aa r X i v : . [ m a t h . L O ] J a n Turing degrees of hyperjumps
Hayden R. JananthanDepartment of MathematicsVanderbilt UniversityNashville, TN 37203, USA https://my.vanderbilt.edu/[email protected]
Stephen G. SimpsonDepartment of MathematicsVanderbilt UniversityNashville, TN 37203, USA
First draft: August 25, 2019This draft: June 29, 2020
Abstract
The Posner-Robinson Theorem states that for any reals Z and A suchthat Z ⊕ ′ ≤ T A and 0 < T Z , there exists B such that A ≡ T B ′ ≡ T B ⊕ Z ≡ T B ⊕ ′ . Consequently, any nonzero Turing degree deg T ( Z ) isa Turing jump relative to some B . Here we prove the hyperarithmeticalanalog, based on an unpublished proof of Slaman, namely that for anyreals Z and A such that Z ⊕ O ≤ T A and 0 < HYP Z , there exists B suchthat A ≡ T O B ≡ T B ⊕ Z ≡ T B ⊕ O . As an analogous consequence, anynonhyperarithmetical Turing degree deg T ( Z ) is a hyperjump relative tosome B . Contents Σ Classes 33 Posner-Robinson for Turing Degrees of Hyperjumps 10
Open Problems 17
Our starting point is the
Friedberg Jump Theorem : Theorem 1.1 (Friedberg Jump Theorem) . [10, Theorem 13.3.IX, pg. 265]Suppose A is a real such that ′ ≤ T A . Then there exists B such that A ≡ T B ′ ≡ T B ⊕ ′ . There are several refinements of the Friedberg Jump Theorem. One suchextension shows that B can be taken to be an element of any special Π class P ⊆ { , } N . Here special means that P is nonempty and has no recursvieelements. Theorem 1.2. [6, following Theorem 3.1, pg. 37] Suppose P ⊆ { , } N is aspecial Π class and A is a real such that ′ ≤ T A . Then there exists B ∈ P suchthat A ≡ T B ′ ≡ T B ⊕ ′ . Another refinement is the
Posner-Robinson Theorem : Theorem 1.3 (Posner-Robinson Theorem) . [8, Theorem 1, pg. 715] [5, Theo-rem 3.1, pg. 1228] Suppose Z and A are reals such that Z ⊕ ′ ≤ T A and < T Z .Then there exists B such that A ≡ T B ′ ≡ T B ⊕ Z ≡ T B ⊕ ′ . In this paper we prove hyperarithmetical analogs of Theorem 1.2 and Theorem 1.3.The hyperarithmetical analog of Theorem 1.1 is due to Macintyre [7, Theorem3, pg. 9]. In these hyperarithmetical analogs, the Turing jump operator X ↦ X ′ is replaced by the hyperjump operator X ↦ O X and Π classes are replaced byΣ classes. A feature of [7, Theorem 3, pg. 9] and of our results is that theyinvolve Turing degrees rather than hyperdegrees, so for instance O B is not onlyhyperarithmetically equivalent to A , but in fact Turing equivalent to A .Here is an outline of this paper:In § classes K ⊆{ , } N . Theorem 2.1.
Suppose K ⊆ { , } N is an uncountable Σ class and Z and A are reals such that Z ⊕ O ≤ T A and < HYP Z . Then there exists B ∈ K suchthat A ≡ T O B ≡ T B ⊕ O and Z ≰ HYP B . In § heorem 3.1. Suppose Z and A are reals such that Z ⊕ O ≤ T A and < HYP Z .Then there exists B such that A ≡ T O B ≡ T B ⊕ Z ≡ T B ⊕ O . The remainder of this section fixes notation and terminology. g ∶ ⊆ A → B denotes a partial function with domain dom g ⊆ A and codomain B . For a ∈ A , if a ∈ dom g then we say ‘ g ( a ) converges’ or ‘ g ( a ) is defined’ andwrite g ( a )↓ . Otherwise, we say ‘ g ( a ) diverges’ or ‘ g ( a ) is undefined’ and write g ( a ) ↑ . If f and g are two partial functions ⊆ A → B and a ∈ A , then f ( a ) ≃ g ( a ) means ( f ( a ) ↓ ∧ g ( a ) ↓ ∧ f ( a ) = g ( a )) ∨ ( f ( a ) ↑ ∧ g ( a ) ↑ ) . We write f ( a ) ↓ = b tomean that f ( a ) ↓ and f ∶ a ↦ b . N N and { , } N denote the Baire and Cantor spaces, respectively, whoseelements we sometimes call reals . We identify { , } N and the powerset P( N ) in the usual manner.If S is a set, then S ∗ is the set of strings of elements from S . If s , . . . , s n − ∈ S , then σ = ⟨ s , . . . , s n − ⟩ ∈ S ∗ denotes the string of length ∣ σ ∣ ∶ = n definedby σ ( k ) = s k . If ⟨ s , . . . , s n − ⟩ , ⟨ t , . . . , t m − ⟩ ∈ S ∗ , then their concatenation is ⟨ s , . . . , s n − ⟩ ⌢ ⟨ t , . . . , t m − ⟩ ∶ = ⟨ s , . . . , s n − , t , . . . , t m − ⟩ . If σ, τ ∈ S ∗ , then σ isan initial segment of τ (equivalently, τ is an extension of σ ) written σ ⊆ τ , if τ ↾∣ σ ∣ = σ . If f ∶ N → S then σ ∈ S ∗ is an initial segment of f (equivalently, f is an extension of σ ), written σ ⊂ f , if f ↾∣ σ ∣ = σ . σ, τ ∈ S ∗ are incompatible if neither is an initial segment of the other. If ≤ is a partial order on S , thenthe lexicographical ordering ≤ lex on S ∗ is defined by setting σ ≤ lex τ if σ ⊆ τ or,where k is the least index at which σ ( k ) ≠ τ ( k ) , then σ ( k ) < τ ( k ) . ϕ ( k ) e denotes the e -th partial recursive function ⊆ N k → N ; e is called an index of ϕ ( k ) e . Likewise, if f ∈ N N then ϕ ( k ) ,fe denotes the e -th partial function ϕ ( k ) ,fe ∶ ⊆ N k → N which is partial recursive in f ; e is again called an index of ϕ ( k ) ,fe , while f is called an oracle of ϕ ( k ) ,fe . ≤ T denotes Turing reducibility while ≡ T denotes Turing equivalence. ≤ HYP denotes hyperarithmetical reducibility while ≡ HYP denots hyperarithmetical equiv-alence. For X ∈ { , } N , X ′ denotes the Turing jump of X and O X denotes thehyperjump of X . O denotes Kleene’s O . For f, g ∈ N N , their join f ⊕ g ∈ N N isdefined by ( f ⊕ g )( n ) = f ( n ) and ( f ⊕ g )( n + ) = g ( n ) . P e denotes the e -th Π set { f ∈ N N ∣ ϕ ( ) ,fe ( )↓} ⊆ N N . P ∗ e denotes the e -thΣ class { X ∈ { , } N ∣ ∃ f ( f ⊕ X ∈ P e )} . Σ Classes
The following theorem includes the Gandy Basis Theorem [11, Theorem III.1.4,pg. 54], the Kreisel Basis Theorem for Σ Classes [11, Theorem III.7.2, pg. 75],and Macintyre’s Hyperjump Inversion Theorem [7, Theorem 3, pg. 9].
