Generics for Mathias forcing over general Turing ideals
aa r X i v : . [ m a t h . L O ] J u l GENERICITY FOR MATHIAS FORCING OVER GENERALTURING IDEALS
PETER A. CHOLAK, DAMIR D. DZHAFAROV, AND MARIYA I. SOSKOVA
Abstract.
In Mathias forcing, conditions are pairs (
D, S ) of sets of natu-ral numbers, in which D is finite, S is infinite, and max D < min S . TheTuring degrees and computational characteristics of generics for this forcingin the special (but important) case where the infinite sets S are computablewere thoroughly explored by Cholak, Dzhafarov, Hirst, and Slaman [2]. Inthis paper, we undertake a similar investigation for the case where the sets S are members of general countable Turing ideals, and give conditions underwhich generics for Mathias forcing over one ideal compute generics for Mathiasforcing over another. It turns out that if I does not contain only the com-putable sets, then non-trivial information can be encoded into the generics forMathias forcing over I . We give a classification of this information in terms ofcomputability-theoretic properties of the ideal, using coding techniques thatalso yield new results about introreducibility. In particular, we extend a re-sult of Slaman and Groszek and show that there is an infinite ∆ set with nointroreducible subset of the same degree. Introduction
Mathias forcing gained prominence in set theory in the article [9], for whoseauthor it has come to be named. In a restricted form, it was used even earlierby Soare [12], to build an infinite set with no subset of strictly higher Turingdegree. In computability theory, it has subsequently become a prominent tool forconstructing infinite homogeneous sets for computable colorings of pairs of integers,as in Seetapun and Slaman [10], Cholak, Jockusch, and Slaman [3], and Dzhafarovand Jockusch [5]. It has also found applications in algorithmic randomness, inBinns, Kjos-Hanssen, Lerman, and Solomon [1]. Dorais [4] has studied a variant ofMathias forcing that behaves nicely with respect to reverse mathematics.The conditions in Mathias forcing are pairs (
D, S ), where D is a finite subset of ω , S is an infinite such subset, and max D < min S . A condition ( D ∗ , S ∗ ) extends ( D, S ) if D ⊆ D ∗ ⊆ D ∪ S and S ∗ ⊆ S . We think of the finite set D as representinga commitment of information, positive and negative, about a (generic) set to beconstructed, and E as representing a commitment of negative information alone.In computability theory, the interest is typically in Mathias forcing with animposed effectivity restriction on the conditions used. For instance, in [3, Theorem4.3], the sets S are restricted to be computable, whereas in [10, Theorem 2.1], thesets are restricted to be members of a Scott set. Many other variants have appeared The second author was partially supported by an NSF Postdoctoral Fellowship and by NSFgrant DMS-1400267. The third author was partially supported by an FP7-MC-IOF grant STRIDE(298471). All three authors were additionally partially supported by NSF grant DMS-1101123,which made their initial collaboration possible. The authors thank Rose Weisshaar and the anony-mous referee for a number of valuable comments. in the literature. The most general requirement of this form is to restrict the sets S to be members of a fixed countable Turing ideal I .Our interest in this paper will be in the computability-theoretic properties ofgenerics for the above forcing. A similar analysis for the special case when I consists just of the computable sets was undertaken by Cholak, Dzhafarov, Hirst,and Slaman [2]. As we shall see, this situation differs from the general one in anumber of important ways. For instance, given a non-computable set A , it is alwayspossible to choose a generic for Mathias forcing over the computable ideal that doesnot compute A . We characterize those sets A and ideals I that have this property,and construct ones that do not. Thus, we show there are ideals I over which everyMathias generic contains some common non-computable information.The paper is organized as follows. In Section 2 we give formal definitions ofMathias forcing, and in particular, of how we choose to represent conditions. Sec-tion 3 establishes some basic results and constructions. In Section 4, we turn tothe computational strength of generics for Mathias forcing, focusing on the prob-lem mentioned above, of which sets are necessarily computed by them. Finally, inSection 5, we compare generics across different ideals.2. Background
Throughout, sets will refer to subsets of ω . We shall use standard terminologyfrom computability theory, and refer the reader to Soare [13] for background. Fora general introduction to forcing in arithmetic, see Shore [11, Section 3]. Definition 2.1. A (Turing) ideal is a collection I of sets closed under ≤ T and ⊕ .In this paper we shall be looking at countable ideals only, and so shall avoidexplicitly mentioning so henceforth. The simplest ideal is COM P , consisting ofall the computable sets, which is of course a sub-ideal of every other ideal. Thenext simplest example of an ideal is a principal one, consisting of all A -computablesets for some set A . We denote this ideal by [ A ] ≤ T (so that COM P = [ A ] ≤ T forany computable set A ), and note that it can be identified with the set of indicesof total A -computable functions, so membership in it is a Σ ( A ) relation of sets.Though not every ideal is principal, more general ideals can be presented in a similarway. Theorem 2.2 (Kleene and Post [8]; Spector [15]) . Every ideal I has an exact pair,i.e., a pair of sets A , A so that a set S belongs to I if and only if S ≤ T A and S ≤ T A . Membership in such an ideal is then Σ ( A ⊕ A ), and the ideal can be identifiedwith the set of pairs of indices h e , e i so that Φ A e and Φ A e are both total andequal. The choice of A and A here is not canonical in any way, but in the sequelwe shall implicitly assume that whenever an ideal I is mentioned a choice of A and A has been made. (Formally, this means different choices of exact pairs givedifferent ideals, even if they are the same as collections of sets.) For simplicity,we suppress mention of A and A when possible: for instance, given n ∈ ω , wewrite Σ n ( I ) instead of Σ n ( A ⊕ A ). In a similar spirit, we call I arithmetical if A and A are arithmetical sets.Recall the following standard terminology. Definition 2.3.
ENERICITY FOR MATHIAS FORCING OVER GENERAL TURING IDEALS 3 (1) A
Mathias condition is a pair of sets (
D, S ) with D finite, S infinite,and max D < min S .(2) A condition ( e D, e S ) extends a condition ( D, S ), written ( e D, e S ) ≤ ( D, S ),if D ⊆ e D ⊆ D ∪ S and e S ⊆ S .(3) A set A satisfies a condition ( D, S ) if D ⊆ A ⊆ D ∪ S .(4) A set A meets a collection C of conditions if it satisfies a condition in C .(5) A set A avoids a collection C of conditions if it satisfies a condition havingno extension in C .We next define the restriction of Mathias forcing to a particular ideal. Definition 2.4.
