that no finite subset of α is a sufficiency set for a. Theorem 7.1 (Low Anti-basis Theorem). Any Π01 class that contains a member of every low degree contains a member of every degree. Proof. Let j be such that [Λj ] does not contain a member of every degree, and for every i, τ let σ(i,  j, τ ) be defined as in the proof of Theorem 4.5. We define non-computable A = i σi which is of low degree and such that for each i, if Ψi (A) is total and non-computable, then it is not an element of [Λj ]. The fact that A is of low degree follows because we run the construction using an oracle for ∅′ and decide whether Ψi (A; i) ↓ at each stage 2i + 2. Stage 0. Define σ0 = λ. Stage 2i + 1. Define σ2i+1 = σ(i, j, σ2i ). Stage 2i + 2. If there exists σ ⊃ σ2i+1 such that Ψi (σ; i) ↓, then define σ2i+2 to be some extension of σ which is not an initial segment of Ψi (∅). Otherwise define σ2i+2 to be some extension of σ2i+1 which is not an initial segment of Ψi (∅).  Now if a is not low, then α, which is the set of all low degrees, is a sufficiency set for a, such that no finite subset is a sufficiency set for a (since if a is low, then {0, a} ∈ P). License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 5316 THOMAS KENT AND ANDREW E. M. LEWIS A modified version of the proof of Theorem 7.1 can also be made to work below any degree which is non-GL2 : Theorem 7.2. If b is non-GL2 , then, for any Π01 class P which does not contain a member of every degree, there exists a non-zero a ≤ b such that P does not contain any member of non-zero degree below a. Proof. (Sketch) We simply use the fact that, since b is non-GL2 , for every function f of degree below 0′ , there exists some function g of degree b which is not dominated by f . Suppose we are given b which is non-GL2 and j such that [Λj ] does not contain a member of every degree. Suppose that in the proof of Theorem 7.1 we had not been concerned with ensuring that A is of low degree (but just of degree below 0′ ), so that we acted at stages 2i + 1 in order to satisfy requirement: Θ2i+1 : Ψi (A) is partial or computable, or is not in [Λj ], and we acted at even stages 2i + 2 in order to satisfy requirement: Θ2i+2 : A = Ψi (∅). Rather than using an oracle for ∅′ directly, we could have used some function f to bound the number of steps for which we need to perform searches at each stage i of the construction in order to define σi . The function f , then, would be used to bound the number of steps we search for in order to compute Ψi (∅) or in order to determine whether any string is a leaf of T as in the definition of σ(i, j, σ2i ) from the proof of Theorem 4.5. Let f0 be a function of degree below 0′ which grows sufficiently quickly that if f0 is used to bound the searches at each stage, then each σi will be defined correctly. Let f1 be a function of degree below 0′ which grows sufficiently quickly that if f1 is used to bound the searches at any particular stage i, then σi will be defined so as to satisfy Θi , no matter how many steps we have searched for at previous stages i′ in order to define σi′ . Finally, consider a modified construction which looks to ensure satisfaction of all requirements Θi′ such that i′ ≤ i at any stage i, and let f2 be a function of degree below 0′ such that if this modified construction searches for f2 (i) many steps at any particular stage i, then this is sufficient to satisfy all requirements of priority higher than Θi , no matter how many steps we have searched for at previous stages i′ in order to define σi′ . Let g be a function of degree b which is not dominated by f2 . If the function g is used in order to bound the searches at each stage of the construction, then all  requirements Θi will eventually be satisfied. 8. (P, <) is a lattice We noted in the Introduction that the intersection of the degree spectra of two Π01 classes need not be the degree spectrum of a Π01 class. The following result suffices to show, however, that (P, <) is a lattice. Theorem 8.1. The intersection of the degree spectra of two Π01 classes is the degree spectrum of a Π01 class if it is the superset of the degree spectrum of a non-empty Π01 class. Proof. Let P0 and P1 be Π01 classes with degree spectra α and β, respectively, and suppose that P2 is a non-empty Π01 class with degree spectrum γ ⊆ α ∩ β. For each i ≤ 2 let Λi be a downward closed and computable set of strings with [Λi ] = Pi . For each (i, j) we define Qi,j = {A ∈ P0 : Ψi (A) ∈ P1 and Ψj (Ψi (A)) = A}. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ON THE DEGREE SPECTRUM OF A Π01 CLASS 5317 In order to define a downward closed and computable set of strings Λ such that P = [Λ] satisfies S(P) = α ∩ β, we proceed as follows. We begin by putting all strings in Λ2 into Λ. Let us say σ ∈ / Λ2 is terminal in Λ2 if all proper initial segments of σ are in Λ2 . We consider strings to be ordered first by length and then from left to right. All strings which are terminal in Λ2 we place in Λ and above the xth terminal string we place a Π01 class Pi,j with degree spectrum S(Qi,j ) ∪ γ, where x = i, j. It remains only to define the classes Pi,j , and in order to do so we define downward closed and computable sets of strings Λi,j . The rough idea behind the construction of Λi,j is that we shall put strings into this set which code pairs of strings, one from each of Λ0 and Λ1 , and which compute each other via Ψi and Ψj . While we wait to find convergent computations we place copies of Λ2 to fill the time. We shall actually construct Λi,j as a subset of 6<ω (the set of all finite strings of elements of {0, 1, 2, 3, 4, 5}) whose downward closure is computable (while Λi,j itself is not necessarily downward closed). Of course it is easy then to define Λ′i,j from Λi,j , which is a downward closed and computable set of finite binary strings and which has the same degree spectrum. Before describing the construction of Λi,j precisely, some simple definitions are required. For any i, n ∈ ω and any σ, we assume that Ψi (σ; n) is only defined if this computation converges in less than |σ| many steps, n < |σ|−1, and only if Ψi (σ; n′ ) is defined for all n′ < n. Thus if Ψi (σ) = σ ′ , then |σ| > |σ ′ | unless |σ ′ | = 0. For any σ ∈ 2<ω we define g0 (σ) to be the string obtained from σ by replacing each 0 with a 2 and replacing each 1 with a 3. We define g1 (σ) to be the string obtained from σ by replacing each 0 with a 4 and replacing each 1 with a 5. Suppose we are given ψ ∈ 6<ω and let ψ be written as σ0 ∗ τ0 ∗ φ0 ∗ · · · ∗ σk ∗ τk ∗ φk , where each σk′ is a string of elements of {0, 1}, each τk′ is a string of elements of {2, 3} and each φk′ is a string of elements of {4, 5}. We define f0 (ψ) = g0−1 (τ0 ∗ · · · ∗ τk ) and f1 (ψ) = g1−1 (φ0 ∗ · · · ∗ φk ). We shall think of ψ as coding a string in Λ0 via f0 and coding a string in Λ1 via f1 . For any ψ ∈ 6<ω we define the coding part of ψ to be the shortest ψ ′ ⊆ ψ such that ψ(n) < 2 for all n with |ψ ′ | ≤ n < |ψ|. We say that ψ ∈ 6<ω requires P0 extension if |f0 (ψ)| ≤ |f1 (ψ)|, and otherwise we say that ψ requires P1 extension. Stage 0. For each τ ∈ Λ0 of length 1, enumerate g0 (τ ) into Λi,j . Stage s + 1. For every string ψ enumerated into Λi,j at stage s, proceed as follows. Let ψ = ψ0 ∗ σ, where ψ0 is the coding part of ψ. For each σ ′ which is a one element extension of σ and such that σ ′ ∈ Λ2 , enumerate ψ0 ∗ σ ′ into Λi,j . Let |σ| = l, let f0 (ψ) = τ and let f1 (ψ) = φ. If ψ requires P0 extension, then check to see if there is τ ∗ τ ′ in Λ0 of length l such that φ ⊆ Ψi (τ ∗ τ ′ ) and such that this is not true of any proper initial segment of τ ∗ τ ′ . If so, and if σ is the leftmost string in Λ2 of length l, then enumerate ψ ∗ g0 (τ ′ ) into Λi,j for each such τ ′ . If ψ requires P1 extension, then check to see if there is φ ∗ φ′ in Λ1 of length l such that τ ⊆ Ψj (φ ∗ φ′ ) and such that this is not true of any proper initial segment of φ ∗ φ′ . If so, and if σ is the leftmost string in Λ2 of length l, then enumerate ψ ∗ g1 (φ′ ) into Λi,j for each such φ′ . Any infinite string has an infinite number of initial segments in Λi,j iff it has an initial segment enumerated into Λi,j at every stage of the construction iff it is an infinite path through the downward closure of Λi,j . It is clear that the downward closure of Λi,j is a computable set of strings. For each A ∈ Qi,j there is a unique License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 5318 THOMAS KENT AND ANDREW E. M. LEWIS element of [Λi,j ], π(A) say, such that f0 (π(A)) = A. Uniqueness here follows from the fact that when we enumerate ψ ∗ g0 (τ ′ ) into Λi,j at stage s + 1, for example, we insist that (using the terminology of that stage) τ ∗ τ ′ in Λ0 is of length |σ|, φ ⊆ Ψi (τ ∗ τ ′ ) and that φ ⊆ Ψi (τ ′′ ) for any τ ′′ ⊂ τ ∗ τ ′ . Since π(A) is computable in A it must hold that A ≡T π(A). Any element of [Λi,j ] which is not π(A) for some A ∈ Qi,j is a finite string ψ concatenated with an element of Λ2 , and so is of degree in γ. Thus S([Λi,j ]) = S(Qi,j ) ∪ γ, as required.  Corollary 8.1. (P, <) is a lattice. Acknowledgements The authors would like to thank Doug Cenzer, Steve Simpson, Antonin Kucera and especially Carl Jockusch for their interest and for many very useful comments and suggestions. References D. Cenzer, Π01 classes, in the Handbook of Computability Theory, Studies in logic and the foundations of mathematics, vol 140, Elsevier, 1999. CS. D. Cenzer and R. 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MR2258714 (2007h:03088) SS. S. Simpson, An extension of the recursively enumerable Turing degrees, Journal of the London Mathematical Society, 75, pp. 287-297, 2007. MR2340228 (2008d:03041) SS2. S. Simpson, Mass problems and randomness, Bulletin of Symbolic Logic, 11, pp. 1-27, 2005. MR2125147 (2005k:03097) Department of Mathematics, Marywood University, Scranton, Pennsylvania 18509 E-mail address: tfkent@marywood.edu Department of Mathematics, University of Leeds, Leeds, England LS2 9JT E-mail address: aemlewis@aemlewis.co.uk License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use