Theorem 2.1.
Suppose K ⊆ { , } N is an uncountable Σ class and Z and A are reals such that Z ⊕ O ≤ T A and < HYP Z . Then there exists B ∈ K such hat A ≡ T O B ≡ T B ⊕ O and Z ≰ HYP B . To prove Theorem 2.1 we use Gandy-Harrington forcing (first introduced byHarrington in an unpublished manuscript [2]; see, e.g., [11, Theorem IV.6.3, pg.108]), forming a descending sequence of uncountable Σ classes K = K ⊇ K ⊇ ⋯ ⊇ K n ⊇ ⋯ where an element of the intersection ⋂ ∞ n = K n has the desired property. Unlikein the case of Π subsets of { , } N , compactness cannot be used to easily showthat the intersection ⋂ ∞ n = K n is nonempty. Instead, some care must be takento show that this is the case. Proposition 2.2. (a) Given a Σ predicate K ⊆ { , } N × N k , there is a primitive recursive func-tion f ∶ N k → N such that P ∗ f ( x ,...,x k ) ( X ) ≡ K ( X, x , . . . , x k ) . (b) Suppose X ∈ { , } N . Then { e ∈ N ∣ X ∉ P ∗ e } ≡ T O X .(c) { e ∈ N ∣ P ∗ e = ∅ } ≡ T O .Proof. Straight-forward.
Corollary 2.3.
There exist primitive recursive functions v , u , and U such thatthat for all n, m ∈ N and σ, τ ∈ N ∗ and I ∈ P fin ( N ) , P ∗ v ( n,m ) = P ∗ n ∩ P ∗ m ,P ∗ u ( e,σ,τ ) = P ∗ e [ σ, τ ] = { X ∈ { , } N ∣ σ ⊂ X ∧ ∃ g ( X ⊕ g ∈ P e ∧ τ ⊂ g )} ,P ∗ U ( I,σ, ⟨ τ ,...,τ n − ⟩) = ⋂ k ∈ I ∧ k < n P ∗ k [ σ, τ k ] . Proposition 2.4.
The following partial functions are O -recursive:(a) The partial function ρ ( σ, e ) ≃ ⟨ σ , σ ⟩ where σ , σ are minimal incompati-ble extensions of σ which have extensions in P ∗ e and σ is lexicographicallyless than σ , whenever σ has at least two extensions in P ∗ e , otherwise di-verging.(b) The partial function ext (⟨ e , . . . , e N ⟩ , σ, ⟨ τ , . . . , τ N ⟩) ≃ ( ˜ σ, ⟨ ˜ τ , . . . , ˜ τ N ⟩) where ( ˜ σ, ⟨ ˜ τ , . . . , ˜ τ N ⟩) is the lexicographically least pair such that1. σ ⊂ − − ˜ σ and τ k ⊂ − − ˜ τ k for ≤ k ≤ N and2. ⋂ Nk = P ∗ e k [ ˜ σ, ˜ τ k ] ≠ ∅ henever ⋂ Nk = P ∗ e k [ σ, τ k ] ≠ ∅ , otherwise diverging.Proof. (a) Using O , search for the first string ν such that P ∗ e [ σ ⌢ ν ⌢ ⟨ i ⟩ , ⟨⟩] ≠ ∅ for i = ,
1. Once such ν has been found, ρ ( σ, e ) ↓ = ⟨ σ ⌢ ν ⌢ ⟨ ⟩ , σ ⌢ ν ⌢ ⟨ ⟩⟩ .(b) Using O , search for the first of i = , ⋂ Nk = P ∗ e k [ σ ⌢ ⟨ i ⟩ , τ k ] ≠ ∅ ,then search for the lexicographically least ⟨ j , . . . , j N ⟩ ∈ { , } N such that ⋂ Nk = P ∗ e k [ σ ⌢ ⟨ i ⟩ , τ k ⌢ ⟨ j k ⟩] ≠ ∅ . If no such i or j , . . . , j N are found, thendiverge. Otherwise,ext (⟨ e , . . . , e N ⟩ , σ, ⟨ τ , . . . , τ N ⟩) ↓ = ( σ ⌢ ⟨ i ⟩ , ⟨ τ ⌢ ⟨ j ⟩ , . . . , τ N ⌢ ⟨ j N ⟩) . Let ρ , ρ be defined by ρ ( σ, e ) ≃ ⟨ ρ ( σ, e ) , ρ ( σ, e )⟩ . We use the ordinal notation description of O (and, more generally, O Y for Y ∈ { , } N ) described in [11] and use the following well-known lemma to describehyperarithmetical reducibility in terms of H -sets. Notation.
For X ∈ { , } N and n ∈ N , define ( X ) n ∶ = { x ∈ N ∣ n ⋅ x ∈ X } . Lemma 2.5.