Let I be an ideal.(1) An I -condition is a Mathias condition ( D, S ) such that S ∈ I .(2) For each n ∈ ω , a set G is n - I -generic if G meets or avoids every Σ n ( I )-definable collection of I -conditions.(3) A set G is I -generic if it is n - I -generic for all n ∈ ω .Cholak, Dzhafarov, Hirst, and Slaman [2, Proposition 2.4] showed that for each n ≥
2, there exists an n - COM P -generic set G with G ′ ≤ T ∅ ( n ) . The argument easilylifts to more general ideals I , yielding n - I -generics with G ′ ≤ T I ( n ) . We leave theverification to the reader. As discussed in Section 2 of [2], it is generally only ofinterest to consider n -generics for n ≥ I is the usual language of second-order arith-metic augmented by a parameter ˙ G for the generic set, but with the modificationthat it also includes a parameter for I , which is to say, for the exact pair chosenfor I . The forcing relation can then be defined in the usual manner (see, e.g., [2],Section 3). 3. Basic results
We begin by collecting a few basic facts about Mathias generics over arbitraryideals, which also serve as preliminary results that we shall expound upon in sub-sequent sections. To begin, we have the following generalization of the well-knownfact that Mathias generics are high (see, e.g., [1], Lemma 6.6, or [3], Section 5.1).For a set A , let p A denote the principal function of A . Proposition 3.1.
Let I be an ideal and G a - I -generic. Then p G dominatesevery function in I .Proof. Fix any function f ∈ I , and let C be the collection of all I -conditions ( D, S )such that p S majorizes f . Note that C is Σ ( I )-definable, and that it is in factdense. (To see this, consider any I -condition ( D, S ). Define e S ⊆ S inductivelyby letting e S ( i ) be the least element of S larger than f ( i ) and all e S ( j ) for j < i .Then e S ≤ T f ⊕ S , so e S belongs to I , and ( D, e S ) belongs to C .) We conclude thatany 3- I -generic set G meets C , so p G dominates f . (cid:3) Our next results concern the computational strength of generics, which shall bethe focus of Section 4.
Proposition 3.2.
Let I be an ideal and A any set not in I . Then there exists aMathias I -generic set that does not compute A . PETER A. CHOLAK, DAMIR D. DZHAFAROV, AND MARIYA I. SOSKOVA
Proof.
We obtain a generic G by building a sequence of I -conditions ( D , S ) ≥ ( D , S ) ≥ · · · with lim s | D s | = ∞ , and setting G = S s D s . Let ( D , S ) = ( ∅ , ω ),and assume that for some s ≥ D s , S s ).If s is even, we work to make G not compute A . Say s = 2 e . Ask if thereexists a finite set F ⊆ S s and an x such that Φ D s ∪ Fe ( x ) ↓6 = A ( x ). If so, choosesome such F and x and let D s +1 = D s ∪ F and S s +1 = S s − { y : y < ϕ D s ∪ Fe ( x ) } .(Here, ϕ D s ∪ Fe ( x ) denotes the use of the computation Φ D s ∪ Fe ( x ).) Now G will sat-isfy ( D s +1 , E s +1 ), and thus Φ Ge will converge on x and differ from A ( x ), ensuringthat G does not compute A via Φ e . If, on the other hand, no such F and x ex-ist, let ( D s +1 , S s +1 ) = ( D s , S s ). In this case, it cannot be that Φ Ge is total, asotherwise S s would compute A , contradicting that A / ∈ I .If s is odd, we work to make G be I -generic. Say s = 2 e + 1, and let C bethe e th member in some fixed listing of all arithmetical collections of I -conditions.Let ( D s +1 , S s +1 ) be any extension of ( D s , S s ) in C , if such exists, and otherwiselet ( D s +1 , S s +1 ) = ( D s , S s ). Thus, G will either meet or avoid C , as needed. (cid:3) We pause here to present a classical computability-theoretic consequence of theabove results. Recall that by Martin’s high domination theorem, a set A is high,i.e., satisfies A ′ ≥ T ∅ ′′ , if and only if p A dominates every computable function.The left-to-right direction of this result relativizes in the following simple form:for any set L , if A ′ ≥ T L ′′ then p A dominates every L -computable function. Theconverse is false, as can be shown directly, but we obtain here the following newsimple proof of this fact. (The correct relativization of the right-to-left directionof Martin’s result is as follows: for any set L , if p A dominates every L -computablefunction then ( A ⊕ L ) ′ ≥ T L ′′ .) Corollary 3.3.
There exist sets L and G such that G dominates every L -computablefunction, but G ′ (cid:3) T L ′′ .Proof. Suppose not. Let L be any low c.e. set, and I the principal ideal below L .Let M be any low c.e. set not computable from L , and by Proposition 3.2, let G beany 3- I -generic that does not compute M . By the limit lemma, fix a computablefunction ˆ M in two arguments that approximates M in the limit.By Proposition 3.1, G dominates every L -computable function, so by assump-tion, G ′ ≥ T L ′′ . But as L and M are low, L ′′ ≡ T ∅ ′′ ≡ T M ′′ , and so G ′ ≥ T M ′′ .It follows that G dominates every M -computable function, and so in particular G dominates the weak modulus of ˆ M , defined by w ( x ) = ( µs )[ ˆ M ( x, s ) = M ( x )]for all x . As M is c.e., this means M is computable from G , a contradiction. (cid:3) One motivation for us in this paper is the question of which ideals satisfy theconverse of Proposition 3.2: that is, for which ideals I is it the case that for some n ,every n - I -generic computes every set A ∈ I ? Definition 3.4.
Fix n ∈ ω . An ideal I is n -generically-coded if every n - I -genericset G computes every A ∈ I .We can find many natural examples of generically-coded ideals. Recall the followingdefinition: a function f is called a modulus for a set A if A is computable from everyfunction g that majorizes f . By a result of Solovay [14, Theorem 2.3], a set has amodulus if and only if it is hyperarithmetic. ENERICITY FOR MATHIAS FORCING OVER GENERAL TURING IDEALS 5
Lemma 3.5 (Folklore) . If A is any ∆ n set, n ≥ , then A has a ( ∅ ( n − ⊕ A ) -computable modulus.Proof. Fix a ∅ ( n − -computable function ˆ A in two arguments that approximates A in the limit. Let f be the ( ∅ ( n − ⊕ A )-computable function mapping each n to theleast s ≥ n such that ˆ A ( x, s ) = A ( x ) for all x ≤ n . Now from any g ≥ f we cancompute A as follows. Given n , we search for an s ≥ n such that ˆ A ( n, t ) = A ( n, s )for all t with s ≤ t ≤ g ( s ). This exists, since for instance any s after the stage atwhich ˆ A has settled on n would do. In particular, s ≤ f ( s ) ≤ g ( s ), so ˆ A ( n, s ) =ˆ A ( n, f ( s )). But since n ≤ s , we have ˆ A ( n, f ( s )) = A ( n ), so ˆ A ( n, s ) = A ( n ). Weconclude that f is a modulus for A . (cid:3) Proposition 3.6.
Let I be an ideal and G a - I -generic. If ∅ ( n − ∈ I forsome n ≥ , then G computes every ∆ n set A ∈ I .Proof. Assume that G computes ∅ ( m − for some m ≤ n , and suppose A ∈ I is ∆ m . By Lemma 3.5, A has a ( ∅ ( m − ⊕ A )-computable modulus for A . Since ∅ ( m − ≤ T ∅ ( n − and ∅ ( n − ∈ I , it follows that this modulus belongs to I . ByProposition 3.1, G dominates this modulus and so computes A . If m < n , thismeans in particular that G computes ∅ ( m − , so by induction, G computes ∅ ( n − .Hence, G computes every ∆ n set in I , as desired. (cid:3) Corollary 3.7. If I is the ideal generated by a union of principal ideals [ A ] ≤ T suchthat A satisfies ∅ ( n ) ≤ T A ≤ T ∅ ( n +1) for some n ∈ ω , then I is -generically-coded.Proof. Let A be any member of I . Then there are sets A , . . . , A k − such that A ≤ T A ⊕ · · · ⊕ A k − , where [ A ] ≤ T , . . . , [ A ] ≤ T k − are among the principal idealsthat generate I . Let n be largest such that ∅ ( n ) ≤ T A i ≤ T ∅ ( n +1) for some i < k .Then ∅ ( n ) ∈ I and A ≤ T ∅ ( n +1) , so by the preceding proposition, we have thatevery 3- I -generic computes A . (cid:3) Corollary 3.8.
Every ∆ ideal is -generically-coded, as is every principal ideal ofthe form [ ∅ ( n ) ] ≤ T , n ∈ ω . The next section is devoted to proving that the last corollary cannot be extendedto general ∆ n ideals if n ≥ Generics and computation
We address the question of which other ideals are generically-coded. That thisis not so for all ideals can be seen as follows. Soare [12] exhibited a set A notcomputable from any of its co-infinite subsets. Let I = [ A ] ≤ T , and let G beany I -generic satisfying the condition ( ∅ , A ). Then G ⊆ A , and genericity ensuresthat A − G is infinite, so G does not compute A , as desired. Unfortunately, thisexample leaves a considerable gap between the complexities of ideals that are andare not generically-coded. Indeed, no arithmetical set satisfies Soare’s theorem,as the degrees of subsets of infinite arithmetical set are closed upwards (see, e.g.,Jockusch [7], Lemma 1.) Hence, the resulting ideal is not arithmetical either.In what follows, we give an example of a non-generically-coded ∆ ideal. Weshall employ the following definition. PETER A. CHOLAK, DAMIR D. DZHAFAROV, AND MARIYA I. SOSKOVA
Definition 4.1.
Let I be an ideal and A any set. Say a set S is I -hereditarilyuniformly A -computing if the infinite subsets of S in I uniformly compute A , i.e.,if there is an e ∈ ω such that Φ Te = A for every infinite T ⊆ S in I .The notion is motivated by the concept of uniform introreducibility introduced byJockusch [6, Section 2]. Recall that an infinite set is (uniformly) introreducible if itis (uniformly) computable from each of its infinite subsets. Every degree contains auniformly introreducible set, namely, the set of codes of all initial segments of anymember of that degree. Proposition 4.2.
Let I be an ideal and A any ∆ n ( I ) set, n ≥ . If G is n - I -generic and G ≥ T A then G is contained in an I -hereditarily uniformly A -computing set S ∈ I .Proof. Fix e ∗ ∈ ω such that Φ Ge ∗ = A . Since n ≥ G must satisfy some condition( D ∗∗ , S ∗∗ ) forcing Φ ˙ Ge ∗ to be total, which means that ( D ∗∗ , S ∗∗ ) avoids the set ofconditions ( D, S ) such that ∃ x ∀ finite sets D ′ ∀ s [ D ⊆ D ′ ⊆ D ∪ S → Φ D ′ e ∗ ,s ( x ) ↑ ] . Let C be the collection of all conditions ( D, S ) such that ∃ x ∃ s [Φ De ∗ ,s ( x ) ↓6 = A ( x )] , which is Σ n ( I )-definable since A is ∆ n ( I ). By genericity, G must consequentlyavoid this collection, say via a condition ( D ∗ , S ∗ ). Without loss of generality, wemay assume ( D ∗ , S ∗ ) ≤ ( D ∗∗ , S ∗∗ ). Let S = D ∗ ∪ S ∗ , and let e ∈ ω be the indexof the functional that, on input x , searches its oracle for the least finite subset F such that min F ≥ min S ∗ and Φ D ∗ ∪ Fe ∗ ( x ) ↓ = v for some value v ∈ { , } , and then returns this value. We claim that S is I -hereditarily uniformly A -computing, as witnessed by e . Since G ⊆ S and S ∈ I ,this gives the desired conclusion.To prove the claim, fix any infinite T ⊆ S in I . If for some x there were no finitesubset F of T with min F ≥ min S ∗ and Φ D ∗ ∪ Fe ∗ ( x ) ↓ , then ( D ∗ , T ∩ S ∗ ) would bean I -extension of ( D ∗ , S ∗ ) (and hence of ( D ∗∗ , S ∗∗ )) forcing Φ ˙ Ge ∗ ( x ) ↑ , which cannotbe. Hence, Φ Te is total, by definition of e . Similarly, if for some x there were suchan F with Φ D ∗ ∪ Fe ∗ ( x ) ↓6 = A ( x ) then ( D ∗ ∪ F, T ∗ ), where T ∗ is the set of elementsof T ∩ S ∗ larger than max F and the use of Φ D ∗ ∪ Fe ∗ ( x ), would be an extension of( D ∗ , S ∗ ) in C . Hence, we also have Φ Te = A . (cid:3) The following corollary extends Proposition 2.8 of [2], where it appears for thespecial case of I = COM P . (Compare also with Proposition 3.2.)
Corollary 4.3.