Suppose X and Y are reals in { , } N . Then X ≤ HYP Y if andonly if there exists b ∈ O Y and n ∈ N such that X = ( H Yb ) n .Proof. Suppose X ≤ HYP Y , so that there is b ∈ O Y such that X ≤ T H Yb . Let e be the index of such a Turing reduction, i.e., let e be such that X = ϕ ( ) ,H Yb e .By definition [11], 2 b ∈ O Y and H Y b ∶ = { n x ∣ ϕ ( ) ,H Yb n ( x ) ↓ } . Let f be an index such that ϕ ( ) ,H Yb f ( x ) ↓ ⇐⇒ ϕ ( ) ,H Yb e ( x ) ↓ = ( H Y b ) f = { x ∈ N ∣ ϕ ( ) ,H Yb f ( x ) ↓ } = { x ∈ N ∣ ϕ ( ) ,H Yb e ( x ) ↓ = } = X b ∈ O Y and n ∈ N such that X = ( H Yb ) n . Let e be an index such that ϕ ( ) ,Ze ( x ) = ⎧⎪⎪⎨⎪⎪⎩ x ∈ ( Z ) n x ∉ ( Z ) n for any Z ∈ { , } N . Then ϕ ( ) ,H Yb e = X , showing that X ≤ T H Yb . Proof of Theorem 2.1.
By the Gandy Basis Theorem [11, Theorem III.1.4, pg.54], assume without loss of generality that ω Y = ω CK1 for all Y ∈ K .In order to control the hyperjump O B , we choose B to be an element of anintersection of Σ subsets K = K ⊇ K ⊇ ⋯ ⊇ K n ⊇ ⋯ . In order for B to be an element of K n = P ∗ j ( n ) for each n , there must be g n ∈ N N such that B ⊕ g n ∈ P j ( n ) , where j ( n ) is some index of K n . Such g n depend on B . Thus, we additionally define sequences of strings σ ⊆ σ ⊆ ⋯ ⊆ σ n ⊆ ⋯ τ , ⊆ τ , ⊆ ⋯ ⊆ τ n, ⊆ ⋯ τ , ⊆ τ , ⊆ ⋯ ⊆ τ n, ⊆ ⋯ τ , ⊆ τ , ⊆ ⋯ ⊆ τ n, ⊆ ⋯⋮ ⋮ ⋱ ⋮ ⋱ so that B = ⋃ n ∈ ω σ n and g k = ⋃ n ∈ ω τ n,k . We also define a sequence of finitesubsets of N I ⊆ I ⊆ ⋯ ⊆ I n ⊆ ⋯ encoded as finite sequences { e , . . . , e N } ↦ ⟨ e , . . . , e N ⟩ which keep track ofthe indices e of Σ classes we have committed to intersecting, so that K n = ⋂ k ∈ I n P ∗ k [ σ n , τ n,k ] . A function j ∶ N → N keeps track of the index of K n , i.e., K n = P ∗ j ( n ) . In the course of the proof, we assume that j encodes all of the information fromprevious steps (i.e., a course-of-value computation) though we avoid making thisprecise to ease the burden of notation.To ease in the notation and exposition, we set the following temporary def-initions. An intersection system consists of the following data:(i) a finite subset I ⊆ N ,(ii) a string σ , and(iii) a sequence of strings ⟨ τ k ∣ k ∈ I ⟩ subject to the constraint that ⋂ k ∈ I P ∗ k [ σ, τ k ] is nonempty. If k ∉ I , then weassign the value ⟨⟩ to τ k .By adding P ∗ e to the intersection system I, σ, ⟨ τ k ∣ k ∈ I ⟩ , we mean thefollowing procedure, where K = ⋂ k ∈ I P ∗ k [ σ, τ k ] :6 ase : K ∩ P ∗ e = ∅ . Let ˜ I = I , ˜ K = K , ˜ σ = σ , and ˜ τ k = τ k for each k . Case : K ∩ P ∗ e ≠ ∅ . Let ˜ I = I ∪ { e } , and let ˜ σ and, simultaneously for all k ∈ ˜ I , ˜ τ k be the lexicographically least proper extensions of σ and τ k ,respectively, such that ⋂ k ∈ ˜ I P ∗ k [ ˜ σ, ˜ τ k ] ≠ ∅ .The resulting intersection system is ˜ I, ˜ σ, ⟨ ˜ τ k ∣ k ∈ ˜ I ⟩ . Note that from I, σ, ⟨ τ k ∣ k ∈ I ⟩ and e , the new intersection system ˜ I, ˜ σ, ⟨ ˜ τ k ∣ k ∈ ˜ I ⟩ can be determinedin a uniform way recursively in O : representing I as ⟨ e , . . . , e N ⟩ and writing e N + = e , then ˜ I = ⎧⎪⎪⎨⎪⎪⎩⟨ e , . . . , e N , e N + ⟩ if K ∩ P ∗ e ≠ ∅ , I otherwise, ( ˜ σ, ⟨ ˜ τ k ∣ k ∈ ˜ I ⟩) = ⎧⎪⎪⎨⎪⎪⎩ ext ( ˜ I, σ, ⟨ τ e , . . . , τ e N , ⟨⟩⟩) if K ∩ P ∗ e ≠ ∅ , ( σ, ⟨ τ k ∣ k ∈ I ⟩) otherwise.In particular, the index U ( ˜ I, ˜ σ, ⟨ ˜ τ k ∣ k < max I ⟩) of ˜ K can be determined uni-formly from the intersection system I, σ, ⟨ τ k ∣ k ∈ I ⟩ using O as an oracle.Now we proceed with the construction. As K is Σ , there is e such that K = P ∗ e . Stage n = : Define K ∶ = K, σ ∶ = ⟨⟩ , τ ,k ∶ = ⟨⟩ , j ( ) ∶ = e , I ∶ = { e } . Note that P ∗ j ( ) = K = ⋂ k ∈ I P ∗ k [ σ , τ ,k ] . Stage n = e + : Let I n , σ n , ⟨ τ n,k ∣ k ∈ I n ⟩ be the result of adding P ∗ e to the in-tersection system I n − , σ n − , ⟨ τ n − ,k ∣ k ∈ I n − ⟩ , let K n ∶ = ⋂ k ∈ I n P ∗ k [ σ n , τ n,k ] ,and finally let j ( n ) be an index for K n . Stage n = e + : At this stage we encode A ( e ) into B .By construction, P ∗ j ( n − ) = K n − = ⋂ k ∈ I n − P ∗ k [ σ n − , τ n − ,k ] ≠ ∅ . As K n − is uncountable, there are infinitely many pairwise-incompatibleextensions of σ n − which extend to elements of K n − . Thus, let σ n ∶ = ρ A ( e ) ( σ n − , j ( n − )) . Define K n ∶ = ⋂ k ∈ I n − P ∗ k [ σ n , τ n − ,k ] = P U ( σ n ,I n − , ⟨ τ n − , ,...,τ n − ,n − ⟩) ,τ n,k ∶ = τ n − ,k , ( for all k ) I n ∶ = I n − ,j ( n ) ∶ = U ( σ n , I n − , ⟨ τ n − , , . . . , τ n − ,n − ⟩) . tage n = b + ⋅ e ⋅ f : Suppose b ∈ O . Let m ∈ N be the least natural num-ber for which there are Y , Y ∈ K n − such that ϕ ( ) ,H Y b f ( e ⋅ m ) and ϕ ( ) ,H Y b f ( e ⋅ m ) are both defined and unequal. For i ∈ { , } , let K in − = { Y ∈ K n − ∣ ϕ ( ) ,H Y b f ( e ⋅ m ) ↓ = i } . Because K n − ∩ K n − = ∅ , there is a least k ∈ N and i ∈ { , } such that { Y ∈ K n − ∣ Y ( k ) = i } and { Y ∈ K n − ∣ Y ( k ) ≠ i } are nonempty. Let i = i and i = − i .Let I n , σ n , ⟨ τ n,k ∣ k ∈ I n ⟩ be the result of adding the (uniformly in b , e , f , m , k ,and i , given Z ( m ) ) Σ class { Y ∈ { , } N ∣ ϕ ( ) ,H Yb f ( e ⋅ m ) ↓ ≠ Z ( m ) ∧ Y ( k ) ≠ i Z ( m ) } to the intersection system I n − , σ n − , ⟨ τ n − ,k ∣ k ∈ I n − ⟩ , andlet K n ∶ = ⋂ k ∈ I n P ∗ k [ σ n , τ n,k ] and j ( n ) be an index for K n .If b ∉ O or no such m exists, do nothing, i.e., let K n ∶ = K n − , σ n ∶ = σ n − , τ n,k ∶ = τ n − ,k , j ( n ) ∶ = j ( n − ) , I n ∶ = I n − . All Other Stages n : Do nothing, i.e., let K n ∶ = K n − , σ n ∶ = σ n − , τ n,k ∶ = τ n − ,k , j ( n ) ∶ = j ( n − ) , I n ∶ = I n − . This completes the construction.Define B ∶ = ⋃ n ∈ N σ n and g k ∶ = ⋃ n ∈ N τ n,k . We start by claiming B ∈ ⋂ n ∈ N K n : by construction, for k ∈ ⋂ n ∈ N I n , we have B ⊕ g k ∈ P k , showing B ∈ P ∗ k . Additionally, by construction B ∈ P ∗ k [ σ n , τ n,k ] for every n and every k ∈ ⋂ n ∈ N I n , so B ∈ ⋂ k ∈ I n P ∗ k [ σ n , τ n,k ] = K n . Thus, B ∈ ⋂ n ∈ N K n . In particular, B ∈ K = K , so ω B = ω CK1 .If Z ≤ HYP B , then Theorem 2.5 shows there are c ∈ O B and e ∈ N suchthat Z = ( H Bb ) e . Because ω B = ω CK1 , there exists b ∈ O such that ∣ b ∣ = ∣ c ∣ and hence H Bb ≡ T H Bc by Spector’s Uniqueness Theorem [11, Corollary II.4.6,pg. 40]. Let f be an index such that ϕ ( ) ,H Bb f = H Bc , so that Z = ( ϕ ( ) ,H Bb f ) e .By construction, at Stage n = b + ⋅ e ⋅ f it must have been the case thatno m and Y , Y ∈ K n − existed with ϕ ( ) ,H Y b f ( e ⋅ m ) and ϕ ( ) ,H Y b f ( e ⋅ m ) both defined and unequal. In particular, ϕ ( ) ,H Bb f is a Σ singleton, and sohyperarithmetical. But then H Bc ≡ T H Bb is hyperarithmetical, hence Z = ( H Bc ) e is hyperarithmetical, a contradiction. Thus, Z ≰ HYP B .We now make the following observations: assuming j ( n − ) is known (andutilizing the implicit course-of-values procedure to yield I n − , σ n − , ⟨ τ n − ,k ⟩ k ∈ N ),then. . . 8 . . in Stage n = e +
1, the determination of I n , σ n , ⟨ τ n,k ⟩ k ∈ N (and hence also j ( n ) ) is recursive in O by Theorem 2.4.. . . in Stage n = e +
2, the determination of I n , σ n , ⟨ τ n,k ⟩ k ∈ N (and hence also j ( n ) ) is recursive in A (by construction) or B ⊕ O (by determining theunique i for which ρ i ( σ n − , j ( n − )) ⊂ B ) by Theorem 2.4.. . . in Stage n = b + ⋅ e ⋅ f , the determination of I n , σ n , ⟨ τ n,k ⟩ k ∈ N (and hencealso j ( n ) ) is recursive in B ⊕ O (the determination of whether b ∈ O andwhether there exists an m and Y , Y ∈ K n − for which ϕ ( ) ,H Y b f ( e ⋅ m ) and ϕ ( ) ,H Y b f ( e ⋅ m ) are both defined and unequal may be performedrecursively in O since it corresponds to checking