Let I be an ideal and G an n - I -generic, n ≥ . Then every ∆ n ( I ) set computable from G belongs to I .Proof. By the preceding proposition, if A ≤ T G is ∆ n then in particular I containsan I -hereditarily uniformly A -computing set, so A ∈ I . (cid:3) We now have the following somewhat technical lemma. For simplicity of no-tation, we shall assume here that all computations are { , } -valued, and that all ENERICITY FOR MATHIAS FORCING OVER GENERAL TURING IDEALS 7 domains of convergence are closed downwards. Given a condition (
D, S ), a fi-nite set F , and e ∈ ω , we shall write F ⊆ Φ ( D,S ) e to mean that Φ De convergeson ω ↾ max F + 1 with use bounded by min S , and that F is contained in the fi-nite set given by this computation. So, if F ⊆ Φ ( D,S ) e then F ⊆ Φ ( e D, e S ) e for every( e D, e S ) ≤ ( D, S ), and also F ⊆ Φ Ue for any U satisfying ( D, S ) such that Φ Ue is total. Lemma 4.4.
Let I be an ideal, ( D ∗ , S ∗ ) an I -condition, and F ∗ a finite set suchthat F ∗ ⊆ Φ ( D ∗ ,S ∗ ) e for some e ∈ ω . Fix i ∈ ω and let C be the collection ofconditions ( D, S ) satisfying one of the following properties:(1) ( D, S ) forces that Φ ˙ Ge is not total, not infinite, or not contained in ˙ G ;(2) there is an x such that Φ F ∗ ∪ Fi ( x ) ↑ for all finite F ⊆ S ;(3) there is an x ≤ max D and a finite F ⊆ Φ ( D,S ) e with max F ∗ < min F suchthat Φ F ∗ ∪ Fi ( x ) ↓6 = D ( x ) .Then C is dense below ( D ∗ , S ∗ ) . Moreover, if I = COM P , then an index foran extension in C of a given ( D, S ) ≤ ( D ∗ , S ∗ ) , along with which of the abovealternatives it satisfies, can be determined uniformly ∅ ′′ -computably from an indexfor ( D, S ) .Proof. Call a condition (
D, S ) good if the following facts are true: • for all finite F ⊆ S and all x ∈ ω , if Φ D ∪ Fe ( x ) ↓ = 1 then x ∈ D ∪ F ; • for each s ∈ ω , there is a finite F ⊆ S and an x > s with Φ D ∪ Fe ( x ) ↓ = 1.Note that if ( D, S ) is good, then the F and x in the second item above can be founduniformly S -computably, and F may be chosen to be non-empty. By iterating thisconstruction, we can thus S -computably construct an infinite subset T of S suchthat Φ D ∪ Te is infinite and contained in D ∪ T . We denote this S -computable subset T of S by S D .If ( D, S ) is not good, we can define an extension of it satisfying alternative 1 inthe definition of C , as follows. If there is a finite set F ⊆ S such that Φ D ∪ Fe ( x ) ↓ = 1for some x / ∈ D ∪ F , let e D = D ∪ F , and let e S consist of the elements of S largerthan x , the maximum of D ∪ F , and the use of the computation Φ D ∪ Fe ( x ). Then( e D, e S ) extends ( D, S ) and forces that Φ ˙ Ge is not contained in ˙ G . If, instead, thereis an s such that either Φ D ∪ Fe ( x ) ↑ or Φ D ∪ Fe ( x ) ↓ = 0 for all finite sets F ⊆ S and x > s , then ( D, S ) itself forces that Φ ˙ G is not total or not infinite. In this case, let( e D, e S ) = ( D, S ).Now to prove the density of C , fix any condition ( D, S ) ≤ ( D ∗ , S ∗ ). We wish toexhibit an extension of ( D, S ) in C , and so, by the preceding observation, we mayassume ( D, S ) and all its extensions are good. (This is equivalent to saying that(
D, S ) forces that Φ ˙ Ge is total, infinite, and contained in ˙ G .)Since all extensions of ( D, S ) are good, we can define the following sequence ofinfinite subsets of S : • S = S D ; • S = Φ D ∪ S e − { x ∗ } , where x ∗ is the least x > max D in Φ D ∪ S e ; • S = ( S ) D ; • S = Φ D ∪ S e .Thus, we have S ⊇ S ⊇ S ⊇ S ⊇ S , PETER A. CHOLAK, DAMIR D. DZHAFAROV, AND MARIYA I. SOSKOVA and the number x ∗ belongs to S (since Φ D ∪ S e ⊆ S ) but not to S . In other words,for v ∈ { , } , we have S v ( x ∗ ) = 1 − v . There are now two cases. Case 1. There is no finite F ⊆ S such that Φ F ∗ ∪ Fi ( x ∗ ) ↓ . Then (
D, S ) is anextension of ( D, S ) satisfying alternative 2 in the definition of C , witnessed by x ∗ . Case 2. Otherwise.
In this case, choose F ⊆ S and v ∈ { , } such that Φ F ∗ ∪ Fi ( x ∗ ) ↓ = v . Thus, Φ F ∗ ∪ Fi ( x ∗ ) ↓ = v = S v ( x ∗ ) = ( D ∪ S v )( x ∗ ) , with the last equality holding because x ∗ is bigger than max D . Since ( D, S v ) isgood, there is an initial segment e D of D ∪ S v long enough so that • x ∗ ≤ max e D and e D ( x ∗ ) = S v ( x ∗ ), and • Φ e De converges on ω ↾ max F + 1.Let e S consist of the elements of S v larger than max e D and the use of Φ e De on ω ↾ max F + 1. Thus, F ⊆ Φ ( e D, e S ) e . Finally, since ( D, S ) extends ( D ∗ , S ∗ ) and is good,we have F ∗ ⊆ Φ ( D,S ) e ↾ max F ∗ + 1 ⊆ D, so max F ∗ < min F since F ⊆ S ⊆ S . We conclude that x ∗ and F witness that( e D, e S ) satisfies alternative 3 in the definition of C .To complete the proof, suppose I = COM P . Now ∅ ′′ can determine whetheror not a condition is good, and can differentiate between Cases 1 and 2. Thus, theonly step in the argument that cannot be performed by ∅ ′′ is the determinationof whether or not all extensions of ( D, S ) are good, which we just assumed above.We can modify the proof to fix this difficulty as follows. Initially, we ask if (
D, S )is good, and if not, we can define an appropriate extension of (
D, S ) that meetsalternative 1 in the definition of C ), and are done. Otherwise, we can define S and S as before. Now we ask if ( D, S ) is good, and if not, we can argue as before, andare again done. Otherwise, we can define S and S , and the rest of the argumentis unchanged. It is easy to see that the proof is thus uniform in ∅ ′′ . (cid:3) Theorem 4.5.
Let I be an ideal and G a - I -generic. Then no - [ G ] ≤ T -genericset contained in G computes G .Proof. We show that each infinite G -computable subset of G has an infinite subsetthat is computable from G , but does not compute G via a prescribed functional.That is, for all e, i ∈ ω , we build a set A e,i ≤ T G to satisfy the following requirement: R e,i : if Φ Ge is an infinite subset of G , then A e,i is an infinite subset of Φ Ge andΦ A e,i i = G .In other words, the set G has no [ G ] ≤ T -hereditarily uniformly G -computing sub-set in the ideal [ G ] ≤ T . By Proposition 4.2, it follows that no 3-[ G ] ≤ T -generic setsatisfying the [ G ] ≤ T -condition ( ∅ , G ) can compute G , which is the desired result.So fix e . If Φ Ge is not infinite or not contained in G , there is nothing to do, andwe can let A e,i = ∅ for all i . Otherwise, by 3- I -genericity of G , we can find an I -condition ( D ∗ , S ∗ ) satisfied by G forcing that Φ ˙ Ge is total, infinite, and containedin ˙ G . In the parlance of Lemma 4.4, this means that ( D ∗ , S ∗ ) and all its extensionsare good. Let F ∗ = ∅ , and let C be the dense collection of conditions of thelemma. This is a Σ ( I )-definable collection, and so G must meet it, say via some ENERICITY FOR MATHIAS FORCING OVER GENERAL TURING IDEALS 9 ( D, S ) ≤ ( D ∗ , S ∗ ). By choice of ( D ∗ , S ∗ ), it cannot be that ( D, S ) satisfies the firstalternative in the definition of C . Hence, one of the following is true: • there an x such that Φ Fi ( x ) ↑ for all finite F ⊆ S ; • there is an x ≤ max D and a finite F ⊆ Φ ( D,S ) e such that Φ Fi ( x ) ↓6 = D ( x ).In the first case, we may take A e,i = { x ∈ Φ Ge : x ≥ min S } , as in this case, Φ Ui cannot be total for any U ⊆ S . In the second case, we may take A e,i = F ∪ { x ∈ Φ Ge : x ≥ min S } , as then we haveΦ A e,i i ( x ) ↓ = Φ Fi ( x ) ↓6 = D ( x ) = G ( x ) . Obviously, A e,i ≤ T Φ Ge , so A e,i is G -computable and thus belongs to [ G ] ≤ T . (cid:3) Taking I = COM P , and letting G be a 3- I -generic computable in ∅ (3) , thetheorem immediately yields the existence of a ∆ ideal, namely [ G ] ≤ T , that isnot n -generically-coded for any n ≥
3. By Corollary 3.8, we cannot improve this to∆ ideals, but we can improve it to ∆ ones by a slightly more careful argument.Thus, we get a sharp dividing line in terms of which arithmetical ideals are and arenot generically-coded. Proposition 4.6.
There is a principal ∆ ideal which is not n -generically-codedfor any n ≥ .Proof. Let I = COM P . The proof is roughly the same as that of Theorem 4.5, andwe build the sets A e,i to satisfy the same requirements as there. The only differenceis that we now need to explicitly construct the set G . To this end, we build asequence of I -conditions ( D , S ) ≥ ( D , S ) ≥ · · · by stages, and let G = S s D s .At stage 0, we let ( D , S ) = ( ∅ , ω ). We define A e,i at stage s + 1 = h e, i i , atwhich we assume we have already defined ( D s , S s ). Applying Lemma 4.4 with( D ∗ , S ∗ ) = ( D s , S s ) and F ∗ = ∅ , we let ( D s +1 , S s +1 ) be the extension of ( D s , S s ) in C given by the lemma. If ( D s +1 , S s +1 ) satisfies the first alternative in the definitionof C , then it forces that Φ ˙ Ge is not total, not infinite, or not contained in ˙ G . It is notdifficult to check that this fact will be true of any set that satisfies ( D s +1 , S s +1 ),and so in particular of G , even though we are not making G even 3-generic. Thus,Φ Ge will not be an infinite subset of G in [ G ] ≤ T , so we need not worry about it. Inthis case, we can set A e,i = ∅ . Otherwise, ( D s +1 , S s +1 ) will satisfy one of the othertwo alternatives in the definition of C , and then the argument can proceed as inthe theorem. By the effectiveness of Lemma 4.4, the construction can be performedcomputably in ∅ ′′ , so G will be ∆ . (cid:3) One further consequence of the above is the following purely computability-theoretic result. Slaman and Groszek (unpublished) showed that there is a ∆ setwith no modulus of the same degree. We obtain the analogous result for introre-ducibility. In fact, it is easy to see that if an infinite set has a modulus of the samedegree then it also has an introreducible subset of the same degree, but of coursenot conversely. Thus, the following is in fact an extension of their result, albeitusing a very different argument. The ∆ bound here is sharp: it is easy to see thatevery ∆ set has a modulus of the same degree. Proposition 4.7.
There is a ∆ set G with no infinite introreducible subset of thesame degree. Proof.
We show that every infinite G -computable subset of G has an infinite subsetof its own that does not compute G . (The latter subsets will only be computablein the construction, and hence in ∅ ′′ , but not necessarily in G .) To this end, wemodify the proof of Proposition 4.6. Instead of building, for each e , a separatesubset A e,i of Φ Ge for each i and ensuring that A e,i does not compute G via Φ i , webuild a single subset A e to satisfy the following requirement: R e : if Φ Ge is an infinite subset of G , then A e is an infintie subset of Φ Ge and G (cid:2) T A e .We build the sets A e by stages along with G , and let F e,s be the initial segment to A e built at stage s . Initially, we let F e, = ∅ for all e . At stage s +1 = h e, i i , we are thengiven ( D s , S s ) along with F e,s , and we assume inductively that F e,s ⊆ Φ ( D s ,S s ) e . Wenow apply Lemma 4.4 with ( D ∗ , S ∗ ) = ( D s , S s ) and F ∗ = F e,s to get an extension( D, S ) of ( D s , S s ) in C . We consider three cases. Case 1: ( D, S ) satisfies the first alternative in the definition of C . In this case, wedo not have to worry about Φ Ge being an infinite subset of G , so we also do not haveworry about making A e infinite. We thus set F e,s +1 = F e,s and ( D s +1 , S s +1 ) =( D, S ). Case 2: ( D, S ) satisfies the second alternative in the definition of C but not thefirst. Since (
D, S ) does not satisfy the first alternative, it does not force that Φ ˙ Ge isnot infinite, so we can find an x > max F e,s and an extension ( e D, e S ) of ( D, S ) suchthat F e,s ∪ { x } ⊆ Φ ( e D, e S ) e . We let F e,s +1 = F e,s ∪ { x } and ( D s +1 , S s +1 ) = ( e D, e S ).Since ( D, S ) satisfies the second alternative, we know that Φ Ui cannot be total forany set U extending F e,s , and so also for any set extending F e,s +1 . Case 3: otherwise.