whether a particular Σ class is nonempty, and once the least such m is found, we may determinethe least k and i ∈ { , } for which { Y ∈ K n − ∣ Y ( k ) = i } and { Y ∈ K n − ∣ Y ( k ) = − i } are nonempty; finally, checking whether B ( k ) = i or B ( k ) = − i determines whether we intersected { Y ∈ { , } N ∣ ϕ ( ) ,H Yb f ( e ⋅ m ) ↓ = ∧ Y ( k ) = i } or { Y ∈ { , } N ∣ ϕ ( ) ,H Yb f ( e ⋅ m ) ↓ = ∧ Y ( k ) = − i } ,respectively) or A (as before, the determination of whether b ∈ O andof the existence of such an m may be done recursively in O ≤ T A , and Z ≤ T A ).. . . in all other Stages n , the determination of I n , σ n , ⟨ τ n,k ⟩ k ∈ N (and hence also j ( n ) ) is recursive.In particular, j ≤ T A and j ≤ T B ⊕ O .We make the following final observations: • A ≤ T j ⊕ O as A ( e ) = i if and only if j ( n ) = U ( ρ i ( σ n − , j ( n − )) , I n − , ⟨ τ n − , , . . . , τ n − ,n − ⟩) , where n = e + • O B ≤ T j ⊕ O as B ∈ P ∗ e if and only if v ( j ( n − ) , e ) ∉ { i ∣ P ∗ i = ∅ } ≡ T O . Thedetermination v ( j ( n − ) , e ) ∉ { i ∣ P ∗ i = ∅ } ≡ T O can be made recursivelyin j ⊕ O .Thus, we find that A ≤ T j ⊕ O ≤ T B ⊕ O ≤ T O B ≤ T j ⊕ O ≤ T A so we have Turing equivalence throughout.The following corollary is originally due to Macintyre [7, Theorem 3, pg. 9]. Corollary 2.6.
Suppose A is a real such that O ≤ T A . Then there exists B such that A ≡ T O B ≡ T B ⊕ O . Corollary 2.7.
Suppose K is a nonempty Σ class. Then there exists B ∈ K such that O ≡ T O B ≡ T B ⊕ O .Proof. If K is uncountable, then we apply Theorem 2.1 with Z = A = O .If K is countable, then its elements are hyperarithmetical [11, TheoremIII.6.2, pg. 72] and so any B ∈ K satisfies O ≡ T O B ≡ T B ⊕ O .We can generalize Theorem 2.1, replacing the real Z by a sequence of reals,as follows. Theorem 2.8.
Suppose K is an uncountable Σ class and Z and A are realssuch that Z ⊕ O ≤ T A and < HYP ( Z ) k for each k ∈ N . Then there exists B ∈ K such that A ≡ T O B ≡ T B ⊕ O and ( Z ) k ≰ HYP B for all k .Proof. The proof of Theorem 2.1 may be adapted by replacing Stage n = b + ⋅ e ⋅ f with n = b + ⋅ e ⋅ f ⋅ k and replacing therein Z with ( Z ) k . Theorem 3.1 (Posner-Robinson for Turing Degrees of Hyperjumps) . Suppose Z and A are reals such that Z ⊕ O ≤ T A and < HYP Z . Then there exists B such that A ≡ T O B ≡ T B ⊕ Z ≡ T B ⊕ O . Theorem 3.1 is essentially due to Slaman [13]. The rest of this section isdevoted to a proof of Theorem 3.1. The key to the proof is a forcing notionknown as Kumabe-Slaman forcing, which was originally introduced in [12].
In order to prove Theorem 3.1, we will use Turing functionals and an associatednotion of forcing to construct the desired B . Definition 3.2 (Turing Functionals) . [12, 9] A Turing functional
Φ is a setof triples ( x, y, σ ) ∈ N × { , } × { , } ∗ (called computations in Φ) such thatif ( x, y , σ ) , ( x, y , σ ) ∈ Φ and σ and σ are compatible, then y = y and σ = σ .A Turing functional Φ is use-monotone if:10i) For all ( x , y , σ ) and ( x , y , σ ) are elements of Φ and σ ⊂ σ , then x < x .(ii) For all x and ( x , y , σ ) ∈ Φ where x > x , then there are y and σ such that σ ⊆ σ and ( x , y , σ ) ∈ Φ. Remark . Despite the terminology, a Turing functional Φ is not assumed tobe recursive or even recursively enumerable.
Definition 3.4 (Computations along a Real) . [12, 9] Suppose Φ is a Turingfunctional and X ∈ { , } N . Then ( x, y, σ ) ∈ Φ is a computation along X if σ ⊂ X , in which case we write Φ ( X )( x ) = y . If for every x ∈ N there exists y ∈ { , } and σ ⊂ X such that ( x, y, σ ) ∈ Φ, then Φ ( X ) defines an element of { , } N (otherwise it is a partial function). Lemma 3.5.
Suppose Φ is a Turing functional, X ∈ { , } N , and Φ ( X ) ∈ { , } N . Then Φ ( X ) ≤ T Φ ⊕ X. Proof.