In this case, (
D, S ) satisfies the third alternative in the definitionof C , so we can fix a finite set F ⊆ Φ ( D s +1 ,S s +1 ) e with max F e,s < min F witnessingthis fact. Since ( D, S ) does not satisfy the first alternative, we can pass to anextension if necessary to add an element to F . Thus, without loss of generality, wemay assume F = ∅ . We let F e,s +1 = F e,s ∪ F and ( D s +1 , S s +1 ) = ( D, S ).It is easy to verify that for each e ∈ ω , the construction at stages s + 1 = h e, i i ensures the satisfaction of requirement R e . And as in our previous result, the entireconstruction can be carried out by ∅ ′′ , so G is ∆ . (cid:3) We finish this section with a converse to Proposition 4.2. We do not knowwhether the level of genericity below can be improved from 4 to 3. This is becausedeciding Σ facts in general requires Mathias 4-genericity (see [2], Lemma 3.3),rather than 3-genericity as in the case of Cohen forcing. Proposition 4.8.
Let I be an ideal and A any set. If G is - I -generic and con-tained in a set S ∈ I such that each infinite subset of S in I has an I -hereditarilyuniformly A -computing subset in I then G ≥ T A .Proof. Suppose every subset of S in I has an I -hereditarily uniformly A -computingsubset in I . Then in particular, A belongs to I , so the formula(1) ∃ e ∀ x [Φ ˙ Ge ( x ) ↓ = A ( x )]is equivalent to a Σ ( I ) one. Since G is 4- I -generic and contained in S , there mustbe some I -condition ( D ∗ , S ∗ ) ≤ ( ∅ , S ) satisfied by G that decides this formula. ENERICITY FOR MATHIAS FORCING OVER GENERAL TURING IDEALS 11
This means S ∗ ⊆ S , so by passing to an extension if necessary we may assume S ∗ is itself I -hereditarily uniformly A -computing subset in I , say witnessed by e ∗ ∈ ω .Let e ∈ ω be the index of the functional defined byΦ Ue = Φ U − D ∗ e ∗ for all oracles U . Now if ( D ∗ , S ∗ ) forced the negation of (1), then some exten-sion ( D ∗∗ , S ∗∗ ) of it would, for some x ∈ ω , either force Φ ˙ Ge ( x ) ↑ or else Φ ˙ Ge ( x ) = A ( x ). In the former case, there could be no finite F ⊆ S ∗∗ such that Φ D ∗∗ ∪ Fe ( x ) ↓ ,so in particular Φ D ∗∗ ∪ S ∗∗ e could not be total. In the latter, it would have to bethat Φ D ∗∗ e ( x ) ↓6 = A ( x ) with use bounded by min S ∗∗ , so also Φ D ∗∗ ∪ S ∗∗ e ( x ) ↓6 = A ( x ).But Φ D ∗∗ ∪ S ∗∗ e = Φ S ∗∗ e ∗ = A , so neither of these cases can be true. We concludethat ( D ∗ , S ∗ ) forces (1), and hence by genericity that G ≥ T A . (cid:3) Corollary 4.9.
Let I be an ideal and A any ∆ n ( I ) set, n ≥ . The following areequivalent for any n - I -generic G :(1) G ≥ T A ;(2) G is contained in an I -hereditarily uniformly A -computing set S ∈ I .Proof. The implication from (1) to (2) is Proposition 4.2, and the implication from(2) to (1) follows by Proposition 4.8 and the fact that being I -hereditarily uni-formly A -computing is closed under infinite subset. (cid:3) Corollary 4.10.
Let I be an ideal and A any ∆ n set, n ≥ . Then the followingare equivalent:(1) every n - I -generic G computes A ;(2) every set in I has an I -hereditarily uniformly A -computing subset in I .Proof. For the implication from (1) to (2), suppose every n - I -generic computes A .Let S be any member of I , and choose an n - I -generic G satisfying ( ∅ , S ). By Propo-sition 4.2, G is contained in some I -hereditarily uniformly A -computing set S ∗ ∈ I .Then S ∩ S ∗ is an I -hereditarily uniformly A -computing subset of S in I . The impli-cation from (2) to (1) is Proposition 4.8 and the fact that every generic is containedin some member of I . (cid:3) Generics for different ideals
In this section we examine relationships between Mathias generic sets for differentideals. We ask how the ideals must relate in order for the generics to relate. Wefirst establish some basic properties of ideals that will be useful to us in this section.Given two sets A and S , define U A,S = { p S ( p A ↾ i q ) : i ∈ ω } , where p A ↾ i q denotes the canonical index of the finite set A ↾ i . Thus, U A,S picksout a subset of S according to indices of initial segments of A . Lemma 5.1.
Let A be any set, and suppose I is an ideal containing a set S .If U A,S has an infinite subset in I , then A belongs to I .Proof. Suppose U ⊆ U A,S belongs to I . Then U is a subset of S , so for each i we can uniformly compute from U ⊕ S an x such that p U ( i ) = p S ( x ). But bythe definition of U A,S , this x must be p A ↾ j q for some j ≥ i . In other words,given i , U ⊕ S can uniformly find a canonical index for a finite initial segment of A of size at least i . Since A is computable from any infinite sequence of its initialsegments, we conclude that A ≤ T U ⊕ S . Since I is an ideal and U, S ∈ I belongto I , it follows that A ∈ I . (cid:3) Proposition 5.2. If I ⊆ J are ideals such that every infinite set in J has aninfinite subset in I , then I = J .Proof. If J = ∅ there is nothing to prove, so assume otherwise and fix any infi-nite A ∈ J . We shall show that A ∈ I . By assumption, we may fix an infinite S ∈ I .Then U A,S is computable from S ⊕ A and hence belongs to the ideal J . By as-sumption, U A,S has an infinite subset in I , so A belongs to I by the lemma. (cid:3) Proposition 5.3. If I ⊆ J are arithmetical ideals and
I 6 = J , then there isan I -generic set that, for some n ∈ ω , computes no n - J -generic set.Proof. Let C be the collection of all J -conditions ( E, T ) such that T has no infinitesubset in I . We claim this set is dense in the J -conditions. Indeed, fix any J -condition ( E, T ), and let A be any infinite set in J − I . Then U A,T is an infinitesubset of T in J , and since A / ∈ I it follows by the lemma that U A,T has no infinitesubset in I . Hence, ( E, U
A,T ) is an extension of (
E, T ) in C .Since I and J are arithmetical, there is some n such that every n - J -generic setmeets C . Our goal, then, is to construct an I -generic set G such that for all e ,either Φ Ge is not an infinite set or Φ Ge avoids C . This G will therefore not computeany n - J -generic, as desired. We shall obtain G in the usual way, as S s D s from asuitably constructed sequence of I -conditions ( D , S ) ≥ ( D , S ) ≥ · · · . List theelements of C as ( E , T ) , ( E , T ) , . . . . Let ( D , S ) = ( ∅ , ω ), and assume that forsome s ≥ D s , S s ).If s is even, we work to make G not compute any n - J -generic. More specifically,say s = 2 h e, i i . We work to either make Φ Ge not be an infinite set, or to make Φ Ge not meet C via satisfying ( E i , T i ). To this end, ask if there exists a finite set F ⊆ S s and an x / ∈ E i ∪ T i such that Φ D s ∪ Fe ( x ) ↓ = 1. If so, choose some such F and x and let D s +1 = D s ∪ F and S s +1 = S s − ϕ D s ∪ Fe ( x ). Now since G willsatisfy ( D s +1 , E s +1 ), we have that if Φ Ge is a set, it will contain x and hence notsatisfy ( E i , T i ). If, on the other hand, no such F and x exist, let ( D s +1 , S s +1 ) =( D s , S s ). In this case, Φ Ge cannot be an infinite set, as otherwise S s would computean infinite subset of T i and so contradict that T i has no infinite subset in I .If s is odd, proceed as in the analogous case in the proof of Proposition 3.2. (cid:3) In the ∆ setting, we can now extend Proposition 5.3 to non-nested ideals. Corollary 5.4. If I and J are ∆ ideals with J * I , then there is an I -genericset that computes no - J -generic.Proof. Fix A ∈ J − I . By Corollary 3.8, every 3- J -generic set computes A , butby Proposition 3.2, there is a I -generic set that does not. (cid:3) The analysis above naturally leads to the following question:
Question 5.5. If J ⊆ I then does every I -generic compute a 3- J -generic?This question, although easily stated, is quite difficult to approach. For onething, one must take into account the complexity of the exact pair representing thesmaller ideal. Even if we restrict J to the ∆ degrees, there are uncountably manyideals of ∆ degrees, and hence J could turn out to have only very complex exact ENERICITY FOR MATHIAS FORCING OVER GENERAL TURING IDEALS 13 pairs. We finish this article with a positive answer to the question above for thesimplest case, when J = COM P , the ideal of all computable sets.
Theorem 5.6.
Let I be an ideal and let n ≥ be a natural number. Every n - I -generic computes an n - COM P -generic set.
We wish to thank Rose Weisshaar, who noticed an omission in the original proofof this theorem. The proof presented below has been corrected.
Proof.
We use the following idea from [2, Theorem 5.2]. For an arbitrary set A we can approximate A ( n ) by approximating iterations of the jump of A . We let A [ s ] = A ↾ s . We fix a uniform way to approximate the jump of a set and byinduction define for all m ≤ n : A ( m ) [ s , . . . , s m ] = ( A ( m − [ s , . . . , s m − ]) ′ [ s m ] . If X computes A then X computes A ( n ) [ s , . . . , s n ] for all n . Furthermore, for all x and e there are arbitrarily large stages s , . . . , s n such that A ( n ) ↾ x = A ( n ) [ s , . . . , s n ] ↾ x and for any c.e. set W : W A ( n ) ↾ x = W A ( n ) [ s ,...,s n ] ↾ x. These observations justify our adopting the following convention: when we write A ( m ) [ s , . . . , s m ] we shall always mean that s < · · · < s m .Let G be an n - I -generic set. We describe a procedure which uses G as an oracle toconstruct an n - COM P -generic set E . The procedure operates as follows: we view G as a sequence of blocks each of size n + 1. Let { G m } m<ω be a partition of G intofinite pieces G m , each of size n +1 and with the property that min G m +1 < max G m .We deal with each block G m in turn and define a finite set D m . The resulting set E will then be S m<ω D m .We fix a list of requirements: {R e } e<ω , where each R e states the following:If R e = W ∅ ( n ) e is a set of conditions then E meets or avoids R e .Consider a block G m = { s , . . . , s n } . We use this piece first to produce anapproximation to the sets ∅ ( p ) for all p ≤ n using the approximating procedure de-scribed above. Let ∅ ( p ) [ m ] denote the set ∅ ( p ) [ s , s , . . . , s n ]. The set D m is obtainedas follows: we construct a sequence α , . . . , α k , where each α i has the structure ofa computable Mathias condition: it consists of a finite set D α i and an index ofa partial computable function e α i . This sequence consists of attempts to satisfythe requirements, performed at previous stages, based on previous approximationsusing earlier blocks. During the current block action we preserve as much of thesequence constructed so far that still seems to consist of actual conditions and pos-sibly extend it one step further to satisfy a new requirement. We will, however,computably ensure that D α i ⊆ D α i +1 for all i and set D m = D α k .The first block G will produce the sequence with only one element α = ( ∅ , N ),or to be precise ( ∅ , e ), where e is some fixed index of the characteristic function of N . Suppose that the ( m − α , . . . , α k and outputsthe set D m − . We now use the sparseness of G to check whether we still believethat α , . . . , α k is a sequence of conditions: Let t be the largest