Obvious from the definition of Φ ( X ) . Definition 3.6 (Kumabe-Slaman Forcing) . [12, 9] Define the poset ( P , ≤ ) asfollows:(i) Elements of P are pairs ( Φ , X ) where Φ is a finite use-monotone Turingfunctional and X is a finite subset of { , } N .(ii) If p = ( Φ p , X p ) and q = ( Φ q , X q ) are in P , then p ≤ q if(a) Φ p ⊆ Φ q and for all ( x q , y q , σ q ) ∈ Φ q ∖ Φ p and all ( X p , y p , σ p ) ∈ Φ p ,the length of σ q is greater than the length of σ p .(b) X p ⊆ X q .(c) For every x , y , and X ∈ X , if Φ q ( X )( x ) = y , then Φ p ( X )( x ) = y .In other words, a stronger condition than p can add longer computationsto Φ p , provided they don’t apply to any element of X p .In the remainder of §
3, we will be discussing Kumabe-Slaman forcing overcountable ω -models of ZFC . Unlike in the forcing constructions in axiomaticset theory, it will be important here that the countable ground model M is not well-founded. We now introduce some conventions for discussing such models.Let M be a countable non-well-founded ω -model of ZFC . Let θ ( x , . . . , x n ) be a sentence in the language of ZFC with parameters x , . . . , x n from M . Wewrite θ M ( x , . . . , x m ) or M ⊧ θ ( x , . . . , x n ) to mean that θ ( x , . . . , x n ) holds in M . In particular, x ∈ M x means that M ⊧ x ∈ x , etc. We tacitly identitythe natural number system of M with the standard natural number system, the Here
ZFC denotes Zermelo-Fraenkel Set Theory with the Axiom of Choice. However, forthe purposes of this paper, our ω -models need not satisfy ZFC but only a small subsystem of
ZFC or actually of second-order arithmetic. M with standard reals, etc. In particular, let P M be the set of pairs ( Φ , X ) such that M ⊧ “ ( Φ , X ) is a Kumabe-Slaman forcing condition”. In thiscase, Φ is identified with a finite Turing functional, X is identified with a finiteset of reals belonging to M , etc., so ( Φ , X ) actually is a Kumabe-Slaman forcingcondition.The key property of Kumabe-Slaman Forcing is the following: Lemma 3.7. [9, based on Lemma 3.10, pg. 23] Suppose M is an ω -model of ZFC , D ∈ M is dense in P M , and X , . . . , X n ∈ { , } N . Then for any p ∈ P M ,there exists q ≥ p such that q ∈ D and Φ q does not add any new computationsalong any X k .Proof. Suppose p = ( Φ p , X p ) ∈ P M . Say that an n -tuple of strings ⃗ τ is essential for ( p, D ) if q > p and q ∈ D implies the existence of ( x, y, σ ) ∈ Φ q ∖ Φ p such that σ is compatible with some component of ⃗ τ , i.e., any extension of p in D adds acomputation along a string compatible with a component of ⃗ τ . Being essentialfor ( p, D ) is definable in M .Define T n ( p, D ) ∶ = {⃗ τ ∈ ({ , } ∗ ) n ∣ ⃗ τ is essential for ( p, D ) and ∣ τ ∣ = ⋯ = ∣ τ n ∣} . Being essential for ( p, D ) is closed under taking (component-wise) initial seg-ments, so T n ( p, D ) is a finitely branching tree in M .Suppose for the sake of a contradiction that for every q > p , either q ∉ D orelse q adds a new computation along some X k . We claim that ⟨ X ↾ m, . . . , X n ↾ m ⟩ is essential for ( p, D ) for all m ∈ N . Given q > p with q ∈ D , by hypothesis thereis some computation ( x, y, σ ) ∈ Φ q ∖ Φ p along some X k . This means that σ ⊂ X k (outside of M ), so σ is compatible with X k ↾ m .This shows that T n ( p, D ) is infinite. As M is a model of ZFC , it follows that T n ( p, D ) has a path through it. The requirement that the components of anyelement of T n ( p, D ) are of the same length implies that such a path is of theform ( Y , . . . , Y n ) for Y , . . . , Y n ∈ M ∩ { , } N .Consider p = ( Φ p , X p ∪ { Y , . . . , Y n }) . Suppose q ≥ p and q ∈ D . Each n -tuple ⟨ Y ↾ m, . . . , Y n ↾ m ⟩ is essential for ( p, D ) for each m , so there exists ( x m , y m , σ m ) ∈ Φ q ∖ Φ p such that σ m is compatible with Y k ↾ m for some k . AsΦ q is finite, letting m be sufficiently large shows that there is ( x, y, σ ) ∈ Φ q ∖ Φ p for which σ is an initial segment of Y k for some k . However, this is not possiblesince q ≤ p implies Y k ∈ X q . This provides the needed contradiction.Suppose G is an M -generic filter for P M . Then for every XX ⊆ M N ⇐⇒ there is p ∈ G with X ∈ X p since for any X ⊆ M N , the set { p ∈ P M ∣ ( ∅ , { X }) ≤ p } M is a dense open subsetof P M in M . Thus, the essential parts of an M -generic filter G are the Turingfunctionals Φ p for p ∈ G . 12 efinition 3.8. A Turing functional Φ is M -generic for P M if and only ifthere exists an M -generic filter G such that ( x, y, σ ) ∈ Φ ⇐⇒ there exists p ∈ G such that ( x, y, σ ) ∈ M Φ p . Φ may be identified with an element ( ˙Φ ) G in M [ G ] , where M ⊧ ˙Φ = {( p, ˙ c ) ∣ p ∈ P M ∧ c ∈ Φ p } and ˙ c is a canonical name for c ∈ M , defined by transfinite recursion in M to bethe unique element in M for which M ⊧ ˙ c = P M × { ˙ b ∣ b ∈ c } . Lemma 3.9.
The following are equivalent for a Turing functional Φ :(i) Φ is an M -generic Turing functional for P M .(ii) For every dense open subset D ⊆ M P M , there exists p ∈ M D such that ( x, y, σ ) ∈ M Φ p Ô⇒ ( x, y, σ ) ∈ Φ . Proof. ( i ) Ô⇒ ( ii ) Let G be an M -generic filter for P M such that ( x, y, σ ) ∈ Φ ⇐⇒ there exists p ∈ G such that ( x, y, σ ) ∈ M Φ p . Suppose D ⊆ M P M is dense open. By definition, there exists p ∈ G suchthat p ∈ M D . Then by definition, ( x, y, σ ) ∈ M Φ p Ô⇒ ( x, y, σ ) ∈ Φ . ( ii ) Ô⇒ ( i ) For p ∈ M P M , temporarily write p < Φ if ( x, y, σ ) ∈ M Φ p implies ( x, y, σ ) ∈ Φ. Define G ∶ = { q ∣ ∃ p ( p < Φ ∧ M ⊧ ( p ≤ q ))} . We claim that G is an M -generic filter for P M . Upwards closed:
Suppose q ∈ G and M ⊧ ( q ≤ q ′ ) . Let p < Φ be suchthat M ⊧ p ≤ q . Then M ⊧ p ≤ q ′ since M ⊧ ( ≤ is transitive ) , so q ′ ∈ G . Downwards directed:
Suppose q, q ′ ∈ G . Let p, p ′ < Φ be such that M ⊧ ( p ≤ q ∧ p ′ ≤ q ′ ) . Then the unique p ′′ for which M ⊧ p ′′ = ( Φ p ∩ Φ p ′ , ∅ ) satisfies p ′′ < Φ and M ⊧ ( p ′′ ≤ q ∧ p ′′ ≤ q ′ ) . M -generic: Suppose D ⊆ M P M is dense open. By hypothesis, there exists p < Φ such that p ∈ M D . By definition of G , p ∈ G .By defining an M -generic Turing functional Φ for P M by means of approx-imations, Theorems 3.7 and 3.10 allow us to meet dense sets without affectingΦ ( Z ) , which can then be arranged independently.13 .2 Proof of Posner-Robinson for Hyperjumps Now we proceed with the proof of Theorem 3.1:
Lemma 3.10.
Suppose Z and A are reals such that Z ⊕ O ≤ T A and < HYP Z .Then there exists a (code for a) countable ω -model M of ZFC such that O M ≡ T A and Z ∉ M .Proof. The set of codes for countable ω -models of ZFC is Σ , so the existenceof a code of such an M follows from Theorem 2.1. Proof of Theorem 3.1:
The main idea of the proof is due to Slaman [13].We shall construct an M -generic Turing functional Φ with B = Φ the desiredreal. Assume without loss of generality that no initial segment of Z is an initialsegment of O . By arranging for Φ ( Z ) ∈ { , } N and Φ ( Z ) = O Φ and Φ ( O ) = A ,this will complete the proof.By Theorem 3.10, there exists a countable ω -model M of ZFC such that O , Z ∉ M and O M ≡ T A . Without loss of generality, M = ⟨ ω, E ⟩ .Let D , D , D , . . . be an enumeration, recursive in A , of the dense opensubsets of P M in M ( M is countable and O M ≡ T A , so this is possible). Toconstruct our M -generic Φ, we approximate it by finite initial segments p ≤ p ≤ ⋯ ≤ p n ≤ ⋯ . During our construction, we alternate between meeting dense sets, arrangingfor Φ ( O ) = A , and arranging for Φ ( Z ) ≡ T O Φ . Stage n = : Define p ∶ = ( ∅ , ∅ ) . Stage n = m : Suppose p n − has been constructed. By Theorem 3.7, there ex-ists q ∈ D n extending p n − which does not add any new computationsalong Z or O . Let p n be the least such condition. Stage n = m ⋅ : We extend p n − to p n by adding ( m, A ( m ) , σ ) where σ ⊂ O is a sufficiently long initial segment of O (i.e., the shortest initial segmentof O which is longer than any existing strings in elements of Φ p n − ). Stage n = m ⋅ : Suppose p n − has been constructed. By construction, thereis no y and σ ⊂ Z such that ( m, y, σ ) ∈ Φ p n − . Now proceed as follows: Substage 1:
Consider the set D (in M ) containig all q ∈ P M such thatone of the following conditions hold:(i) q ⊩ ( m encodes a Φ-recursive linear order on ω ∧ m ∈ O Φ ∧∃ α ( α ∈ Ord M ∧ ∣ m ∣ = α )) ,(ii) q ⊩ ( m encodes a Φ-recursive linear order on ω ∧ m ∉ O Φ ) , or(iii) q ⊩ ¬ ( m encodes a Φ-recursive linear order on ω ) . D is dense. By Theorem 3.7, there exists q ∈ D extending p n − whichdoes not add any new computations along Z or O . Let q be minimalwith that property. 14 ubstage 2: Extend q to p n by adding ( m, y, σ ) , where σ ⊂ Z is a suffi-ciently long initial segment of Z (i.e., the shortest initial segment of Z which is longer than any existing strings in elements of Φ q ) and y depends on the following cases: Case 1: If q ⊩ ( m encodes a Φ-recursive linear order on ω ∧ m ∈ O Φ ∧∃ α ( α ∈ Ord M ∧ ∣ m ∣ = α )) , then we break into two subcases: Case 1a: If α is in the standard part of Ord M , then α is actuallyan ordinal and m does encode a Φ-recursive linear order on ω . Thus, set y ∶ = Case 1b: If α is not in the standard part of Ord M , then α is notactually well-ordered (it is only well-ordered when viewed in M ) so m does not encode a Φ-recursive linear order on ω .Thus, set y ∶ = Case 2: If q ⊩ ( m encodes a Φ-recursive linear order on ω ∧ m ∉ O Φ ) ,then m cannot encode a Φ-recursive well-ordering of ω . Thus,set y ∶ = Case 3: If q ⊩ ¬ ( m encodes a Φ-recursive linear order on ω ) , thenset y ∶ = All Other Stages n : Let p n = p n − .Define Φ to be the unique set such that ( x, y, σ ) ∈ Φ ⇐⇒ there exists n ∈ N such that ( x, y, σ ) ∈ M Φ p n . Thanks to Stages n = m and Theorem 3.9, Φ is an M -generic Turing functional.Thanks to Stages n = m ⋅
3, Φ ( O ) = A . Thanks to Stages n = m ⋅
5, Φ ( Z ) = O Φ .We also note that in the above construction of Φ, (assuming p n − is given). . .. . . Stage n = m is recursive in O M ⊕ Z ⊕ O ≤ T A ,. . . Stage n = m ⋅ O ≤ T A ,. . . Stage n = m ⋅ O ⊕ O M ⊕ Z ≤ T A ,. . . Stage n = m ⋅ Z ≤ T A , and. . . Stage n (for all other n ) is recursive.Thus, Φ ≤ T M ⊕ ( O M ⊕ Z ) ⊕ A ≤ T A. Applying Theorem 3.5 we find A = Φ ( O ) ≤ T O ⊕ Φ ≤ T O Φ ≡ T Φ ( Z ) ≤ T Z ⊕ Φ ≤ T Z ⊕ A ≡ T A so we have Turing equivalence throughout. B = Φ is hence the desired real.Theorem 3.1 can be generalized, replacing the real Z by a sequence of reals.15 heorem 3.11. Suppose Z and A are reals such that Z ⊕ O ≤ T A and < HYP ( Z ) k for every k ∈ N . Then there exists B such that for every k ∈ N A ≡ T O B ≡ T B ⊕ ( Z ) k ≡ T B ⊕ O . Proof.