element in G m andsuppose that t is the x -th element of the oracle G . Let S α i ,t = { y : ϕ e αi ,t ( y ) ↓ = 1 } .For every i , such that 0 < i < k we check whether: (1) ϕ e αi converges on every element y < x in less than t steps with output 0or 1;(2) S α i ,t contains at least x many elements;(3) The maximal element in D α i is smaller than the minimal element in S α i ,t ;(4) S α i ,t ↾ x ⊆ S α i − ,t ;If we reach an index j , such that α j +1 does not satisfy the requirements above,then the sequence defined at block m will be α , . . . , α j , β , where β = ( D m − , e β )and e β is the index of the function which outputs 0 on elements less than max D m − and otherwise behaves like the function with index e j .If the sequence defined at block m − i < k we have that α i satisfies the properties above, then we try to extendit to meet a requirement. Using the block m approximation to ∅ ( n ) , we search fora least number e ≤ m such that • R e is not satisfied by any member of the sequence α , . . . , α k ; • there is an element ( D, e ) ∈ R e [ m ], such that ( D, e ) is a computable Mathiascondition which extends α k according to ∅ ′′ [ m ]; • D m − ⊆ D and for every element x ∈ D − D m − we have that ϕ e αk ( x ) isdefined in t = max G m many steps and is equal to 1.If there is such an e then we set α k +1 = ( D, e ). Otherwise we end the sequence at α k . This completes the block m -action of the procedure.It remains to show that the constructed set E = S m D m really satisfies eachrequirement R e . We use G ’s genericity for this. We can view the procedure thatwe just described as a functional Γ, such that for every finite set F , Γ F is a sequence α , . . . , α k , where if m is the number of size- n + 1 blocks that F can be partitionedin then this sequence is obtained by the block- m action of the procedure with oracle F . Assume inductively that there is a condition ( F , T ), such that F ⊆ G ⊆ T ,such that Γ F is a monotone sequence of true computable Mathias conditions, T is so sparse that it witnesses this and all requirements R i for i < e are satisfied bysome member of this sequence.For every finite monotone sequence ~α = α . . . α k , where α i = ( D α i , e α i ) is apair of a finite set and an index of a partial computable function, consider the set C ~α consisting of all I -conditions ( F, T ) such that T is so sparse that it witnessesthat ~α is a monotone sequence of computable Mathias conditions: for every i ≤ k and every n ≥ | F | , if t is the n -th element of F ∪ T then:(1) ϕ e αi converges on every element y < x in less than t steps with output 0or 1;(2) S α i ,t = { y : ϕ e αi ,t ( y ) ↓ = 1 } contains at least x many elements;(3) the maximal element in D α i is smaller than the minimal element in S α i ,t ;(4) if i > S α i ,t ↾ x ⊆ S α i − ,t ).Note that if ~α is a monotone sequence of computable Mathias conditions, then C ~α is a dense set. In fact, to find an extension of a condition in C ~α we only need tothin the reservoir, i.e. for every ( F, T ) there is an extension (
F, T ′ ) ∈ C ~α .Consider the set C of conditions ( F, T ) such that F ⊆ F and Γ F = ~α is asequence of computable Mathias conditions with a member in R e and T is sparseenough to witness this: ( F, T ) ∈ C ~α . This set of conditions is Σ n ( I )-definable. If G meets C via ( F, T ) then E meets R e via one of the conditions in the sequence ENERICITY FOR MATHIAS FORCING OVER GENERAL TURING IDEALS 15 Γ F , because the sparseness of T guarantees that Γ F is an initial segment of Γ G ↾ l for every extension G ↾ l of F .Suppose that G avoids C via ( F, T ). Without loss of generality we may assume F ⊆ F . Consider Γ F and let α , . . . , α k be the sequence defined by Γ during thecomputation on the last block of F . Let j be the largest index such that α , . . . , α j is an initial segment of all further sequences defined at further blocks with oracle G .Consider the block of G , say G ↾ l , where α j is the last condition in the computationΓ G ↾ l . Suppose that α j has an extension β ∈ R e . Let ( G ↾ l, T ∗ ) be an extensionof ( F, T ) that is in C α ,...α j ,β , such that for the first block s < s < · · · < s n of T ∗ we have that β ∈ R e [ s , s , . . . , s n ] and furthermore for every x ∈ D β , x can beverified to belong to D α i ∪ S α i in s n many steps.Now let ( F ′ , T ′ ) be such that F ′ = G ↾ l ∪ s , . . . , s n and T ′ is obtained from T ∗ by removing all elements less than or equal to s n . It follows that ( F ′ , T ′ ) extends( F, T ) and belongs to C , contradicting our assumptions. Thus α j avoids R e . Thiscompletes the proof. (cid:3) References [1] Stephen Binns, Bjørn Kjos-Hanssen, Manuel Lerman, and Reed Solomon. On a conjec-ture of Dobrinen and Simpson concerning almost everywhere domination.
J. Symbolic Logic ,71(1):119–136, 2006.[2] Peter A. Cholak, Damir D. Dzhafarov, Jeffry L. Hirst, and Theodore A. Slaman. Generics forcomputable Mathias forcing.
Ann. Pure Appl. Logic , 165(9):1418–1428, 2014.[3] Peter A. Cholak, Carl G. Jockusch, and Theodore A. Slaman. On the strength of Ramsey’stheorem for pairs.
J. Symbolic Logic , 66(1):1–55, 2001.[4] Fran¸cois G. Dorais. A variant of Mathias forcing that preserves
ACA . Arch. Math. Logic ,51(7-8):751–780, 2012.[5] Damir D. Dzhafarov and Carl G. Jockusch, Jr. Ramsey’s theorem and cone avoidance.
J.Symbolic Logic , 74(2):557–578, 2009.[6] Carl G. Jockusch, Jr. Uniformly introreducible sets.
J. Symbolic Logic , 33:521–536, 1968.[7] Carl G. Jockusch, Jr. Upward closure and cohesive degrees.
Israel J. Math. , 15:332–335, 1973.[8] S. C. Kleene and Emil L. Post. The upper semi-lattice of degrees of recursive unsolvability.
Annals of Mathematics , 59(3):pp. 379–407, 1954.[9] A. R. D. Mathias. Happy families.
Ann. Math. Logic , 12(1):59–111, 1977.[10] David Seetapun and Theodore A. Slaman. On the strength of Ramsey’s theorem.
Notre DameJ. Formal Logic , 36(4):570–582, 1995. Special Issue: Models of arithmetic.[11] Richard A. Shore. Lecture notes on turing degrees. In
Computational Prospects of InfinityII: AII Graduate Summer School , Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap. WorldSci. Publ., Hackensack, NJ, to appear.[12] Robert I. Soare. Sets with no subset of higher degree.
J. Symbolic Logic , 34:53–56, 1969.[13] Robert I. Soare.
Computability theory and applications . Theory and Applications of Com-putability. Springer, New York, to appear.[14] Robert M. Solovay. Hyperarithmetically encodable sets.
Trans. Amer. Math. Soc. , 239:99–122, 1978.[15] Clifford Spector. On degrees of recursive unsolvability.
Annals of Mathematics , 64(3):pp.581–592, 1956.
Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556U.S.A.
E-mail address : [email protected] Department of Mathematics, University of Connecticut, Storrs, Connecticut U.S.A.
E-mail address : [email protected] Faculty of Mathematics and Infromatics, Sofia University, 5 James Bourchier blvd.,Sofia 1164, Bulgaria
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