The proof of Theorem 3.1 may be adapted by making the following ad-justments. First, we replace the use of Theorem 3.10 with the following lemma:
Lemma 3.12.
Suppose Z and A are reals such that Z ⊕ O ≤ T A and < HYP ( Z ) k for every k ∈ N . Then there exists a (code for a) countable ω -model M of ZFC such that O M ≡ T A and ( Z ) k ∉ M for every k ∈ N .Proof. Replace the usage of Theorem 2.1 in the proof of Theorem 3.10 withTheorem 2.8.This yields a (code for a) countable ω -model M of ZFC such that O , ( Z ) , ( Z ) , . . . ∉ M and O M ≡ T A . We assume without loss of generality that O ≠ ( Z ) k for each k .The adjustments to the construction are the following: • In Stages n = m and n = m ⋅
3, we avoid adding new computations along ( Z ) , . . . , ( Z ) n and O . • Replace Stage n = m ⋅ n = m ⋅ k + , and at the beginningof Stage n = m ⋅ k + , first check if there exists y and σ ⊂ ( Z ) k suchthat ( m, y, σ ) ∈ Φ p n − . If such a y and σ are found, do nothing andproceed to the next stage. Otherwise, proceed as in Stage n = m ⋅ ( Z ) , . . . , ( Z ) n and O as above.Note that it is no longer necessarily the case that Φ (( Z ) k ) = O Φ for every k ∈ N , as early stages may have added computations to Φ which make Φ (( Z ) k ) disagree with O Φ . However, after Stage k , no other stages add new computationsalong ( Z ) k except for those purposely added (i.e., in Stages n = m ⋅ k + ). Itfollows that Φ (( Z ) k ) and O Φ differ only on a finite set of indices, so Φ (( Z ) k ) ≡ T O Φ .In the resulting construction of Φ, (assuming p n − is given). . . Stage n = m is recursive in O M ⊕ ⊕ ni = ( Z ) i ⊕ O ≤ T A ,. . . Stage n = m ⋅ O ≤ T A ,. . . Stage n = m ⋅ k + (Substage 1) is recursive in O ⊕ O M ⊕ ⊕ ni = ( Z ) i ≤ T A ,. . . Stage n = m ⋅ k + (Substage 2) is recursive in ( Z ) k ≤ T A , and. . . Stage n (for all other n ) is recursive.Thus, Φ ≤ T M ⊕ ( O M ⊕ Z ) ⊕ A ≡ T A. The proof concludes as in the proof of Theorem 3.1.16
Open Problems
In light of Theorems 2.1 and 3.1, it is natural to ask whether they can becombined into one theorem. In other words, for which uncountable Σ classes K ⊆ { , } N do the following properties hold? Property 4.1.
Suppose Z and A are reals such that Z ⊕ O ≤ T A and 0 < HYP ( Z ) k for every k ∈ N . Then there exists B ∈ K such that A ≡ T O B ≡ T B ⊕ Z ≡ T B ⊕ O . Property 4.2.
Suppose Z and A are reals such that Z ⊕ O ≤ T A and 0 < HYP ( Z ) k for every k ∈ N . Then there exists B ∈ K such that for every kA ≡ T O B ≡ T B ⊕ ( Z ) k ≡ T B ⊕ O . The following theorem answers some special cases of this problem.
Theorem 4.3.
Let L T = { X ∣ O X ≡ T X ⊕ O } . Suppose K is an uncountable Σ class which is Turing degree upward closed in L T , i.e., whenever X, Y ∈ L T , X ∈ K , and X ≤ T Y , then there is Y ∈ K such that Y ≡ T Y . Then K hasTheorems 4.1 and 4.2.Proof. This theorem is analogous to [3, Lemma 3.3]. By Theorem 2.1, let C besuch that A ≡ T O C ≡ T C ⊕ Z ≡ T C ⊕ O . ( ∗ )Theorem 2.1, relativized to C , yields B ∈ K such that O C ≡ T O B ⊕ C ≡ T B ⊕ O C . ( † )Combining ( † ) and ( ∗ ) shows that O B ⊕ C ≡ T B ⊕ C ⊕ O . As B ≤ T B ⊕ C ,there is B ∈ K such that B ≡ T B ⊕ C by hypothesis. In particular, O C ≡ T O B ≡ T B ⊕ O . Moreover, in combination with ( ∗ ), A ≡ T O B ≡ T B ⊕ O ≡ T B ⊕ Z. This shows that K has Theorem 4.1.To show that K has Theorem 4.2, repeat the above argument using Theorem 2.8instead of Theorem 2.1. Remark . The proof of Theorem 4.3 is easily adapted to prove the same resultwith L T = { X ∣ O X ≡ T X ⊕ O } replaced by L HYP = { X ∣ O X ≡ HYP X ⊕ O } .The hyperarithmetical analog of the Pseudojump Inversion Theorem [4, The-orem 2.1, pg. 601] also remains open. Namely, suppose V Xe is an effectiveenumeration of the Π ,X predicates, uniformly in X . Define the e -th pseudo-hyperjump by HJ e ( X ) ∶ = X ⊕ V Xe . Does the following result hold? 17 onjecture 4.5.
Suppose e ∈ N and A is a real such that O ≤ T A . Then thereexists B such that A ≡ T HJ e ( B ) ≡ T B ⊕ O . (1)Even if Theorem 4.5 holds, this leaves open the question of characterizingthe Σ classes K ⊆ { , } N with the following properties: Property 4.6.
Suppose e ∈ N and A is a real such that O ≤ T A . Then thereexists B ∈ K such that Equation (1) holds. Property 4.7.
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